Texas Government OnlineDeath Penalty Paper ProjectCindy Case.docx

Texas Government Online
Death Penalty Paper Project
Cindy Casey Brown
El Centro College
Summer 2018
Overview
In this presentation you will read about the death penalty in
Texas, see pictures, explore websites and see videos.
You will also learn about the insanity defense in Texas.
You will also be presented with a statute (a law) regarding the
duty to report child abuse.
The purpose of this exercise is to educate you on the laws in
Texas that are significant - and - during you lifetime you may
see the United States Supreme Court take up and comment on.
At the end of my presentation I will present you with a fictional
case study and then you will write me a paper about your
reflections on the work you did and what you saw in this
presentation, whether you think the defendant should be charged
with the death penalty, whether an insanity defense would be
successful, and whether another individual in the case study had
a duty to report child abuse and should be charged with a crime.
The case study is based on a compilation of a few true Texas
cases.
If you want to read the case study first then you can go to the
case study slides and start there to get an idea of what you will
be writing about. More specific details on the paper are at the

end of the power point.
Why are we doing this?
I want you to leave this class being well-versed in the death
penalty and the insanity defense.
Texas executes more people than any other state in the country
(Texas is a big state) – this is a Texas Government class so you
should know about this topic because it is a big topic in this
state. During your lifetime the United States Supreme Court
will take cases on this topic and make comments on it – you
should be familiar with the topic.
The insanity defense – this is a big topic in the news today so I
want you to work with it. With the unfortunate mass public
shootings going on - if the shooter survives – the shooter is
likely to plead the insanity defense – so I want you to
understand it.
I think you learn so much more about these topics by doing this
exercise and critically thinking through this case study than by
reading about it and taking a test on it.
I know this topic can be disturbing and hard to think about – but
in a college course like this it is important to address.
And remember – I am NOT trying to make you think one way or
the other or push any opinion on you – I am trying to present
many sides to a hard topic that will make you think – whatever
your opinion is it does not impact your grade – your grade on
this project is based on your hard work and critical thinking and
analysis and your ability to form opinions and back them up.
Do Not Do This

Assignment Around Children
This is a hard topic and many of these photos are hard to see
and stories are hard to hear. You are a in a college level course
so I know you can handle this material.
But please DO NOT watch these videos or explore these
websites around children or in a public place as some of the
material is not appropriate for everyone.
Thank you!!
The Death Penalty in Texas
Four degrees of murder in Texas
Murder
Intentionally or knowingly causes the death of an individual.
Manslaughter
Recklessly causes the death of an individual.
Example: Two men get into a physical fight and one kills the
other by beating him severely.
Criminally Negligent Homicide
Causes death by criminal negligence.
Death when driver is drunk.
Death when driver is texting.
Capital Murder / The Death Penalty
This is what we are talking about in this presentation.

Capital Murder - Definition
Sec. 19.03. CAPITAL MURDER.
A defendant can be eligible for the offense of capital murder IF
the person commits a murder AND:
(1) murder a police officer or fireman acting in their official
duty AND the defendant KNOWS the person is a police officer
or fireman.
(2) defendant intentionally commits the murder in the course of
committing OR attempting to commit kidnapping, burglary,
robbery, aggravated sexual assault, arson, obstruction or
retaliation, or a terroristic threat.
(3) the defendant commits the murder for remuneration or the
promise of remuneration or employs another to commit the
murder for remuneration or the promise of remuneration;
(4) the defendant commits the murder while escaping or
attempting to escape from a penal institution;
(5) the person, while incarcerated in a penal institution, murders
someone who is employed by the penal institution.
(7) the defendant murders more than one person:
(A) during the same criminal transaction; or
(B) during different criminal transactions but the murders are

committed pursuant to the same scheme or course of conduct;
(8) the defendant murders an individual under 10 years of age;
or
(9) the defendant murders another person in retaliation for or on
account of the service or status of the other person as a judge or
justice of the supreme court, the court of criminal appeals, a
court of appeals, or trial court.
Photos of Death Row Inmates
https://apps.texastribune.org/death-row/
You can go to this link and click on the photos and it will tell
you what the person did – scroll through here and you will
realize there are women and men of all races, ethnicities, and
backgrounds.
Picking a Jury
This can take up to two weeks because potential jurors come in,
fill out a questionnaire, then come back a week or so later to the
courtroom where the lawyers go through the process of VOIR
DIRE – this is the process of questioning the jurors in open
court with the goal of seating a jury that is fair and impartial.
Sample Juror Questionnaire
Given to jurors, attorneys review it, then Voir Dire.
http://www.cebjury.com/copy/sampleQuestionaireCRIMINAL/d
eathPenaltyJuryQuestions.pdf
Seating the Jury
A jury of 12 is seated for the trial and usually 2 alternates are

selected (an alternate will enter deliberations if any of the 12
members cannot finish the trial).
Bifurcated Trial
Trial is bifurcated = meaning there are two parts
Guilt / Innocence Phase
Punishment Phase
In Guilt / Innocent phase if defendant is found innocent then
trial is over.
In Guilt / Innocent phase if found guilty – then Punishment
Phase starts.
Punishment Phase of Trial
If the defendant makes it to the punishment phase of a capital
murder trial (capital murder = death penalty) then there are two
options – 1) life in prison OR 2) the death penalty.
During the punishment phase the jury will hear testimony from
witnesses regarding any mitigating factors that exist.
Mitigating factors are things in the defendant’s life that would
warrant giving life in prison and not the death penalty.
IF you get to the punishment phase of the trial in a death
penalty case, the jury answers TWO questions:
(1) Does the defendant pose a risk to society because of the
probability that he or she will continue to commit criminal acts.
(2) Do mitigating factors exist like mental retardation or child
abuse to warrant a sentence in life in prison instead of death?
The Jury MUST answer YES to (1) and NO to (2).

The Jury MUST be UNANIMOUS. If the jury is not unanimous
it is a life sentence.
If the defendant is found guilty but the jury is not unanimous
then the defendant is given LIFE in prison.
Texas Death Row
Polunsky Unit – Livingston, Texas
https://www.tdcj.state.tx.us/death_row/
Go to this link – if you cannot get there by clicking here then go
to your favorite search engine and type in “Texas Department of
Criminal Justice Death Row” and you will be directed.
Spend some time on this website and explore the offenders, the
crimes committed, the victims of the crimes and the information
about last statements.
I want you to pay close attention to the crimes committed which
you find by clicking on each offender so that you get an idea of
the perspective of the family and friends of the victim(s) in
these cases.
Texas Death Row Cell
Texas Execution Chamber

Texas Death Row
https://www.youtube.com/watch?v=9w1ANeiohXo
You can cut and paste this link into your browser. This is a 45
minute video to watch that discusses Texas Death Row from the
perspective of employees, inmates / those on death row, the
victim’s families, and attorneys. **Some of these facts and
interviews are very disturbing. DO NOT watch this around
children.**
Jim Willett
Retired Texas Prison Warden.
Wrote Book: Warden: Texas Prison Life and Death from the
Inside Out
Oversaw 89 executions in Texas.
He has many videos on You Tube where he has been
interviewed – the link below is to a short video where he
discusses overseeing his first execution:
https://www.youtube.com/watch?v=vzmr0FHBd-4
What does the United States Supreme Court say about the death
penalty?
Gregg v. Georgia (1976)
The Death Penalty is constitutional under the 8th Amendment of
the US Constitution.
The 8th Amendment guarantees that in the United States you
have the right to be free from cruel and unusual punishment –

this case tells us that the death penalty is not considered cruel
and unusual punishment.
Glass v. Louisiana (1985)
Electrocution does not violate the 8th Amendment.
Ford v. Wainwright (1986)
Execution of insane persons is a violation of the 8th
Amendment.
Atkins v. Virginia (2002)
Execution of “mentally retarded” violates 8th Amendment.
Roper v. Simmons (2005)
Execution of someone who committed crime at under age 18 is
violation of 8th.
Baze v. Rees (2008)
Lethal injection does not violate 8th.
Kennedy v. Louisiana (2008)
You have to have a murder to impose the death penalty – crimes
without a murder cannot get the death penalty under the 8th.
Are there innocent
people on death row?
http://www.deathpenaltyinfo.org/innocence-and-death-penalty
There are many websites that explore innocent people on death
row and this is one of the most well-respected in terms of
accuracy. Spend a few minutes exploring this website and form
some impressions about what you find so you can write your
reflections in your paper.

One note on “innocence” – many times someone will be taken
off of death row and given life or a lesser sentence but still be
“guilty” – when you look at statistics of people taken off of
death row look and see if they were found innocent or were
given a lesser sentence for some reason.
Anthony Graves – one example of an innocent man on Texas
Death Row
In 1992 he was charged and convicted of the murder of six
people in Somerville, Texas.
No criminal record.
No motive or physical evidence linking Graves to the crime
scene.
Conviction based on testimony of Robert Carter that they
committed the crime together.
On the eve of his death Carter admitted he committed the crime
on his own – executed the next day.
Graves served 18 years when his conviction was overturned and
he was released.
He received $1.4 million for his time served.
Click the link below for a short news story on Mr. Graves and

what he is doing now:
http://www.nbcnews.com/news/us-news/how-anthony-graves-
went-death-row-overseeing-his-local-crime-n381891
Anthony Graves’ Testimony Before Texas Senate on Texas
Death Row
Click here to watch his testimony:
Please note I am not trying to make you think one way or the
other about Texas Death Row – since Mr. Graves was there and
can tell us about it that is why I want you to watch his
testimony. Most people do not leave Texas Death Row and so
we cannot interview them.
http://www.motherjones.com/politics/2013/05/10-worst-prisons-
america-allan-polunsky-unit-texas-death-row
Ron Williamson
Williamson’s wrongful conviction was the subject of author
John Grisham’s true story An Innocent Man.
One of two men wrongfully convicted in 1988 in Oklahoma in
the rape and murder of Debra Sue Carter.
Both were released 11 years later when DNA evidence cleared
them both from the crime.

City gave defendants $500K and the state settled for an
undisclosed amount.
Williamson maintained his innocence throughout his time on
death row and his mental health rapidly declined – hair turned
white, pulled out his teeth.
Williamson died in a nursing home five years later of cirrhosis
(liver damage).
John Grisham read the obituary and then wrote the book called
An Innocent Man so you may have heard of his story.
The Insanity
Defense in Texas
Why do you want to know this?
You will hear of criminal defendants committing crimes and
claiming the insanity defense.
There are so many gun violence crimes today – people who are
caught may claim an insanity defense so you want to understand
it.
Chapter 46c of
Texas Code of Criminal Procedure
Defendant must file a notice of intent to plead the insanity
defense at least 20 days before trial.

Court then appoints one or more disinterested psychiatric
experts to examine the defendant and render an opinion on
defendant’s insanity.
This same expert(s) may also examine the defendant on the
issue of competency to stand trial.
The defendant may also request an examination by an expert(s)
of his own choosing.
County pays their expert(s) and defendant pays his expert(s).
Expert submits a report within 30 days.
The defendant can choose to be tried by the judge OR a jury.
The trier of fact will find the defendant
Guilty
Not Guilty
Not Guilty by Reason of Insanity
If the defendant is found not guilty by reason of insanity – what
happens?
The judge immediately makes a determination whether the
conduct of the defendant:
Caused serious bodily injury to another person
Placed another person in imminent danger of serious bodily
injury
Consisted of a threat of serious bodily injury to another person
through the use of a deadly weapon.

If the answer to any of those questions is YES then the court
retains jurisdiction over the person until the court:
Terminates jurisdiction and releases the person
OR
The total period of inpatient and/or outpatient treatment equals
the maximum term allowed for punishment under the offense of
which the person was acquitted by reason of insanity.
Standard for Insanity:
What Do Psychiatrists Testify About?
Texas follows the “McNaghten Rule”
Did the Defendant comprehend right from wrong at the time the
offense was committed?
Some kind of mental illness that existed at the time of the
offense prevented the Defendant from understanding that his/her
actions were of a wrongful nature.
The Defendant has to prove INSANITY by “a preponderance of
the evidence” / means “more likely than not.”
EXAMPLE OF
INSANITY DEFENSE
**Note the following is an incredibly difficult case to study but
this is considered “the” example of how to study the insanity

defense in Texas – do not watch these videos or stories around
children**
The Andrea Yates Case
Video Overview
https://www.youtube.com/watch?v=FjJuN9Qfr0I
June 20, 2011 – Andrea Yates feeds breakfast to her children,
her husband Rusty goes to work at NASA in Houston, she fills a
bathtub with water then drowns her five children: John (5), Paul
(3), Luke (2), and Mary (6 months), then Noah (7)
Before she filled the tub she pulled
the keys out of the door to
make sure the kids could not leave.
She admits she had to chase Noah around
the house to catch him because he realized
what was happening.
She called 911 and then
called her husband.
Police arrived and she calmly
confessed and was arrested.

Mental History
She has a history of post partum depression / psychosis post
birth of children
Husband found her shaking and trying to eat her hands in the
past.
Multiple suicide attempts over the years (at least 4)
Nervous breakdown
Had been treated as an inpatient for depression
Had taken drugs for severe depression in the past.
Psychiatrist urged couple not to have any more kids because it
would guarantee a relapse of the postpartum depression.
Medical records note that she will discontinue her medications
because she and husband want to have “as many children as
nature will allow” and psychiatrist disagrees with this.
She conceives her fifth child 7 weeks post discharge.
Seemed to be doing okay until father passes away.
Started mutilating herself, reading Bible feverishly, stopped
feeding baby, was catatonic – put back in hospital.

April 1, 2001 – treated by Dr. Mohammed Saeed after a suicide
attempt – treated and released.
May 3, 2001 – filled bathtub full in middle of the day – tells
doctor the next day and he assumes she did it to drown herself
and puts her in hospital.
Continued in Dr. Saeed’s care until June 20, 2001 when murders
occurred.
While in prison, Andrea stated she had considered killing the
children for two years, adding that they thought she was not a
good mother and claimed her sons were developing improperly.
She told her jail psychiatrist: "It was the seventh deadly sin. My
children weren't righteous. They stumbled because I was evil.
The way I was raising them, they could never be saved. They
were doomed to perish in the fires of hell."
She also told her jail psychiatrist that Satan influenced her
children and made them more disobedient.
Facts That Jury Heard That Show
She Knew What She Was Doing
She was a nurse and told police she knew how long it would
take for someone to drown.
Removed keys from locks and put them up high so kids could
not get them and get out.
Locked up the family dog so it could not interfere.

Started with youngest boys – after drowning them she removed
them from the bath tub and lined them up in her bed side by
side.
Then drowned the baby, Mary – left her floating in tub.
Oldest son comes in and asks what is wrong with Mary – then
he RUNS – she chases him and drowns him and then leaves him
in tub. He fought very hard. Came up for air twice per her
testimony. She testified Mary was floating in the tub when she
drowned the oldest.
Water was very dirty because all in the same water – photos
upset jury greatly. (This is a horrible fact to include for you I
know but this was a huge point for the jury so I did not want to
exclude it).
Did she do it to get back at her husband (prosecution theory).
Trials
Yates admits committing the crime but pleads not guilty by
reason of insanity.
In March 2002 the jury found her guilty, rejected her insanity
defense, and sentenced her to life in prison (she did not get the
death penalty).
Yates appealed the conviction.

The conviction was reversed – in part because of a
psychiatrist’s testimony that she had seen an episode of Law
and Order where a woman drowned her kids (showed motive) –
that was NOT TRUE – she never did b/c no episode existed.
The case was remanded for a new trial.
In July 2006 she was found not guilty by reason of insanity by
new jury and put into a mental institution.
Currently at Kerville State Hospital in Kerville, Texas.
Was Andrea Yates Insane? Would you have accepted the
insanity defense?
Texas follows the “McNaghten Rule”
Did the Defendant comprehend right from wrong at the time the
offense was committed?
Some kind of mental illness that existed at the time of the
offense prevented the Defendant from understanding that his/her
actions were of a wrongful nature.
The Defendant has to prove INSANITY by “a preponderance of
the evidence” / more likely than not.
There are many people who believe she was insane and many
people who believe she was not.

Where is Andrea Yates now?
http://people.com/crime/andrea-yates-now-she-grieves-for-her-
children-15-years-after-drownings/
http://time.com/4375398/andrea-yates-15-years-drown-children/
Because everyone will be curious –What about her husband?
He had been told by psychiatrist not to leave her alone with
kids.
Weekend before murders told family he started to leave her
alone for an hour in AM and PM with kids so she would not get
too dependent on him and her mom for childcare.
He told her brother that all someone depressed needs is a “swift
kick in the pants.”
Testimony Andrea did not want to have more kids and he did.
Where is he now?
Where is Rusty Yates?
http://abcnews.go.com/GMA/story?id=6278872&page=1
https://www.youtube.com/watch?v=zJXgM-mL4RM

YOUR PAPER
For your paper you are going to do the following:
Introduction Paragraph
I want you to write a few paragraphs on your reflections based
on what you just watched and researched in this presentation –
at least 1 page.
Read the case study that follows and tell me in at least 2 pages:
1) Based on the facts presented in the following case study, if
YOU were the District Attorney would YOU file a death penalty
case against the defendant and fully explain your answer and
rationale including your feelings on the death penalty.
2) If a death penalty case were filed against the defendant father
in the following case study do you think the defendant would be
found guilty and get the death penalty – why or why not and
fully explain your answer.
3) If the defendant pled the insanity defense do you think the
jury would accept it or reject it? Explain.
4) Do you think the mother should be charged with a violation
of Texas Family Code 261.101? Fully explain.
Conclusion Paragraph
What will your paper look like?
Make sure you include an introduction and conclusion.
Make sure you address everything you are asked to address.
Make sure you use complete sentences, great grammar, and

perfect capitalization and punctuation.
Make sure you spell check AND proofread.
The paper should be at least three (3) pages typed, double-
spaced, 1 inch margins, 12 point font.
Any pages over three (3) pages are Enhancements – 5 points
extra credit per page.
Turn the paper in on BlackBoard.
The Case Study
The following case study is based on several Texas cases.
Names and certain facts have been changed for this project in
order to allow you to fully study and critically think about the
death penalty, the Texas capital punishment statute, the insanity
defense in Texas, and the law in Texas regarding the mandatory
duty to report suspected child abuse.
Case Study
Jane and John have been married for 10 years. They have a 5
year old son named Patrick and 4 months ago Jane gave birth to
a baby boy named Russell.
John is a lawyer and works many hours each week in a high
stress position.
Jane worked for many years as a realtor and now stays at home
with Patrick and Russell. When Russell was born he was
diagnosed with Down Syndrome. Being a one-income household
is not easy on them financially with the high out-of-pocket
medical bills they must pay and special doctor’s visits they
must pay for but they are trying to make it work.
Neither Jane nor John have a criminal record.
Jane knows that with John’s high-stress job he has a hard time
being patient with the children.
Facts of the Case, Continued….

Jane does not routinely leave the children alone with John and
considers childcare her job now.
Before their oldest son, Patrick, was born John and Jane had too
much alcohol to drink one night and got into an argument. John
hit Jane, threw her against a wall, and threw a coffee table at
her. She was upset and had never see that kind of temper in
John before. She knew that he had a lot on his plate at the time
with trying to finish law school, study for his bar exam, and
begin work for a big law firm. They had lots of student loans
between them and John’s job was going to help ease the
financial burdens. John was very sorry and Jane forgave him.
She never told anyone.
Facts of the Case, Continued…
After that incident sometimes John would raise his voice to Jane
or pound his fists on the counter when he was upset with her
about bills, the house not being clean, if she forgot to pick up
something at the store, or if he was really stressed out at work.
Occasionally John would throw something at Jane. Jane knew
John did not mean to upset her and would never hit her again.
She loved John and thought once they had children he would
relax and realize how important it was to keep your cool.
When Patrick was born he had colic and kept Jane and John up
many nights.
Although Jane was working as a realtor, her job was more
flexible than John’s and so she did not expect John to stay up at
night with the baby. But their house was small and when the
baby cried many nights from 2:00 a.m. – 6:00 a.m. they both
heard it no matter how hard Jane tried to keep the baby quiet.
Some nights John would get upset with Jane, raise his voice to

her to tell her to quiet the baby down, and one night he threw a
baby bottle at her across the room while she was holding the
baby. The bottle hit her and the baby was not harmed. She
never told anyone and hid the bruise on her arm so no one
would see it.
Jane knew that John would never hurt her or the baby – he loved
them – and his job was stressful and he needed rest. She
figured all households had some of this going on and tried her
best to make sure things at home were as stress-free as possible
for John. Her realtor job did not carry the load of the expenses
and John was a good provider for their family of three. She
figured John would be less stressed and have more patience
when he was promoted to partner at his law firm. She knew he
did not mean to loose his temper. She figured everyone did it
from time to time.
As Patrick grew older and John became a more successful
lawyer, Jane noticed that John could be hard on Patrick. Jane
knew the hard work John had engaged in to become successful
and John held himself to a high standard. He did not accept
second best. It was part of what Jane loved about John. But
she worried though John was too hard on Patrick.
John never spanked Patrick in front of Jane and she was the
primary caregiver so she didn’t worry about John harming
Patrick. Jane wished John wouldn’t raise his voice so much
and wouldn’t put so much pressure on their young son to be
quiet, behave, work hard, and be perfect. Jane wanted Patrick
to just be able to be a kid and have fun sometimes - but John
disagreed - he expected Patrick to be on his best behavior at all
times and it caused stress in their household. Jane figured if
that was her biggest problem she should not complain.

Jane did notice that Patrick never wanted to be alone with his
dad when she left the room and seemed really timid around him.
It bothered her but she never questioned John. She didn’t want
to upset her husband.
Jane and John tried for many years to have a second child and
were thrilled when they learned they were having another baby
boy. This was the happiest time Jane can remember with John –
he was so proud to tell everyone he would have two perfect
sons.
When baby boy Russell was born, he was immediately
diagnosed with Down Syndrome and many medical issues
consistent with this diagnosis. This was totally unexpected.
Jane did everything she could to research the best doctors, the
best treatments for her son’s specific issues, and worked hard to
make sure she was fully educated on how to handle everything
in the best way for baby Russell.
John did not accept the diagnosis so easily. He simply could
not accept that their son had Down Syndrome and that it could
not be “fixed.”
John often screamed at Jane that this was “her fault” and
questioned her about what “she did” to make this happen. A
baby with severe medical issues was not what he had planned on
and was not the kind of “perfection” he demanded. It seemed to
Jane that John was angry all the time.
Jane often talked to John the first few months about moving to a
different home in a different school district where home care for

infants with Down Syndrome was provided even at this young
age. The visits from the home health aids would benefit baby
Russell a lot and ease financial burdens. John refused. He said
he was not going to change his home or lifestyle for this baby
and told Jane it was “all her fault” and “her problem.”
As the weeks went on the costs of the baby’s extra medical
expenses, home health aide visits, and special evaluations
caused a lot of stress for the couple. Jane worked as a realtor
for herself and if she was not working she was not making
money. Jane was not comfortable leaving the baby alone with a
babysitter given some of his medical issues so over time she
just did not go back to work. The loss of her income was
something the couple fought about on a daily basis.
John continued to express his anger towards Jane verbally about
the situation they were now in – she no longer was contributing
financially, they had a baby with medical issues that were not
planned on, and the baby was costing them too much money that
they did not have to spend.
Jane felt sad, stressed, and trapped in a hard situation. She just
focused on her two boys and how much she adored them and
figured her husband’s attitude would get better over time.
One Saturday morning Jane got a phone call from her father that
her mother was in the hospital with what appeared to be a heart
attack. Her parents lived an hour away and so she started to
pack up items for herself and the two boys so they could go to
the hospital to be with her parents.
John became furious. This was the one day off he had all week

and there were things he wanted them to accomplish together
around the house. This was not convenient for him – he told
her she should stay at the house for the day to work then she
could go this evening to the hospital. Jane adored her parents
and really wanted to go see her mother and support her father.
She was so worried. John was also angry that taking the kids to
the hospital would make him “look bad” to her family – like he
couldn’t take care of his own kids for a few hours so she had to
drag them along. He said if she insisted on going and leaving
him then she would have to leave the boys as well so he did not
look bad and she needed to be back in time to make dinner.
Jane – feeling stressed and upset about her mother and about her
husband’s behavior – agreed. She left the boys in the care of
her husband and went to the hospital to be with her mother and
father.
Jane arrived at the hospital to check on her mother safely and
tried to reach John a few times to check on thing at the house.
She could never reach him.
At 3 p.m. Jane got a call from John. He sounded stressed and
nervous. He said the baby wasn’t acting normal and was really
lethargic. She questioned whether John had been able to feed
the baby, whether the baby was having stomach trouble
consistent with his medical issues and whether he had a
temperature. John said he would try and feed him and asked
when Jane was coming home. Jane asked to talk to their son,
Russell, and John said Russell was playing in the backyard so
could not come to the phone.
At 5 p.m. John called Jane to say the baby was more lethargic –
he sounded nervous and jittery and very stressed. Jane briefly
felt that maybe this was what her husband needed to “bond”
with their baby and was happy John was finally engaged in the

care of their son. John said he thought he should take the baby
to the local Children’s Hospital and Jane said she would leave
her mother now and be there in one hour.
When John got to the Children’s Hospital the hospital staff
immediately saw signs of abuse and called the local authorities
– as required.
Upon examination the baby immediately went into what
appeared to be cardiac arrest and the hospital started life saving
measures but could not save the baby. The baby died within
minutes of arriving at the hospital.
Local authorities removed Russell from John’s care at the
hospital – as required – and started questioning John as to what
happened.
Jane arrived at the hospital, was told what happened, and had to
be sedated she was so hysterical.
Given the physical signs of abuse on the baby the police began
investigating the baby’s death as a potential homicide and an
autopsy is performed.
The autopsy revealed the following:
The baby died from peritonitis and sepsis either caused by a
defect in the bowel wall or from inflicted trauma (i.e. hitting or
punching) causing the bowel to tear and the contents to empty
and cause sepsis.
The baby had blunt-force injuries including cuts, bruises, a
fractured skull and a rib fracture that are suspicious as being
not accidental.

You are the District Attorney and you must decide whether to
charge this case as a death penalty case. Based on the police
investigation you have everything in your file that is in this
case study to use to make your decision.
FYI – if a District Attorney does not charge a defendant with
capital murder then the DA will charge the defendant with a
lesser form of murder with a lesser punishment – i.e. not death.
You also have to determine whether to charge the mother based
on the following statute:
Mandatory Duty to Report Suspected Child Abuse in the State
of Texas
Texas Family Code 261.101
Texas law requires anyone with knowledge of suspected child
abuse or neglect to report it to the appropriate authorities.
This mandatory reporting applies to all individuals and is not
limited to teachers or health care professionals.
The law even extends to individuals whose personal
communications may be otherwise privileged, such as attorneys,
clergy members, and health care professionals.
The report may be made to (1) any local or state law
enforcement agency; or (2) the Department of Family and
Protective Services.

All persons are required to make the report immediately, and
individuals who are licensed or certified by the state or who
work for an agency or facility licensed or certified by the state
and have contact with children as a result of their normal duties,
such as teachers, nurses, doctors, and day-care employees, must
report the abuse or neglect within 48 hours.
Texas law broadly defines abuse and neglect – so physical AND
mental welfare is covered.
A person who makes a report in good faith is IMMUNE from
criminal and civil liability.
You now have all of the information you need to write your
paper – please make sure you answer the questions listed out on
Slide 51 and proofread your papers!
Thank you for your hard work!
BUS 308 Week 5 Lecture 3
Excel has limited functions available for us to use on this
week’s material. We generally
need to take the output from the other functions and generate
our Effect Size values.
Effect Sizes

One issue many have with statistical significance is the
influence of sample size on the
decision to reject the null hypothesis. If the average difference
in preference for a soft drink was
found to be ½ of 1%; most of us would not expect this to be
statistically significant. And,
indeed, with typical sample sizes (even up to 100), a statistical
test is unlikely to find any
significant difference. However, if the sample size were much
larger; for example, 100,000; we
would suddenly find this miniscule difference to be significant!
Statistical significance is not the same as practical significance.
If for example, our
sample of 100,000 was 1% more in favor of an expensive
product change, would it really be
worthwhile making the change? Regardless of how large the
sample was, it does not seem
reasonable to base a business decision on such a small
difference.
Enter the idea of Effect Size. The name is descriptive but at the
same time not very
illuminating on what this measure does. We will get to specific
measures shortly, but for now,
let’s look at how an Effect Size measure can help us understand
our findings. First, the name:
Effect Size. What effect? What size? In very general terms,
the effect we are monitoring is the
effect that occurs when we change one of the variables. For
example, is there an effect on the
average compa-ratio when we change from male to female.
Certainly, but not all that much, as
we found no significant difference between the average male
and female compa-ratios. Is there

an effect when we change from male to female on the average
salary? Definitely. And it is
much larger than what we observed on the compa-ratio means.
We found a significant
difference in the average salary for males than females – around
$14,000.
The Effect Size measures looks at the impact of the variables on
our outcomes; large
impacts suggest that variables are important, while small
impacts might suggest that the variable
is not particularly important in determining changes in
outcomes. We could, for example, argue
that both male and females in the population had the same
compa-ratio mean and what we
observed in the sample was simply the result of sample error.
Certainly, our test results and
confidence intervals could support this.
Now, when do we look at an Effect Size; that is, when should
we go to the effort of
calculating one. The general consensus is that the Effect Size
measure only adds value to our
analysis if we have already rejected the null hypothesis. This
makes sense, if we found no
difference between the variables we were looking at, why try to
see what effect changing from
one to the other would do. We already know, not much.
When we reject a null hypothesis due to a significant test
statistic (one having a p-value
less than our chosen alpha level), we can ask a question: was
this rejection due to the variable
interactions or was it due to the sample size? If due to a large
sample size, the practical
significance of the outcome is very low. It would often not be

“smart business” to make a
decision based on those kinds of results. If, however, we have
evidence that the null was
rejected due to a significant interaction by the variables, then it
makes more sense to use this
information in making decisions.
Therefore, when looking at Effect Sizes, we tend to classify
them as large, moderate, or
small. Large effects mean that the variable interactions caused
the rejection of the null, and our
results have practical significance. If we have small effect size
measures, it indicates that the
rejection of the null was more likely to have been caused by the
sample size, and thus the
rejection has very little practical significance on daily activities
and decisions.
OK, so far:
• Effect sizes are examined only after we reject the null
hypothesis, they are
meaningless when we do not reject a claim of no difference.
• Large effect size values indicate that variable interactions
caused the rejection of the
null hypothesis, and indicate a strong practical significance to
the rejection decision.
• Small effect size values indicate that the sample size was the
most likely cause of
rejecting the null, and that the outcome is of very limited
practical significance.

• Moderate effect sizes are more difficult to interpret. It is not
clear what had more
influence on the rejection decision and suggests only moderate
practical significance.
These results might suggest a new sample and analysis.
Different statistical tests have different effect size measures and
interpretations of their
values. Here are some that relate to the work we have done in
this course.
• T-test for independent samples. Cohen’s D is found by the
absolute difference
between the means divided by the pooled standard deviation of
the entire data set. A
large effect is .8 or above, a moderate effect is around .5 to .7,
and a small effect is .4
or lower. Interpretation of values between these levels is up to
the researcher and/or
decision maker.
• One-sample T-test. Cohen’s D is found by the absolute
difference between the means
divided by the standard deviation of the tested variable data set.
A large effect is .8 or
above, a moderate effect is around .5 to .7, and a small effect is
.4 or lower.
Interpretation of values between these levels is up to the
researcher and/or decision
maker.
• Paired T-test. Effect size r = square root of (t^2/(t^2 + df)).
A large effect is .4 or
above, a moderate effect is around .25 to .4, and a small effect
is .25 or lower.

• ANOVA. Eta squared equals the SS between/SS total. A
large effect is .4 or above,
a moderate effect is .25 to .40, and a small effect is .25 or
lower.
• Chi Square Goodness of Fit tests (1-row actual tables). It is,
also called Effect size r
= square root (Chi Square statistic/(N * (c -1)), where c equals
the number of columns
in the table. A large effect is .3 or above, a moderate effect is
.3 to .5, and a small
effect is .3 or lower.
• Chi Square Contingency Table tests. For a 2x2 table, use phi
= square root of (chi
square value/N). A large effect is .5 or above, a moderate effect
is .3 to .5, and a
small effect is .3 or lower.
• Chi Square Contingency Table tests. For larger than a 2x2
table, use Cramer’s V =
square root (chi square value/((smaller of R or C)-1)). A large
effect is .5 or above, a
moderate effect is .3 to .5, and a small effect is .3 or lower.
• Correlation. Use the absolute value of the correlation, A large
effect is .4 or above, a
moderate effect is .25 to .4, and a small effect is .25 or lower.
Would using these measures change any of our test
interpretations?

Please ask your instructor if you have any questions about this
material.
When you have finished with this lecture, please respond to
Discussion thread 3 for this
week with your initial response and responses to others over a
couple of days.
BUS308 Week 5 – Lecture 2
This week we are changing focus again. From looking at
differences and relationships,
we are going to look at considering what we can take from our
findings – broadening, so to
speak, our interpretation of what we have found out.
Our focus will take us to two distinctly different views of the
findings. The first topic
will be confidence intervals. We saw something similar to these
when we found which pairs of
means different with our ANOVA outcomes. One of the most
commonly seen example of
confidence intervals comes with opinion polling data. We often
see a result such as 56% of the
population approves of the idea with a margin of error of +/-
3%. This means that the real
percentage if everyone were asked would lie between 53 and
59% - the confidence interval. The
second will be something called Effect Size, a way of
examining how practically significant a
finding is. Effect sizes will be discussed in the third lecture for

this week.
Confidence Intervals
One thing we might have noticed during this class is that
different samples from the same
population (technically, with replacement, putting previously
selected individuals back into the
pool for a chance of being selected again), give us different
results. The sample used in the
lectures was different from the sample used in the homework
assignments, at least with the
salary and compa-ratio values. Yet, even with these sample
differences, the outcomes of the
statistical tests were the same – we rejected or not rejected the
same hypothesis statements with
our slightly different values.
So, many want to know the range of values that would could be
a population’s mean
based on the sample results, particularly the variation present in
the sample. A confidence
interval can provide us with this information. This is basically
a range of values that show us
the possible sampling error in our results. When we construct
them, we are able to specify the
level of confidence we desire that the interval will contain the
actual population mean – 90%,
95%, 99%, etc.
Confidence intervals often provide the added information and
comfort about estimates of
population parameter values that the single point estimates lack.
Since the one thing we do know
about a statistic generated from a sample is that it will not
exactly equal the population

parameter, we can use a confidence interval to get a better feel
for the range of values that might
be the actual population parameter. They also give us an
indication of how much variation exists
in the data set – the larger the range (at the same confidence
level), the more variation within the
sample data set – and the less representative the mean would be.
We are going to look at two
different kinds of confidence intervals this week – intervals for
a one sample mean and intervals
for the differences between the means of two samples.
One Sample Confidence Interval for the mean
A confidence interval is simply a range of values that could
contain the actual population
parameter of interest. It is centered on the sample mean, and
uses the variation in the sample to
estimate a range of possible values. To construct a confidence
interval, we use several pieces of
information from the sample, and the confidence level we want.
From the sample we use the mean, standard deviation, and size.
To get the confidence
level – a desired probability, usually set at 95%, that the
interval does, in fact, contain the
population mean – we use a related t value.
One-Sample Intervals
Example. Confidence intervals can be generated manually or
we can use the Excel
Confidence.T function found in the fx or Formulas statistical

list. To compare the two, we will
construct a 95% confidence interval for the female compa-ratio.
Our sample mean value is
1.069, the standard deviation is 0.70, and the sample size is 25
(from week 1 material).
Our confidence interval formula is mean +/- t* standard error of
the mean, which is the
same as: mean +/- t * stdev/sqrt(sample size).
Once we determine the confidence level we want, we use the
associated 2-tail t value to
achieve it. The t-value is found with the fx function t.inv.2t
(Prob, df). For a 95% confidence
interval, we would use t.inv.2t(0.05, 24), this equals 2.064
(rounded). Remember that the df for
a t is n-1, so for a sample of 25 it equals 24.
We now have all the information we need to construct a 95%
confidence interval for the
female compa-ratio mean:
CI = mean +/- t * stdev/sqrt(sample size) = 1.069 +/-
2.064*0.070/sqrt(25) = 1.069 +/- 0.029.
This is typically written as 1.04 to 1.098. Note: the standard
deviation divided by the square root
of the sample size is called the standard error of the mean, and
is the variation measure of the
sample used in several statistical tests, including the t-test and
confidence intervals.
Now, let’s look at how Excel can help us with this. Excel has
two confidence interval
functions, one based on the z distribution (often used for large
samples) and another based on the

t distribution. The t-based interval is becoming more and more
of the standard with the wide
availability of Excel and other statistical programs. So, we will
use it.
The Fx (or Formulas) function Confidence.t will give us the
margin of error (the +/- term)
for any desired interval. Here is a screenshot of this tool, with
our Female Compa-ratio values
filled in. Note that our value from above (0.029 rounded) is
shown even without hitting OK.
The associated 95% CI for males is 1.022 to 1.091. Note that
the ranges overlap. The
female range of 1.04 to 1.098 (with higher endpoint values than
the male range) and the male
range of 1.022 to 1.091(with lower endpoint values) overlap
quite a bit: between 1.04 to 1.091 of
each range is common to both. How do we interpret this
overlap? We cannot immediately say
an overlap means the two means might be equal. We will look
at the reason for this after looking
at confidence intervals for differences below.
In the meantime, there are a couple of guidelines we can use.
• In general, the more overlap, the less significant the difference
is.
• If the bottom quarter of the higher value range overlaps with
the top quarter of the
lower value range, the means would be found to be significantly

different at the alpha
= 0.05 level.
• If the bottom value of the higher value range just touches the
top value of the lower
valuer range, the means would be found to be significantly
different at the .01 alpha
level.
• If the ranges do not overlap at all, the means are significantly
different at an alpha
less than .01 level.
• If the intervals overlap more than at the extreme quarters, the
means are not
significantly different at the alpha = .05 level.
Since these CIs overlap more than a single quarter of their
ranges, our interpretation would be
that they are not significantly different at the alpha = .05 level.
This is what we found in week 1
when we performed the t-test for mean difference.
The Confidence Interval for mean differences.
When comparing multiple samples, it is always best to use all
the possible information in
a single test or procedure. The same is true for confidence
intervals. Rather than calculate 2 CIs
to determine if means differ, we can use a single Confidence
Interval for the difference. If we
are interested in seeing if sample means could be equal, we look
to see if the interval for their
difference contains 0 or not. If so, then the means could be the
same; if not, then the means must
be significantly different. This interpretation of the interval is

the same as we used with the
ANOVA intervals in week 3.
The formula for the mean difference confidence interval is mean
difference +/-
t*standard error. The standard error for the difference of two
populations is found by adding the
variance/sample size (which is the standard error squared) for
each and taking the square root.
For our compa-ratio data set we have the following values:
Female mean = 1.069 Male mean = 1.056 t = t.inv.2t(0.05, 48) =
2.106
Female Stdev = 0.070 Male Stdev = 0.084 Sample size = 50, df
= 48
Female variance = 0.070^2 = 0.0049 Male variance = 0.084^2 =
0.0071
Standard error = sqrt(Variance (female)/25) + Variance
(male)/25) =
Sqrt(0.0049/25 + 0.0071/25) = sqrt(.000196 + 000284)
=0.0219.
This gives us a 95% confidence interval for the difference
equaling:
(1.069-1.056)) +/- 2.011 * 0.0219 = 0.013 +/- 0.044 = -0.031 to
0.057.
Since this confidence interval does contain 0, we are 95%
confident that the male and female

compa-ratio means are not equal – which is the same result we
got from our 2-sample t-test in
week 2. We also now have a sense of how much variation exists
in our measures. One
interpretation of these two intervals is that the averages are
clearly similar rather than just barely
the same.
Side note: The “+/- t* SE” term is often called the margin of
error. We most often hear
this phrase in conjunction with opinion polls – particularly
political polls, “candidate A has 43%
approval rating with a margin of error of 3.5%. While we do
not deal with proportions in the
class, they are calculated the same as an empirical probability –
number of positive replies
divided by the sample size. The construction of these margins
or confidences is conceptually the
same – a t-value and a standard error of the proportion based on
the sample size and results.
Now, let’s go back and look at why two individual confidence
intervals do not provide
the same information on differences as a single CI for
differences. The reason lies in the statistic
used for our confidence level. The male and female individual
mean confidence intervals used a
t value of 2.064 while the CI for the difference used the smaller
value of 2.011, due to the larger
sample size. This difference in the t value between individual
and difference intervals is the
reason why the individual intervals cannot be used for a direct
interpretation of differences; they
are a bit too large for the combined data set.

material.
couple of days before reading the
third lecture for the week.
BUS308 – Week 5 Lecture 1
Years ago, a comedy show used to introduce new skits with the
phrase “and now for
something completely different.” That seems appropriate for
this week’s material.
This week we will look at evaluating our data results in
somewhat different ways. One of
the criticisms of the hypothesis testing procedure is that it only
shows one value, when it is
reasonably clear that a number of different values would also
cause us to reject or not reject a
null hypothesis of no difference. Many managers and
researchers would like to see what these
values could be; and, in particular, what are the extreme values
as help in making decisions.
Confidence intervals will help us here.
The other criticism of the hypothesis testing procedure is that
we can “manage” the
results, or ensure that we will reject the null, by manipulating
the sample size. For example, if

we have a difference in a customer preference between two
products of only 1%, is this a big
deal? Given the uncertainty contained in sample results, we
might tend to think that we can
safely ignore this result. However, if we were to use a sample
of, say, 10,000, we would find
that this difference is statistically significant. This, for many,
seems to fly in the face of
reasonableness. We will look at a measure of “practical
significance,” meaning the likelihood of
the difference being worth paying any attention to, that is called
the effect size to help us here.
Confidence Intervals
A confidence interval is a range of values that, based upon the
sample results, most likely
contains the actual population parameter. The “most likely”
element is the level of confidence
attached to the interval, 95% confidence interval, 90%
confidence interval, 99% confidence
interval, etc. They can be created at any time, with or without
performing a statistical test, such
as the t-test.
A confidence interval may be expressed as a range (45 to 51%
of the town’s population
support the proposal) or as a mean or proportion with a margin
of error (48% of the town
supports the proposal, with a margin of error of 3%). This last
format is frequently seen with
opinion poll results, and simply means that you should add and
subtract this margin of error from
the reported proportion to obtain the range. With either format,
the confidence percent should
also be provided.

Confidence intervals for a single mean (or proportion) are fairly
straightforward to
understand, and relate to t-test outcomes simply. Details on
how to construct the interval will be
given in this week’s second lecture. We want to understand
how to interpret and understand
them in this discussion.
In Week 2, we looked at how to test sample means against a
constant, and we found that
the female compa-ratio mean was not equal to or less than 1.0.
The related confidence interval
for the female compa-ratio mean would be 1.0367 to 1.0977, or
1.0687 +/- 0.0290 (all values
rounded to 4 decimal places). This result relates to possible t-
test outcomes directly. If, again in
the one sample situation, the standard/constant we are
comparing our sample result against is
within this range, then we would NOT reject the null hypothesis
of no difference. If the standard
is outside of this range, as our 1.00 test in Week 2, then we
reject the null and say we have a
significant difference. It is clear in this case, that the female
mean is not even close to the mid-
point value of 1.0 that we looked at.
Confidence intervals allow us to make some informed “gut
level” decisions when more
precise measure may not be needed. For example, if the means
of two variables are fairly close,
the wider confidence interval will have more variation within
the data, and be less consistent.

We could test this with the F-test for variance that we covered
in week 2. While a hypothesis
result of “reject the null hypothesis” or “do not reject the null
hypothesis” with an alpha of 0.05
is definite; it does not convey the “strength” of the rejection.
Comparing the endpoints against
the standard used in our one-sample t-test would give a sense of
how “close” we came to making
the other decision.
Confidence intervals can also be used to examine the difference
between means. The
most direct way is by constructing a confidence interval for the
difference. Again, the details on
how to develop one of these will be presented in the second
lecture for this week. This result is
very similar to the intervals we constructed while doing the
ANOVA comparisons. While we
use a different calculation formula when comparing only two
means (rather than two means at a
time with the ANOVA situation), the interpretation is the same.
If the range contains a 0, then
the population means could be identical and we would not reject
the null hypothesis of no
difference.
If we have two single mean confidence intervals, for example
intervals for the male and
female compa-ratios; using them to determine if the means are
significantly different is a bit
trickier than simply seeing if they might contain the same value
within their range. If, the top ¼
of one interval and the bottom ¼ of the other overlap, then we
have a significant difference at the
alpha = 0.05 level. If the endpoints barely overlap, we have a
significant difference around the

alpha = 0.01 level.
The natural question at this point is why does an overlap when
comparing means show a
significant difference when it does not do so when comparing a
mean against a constant. The
answer lies in how the intervals are constructed. Without
getting too technical, the intervals use
a t-value to establish the level of confidence. And, as the
sample size gets larger, the
corresponding t-value gets smaller for any specific alpha level.
So, in our example of comparing
compa-ratio means, we had samples of 25 when constructing the
individual intervals and used a
slightly larger t-value than we would use with our overall
sample of 50 when comparing the two
groups together. This means the individual intervals are a bit
longer when compared to the
larger sample result, hence why some of the overlap shows a
significant difference rather than
the more “logical” interpretation of only no overlap at all means
they differ.
Effect Size – Practical Importance
A popular saying a few years ago was “if you torture data long
enough, it will confess to
anything.” �� Unfortunately, many regard statistical analysis
this same way. Some think that if
you do not get a rejection of the null hypothesis that you want,
simply repeat the sampling with a
larger group, at some sample size virtually all differences will
be found to be statistically

significant. Note, this is somewhat unethical for professional
researchers; however, those who
feel that proving their point is more important than following
professional guidelines have been
known to do this.
But, does statistical significance mean the findings should be
used in decision making?
If, for example we typically round salary to the nearest
thousand dollars when making decisions,
does a significant difference based on a $500 difference have
any practical importance?
Probably not, even if we could find a sample size large enough
to make this difference
statistically significant.
So, how do we decide the practical importance of a statistically
significant difference?
Once, and this is important, we have rejected the null
hypothesis – and only if we have rejected
the null hypothesis – we calculate a new statistic called the
effect size.
The name comes from the effect changing the variable’s value
would have on the
outcome. To understand this idea, let’s look at the male and
female compa-ratios. We found in
week 2 that the male and female compa-ratio means were not
significantly different. So, the
“effect” of changing from male to female when doing an
analysis with the compa-ratio mean
would not be very big. However, if we switched from the male
to female average salary, we
would expect to see a large effect or difference in the outcome
since their salaries were so
different.

The effect size measure – however it is calculated for different
statistical tests – can be
interpreted in a similar fashion. Effect sizes generally have
their value translated into a “large,”
“moderate,” or “small” label. If we have a large effect, then we
know that the variable
interaction caused the rejection of the null hypothesis, and that
our results have a strong practical
significance. If, however, we have a small effect, then we can
be fairly sure that the sample size
caused the rejection of the null hypothesis and the results have
little to no practical significance
for decision making or research results. A moderate outcome is
less clear, and we might want to
redo the analysis with a different sample. (Note: since we have
already rejected the null,
repeating the experiment with a different sample in this case is
not to manipulate the findings,
but rather to study the effect of the variables in more detail.
This is done in research all the time
– providing evidence that the findings are replicable and
correct.) Examples of different effect
size measures and how to determine what is large, medium, and
small are presented in the third
lecture for this week.
If you have any questions on this material, please ask your
instructor.
After finishing with this lecture, please go to the first
discussion for the week, and engage
in a discussion with others in the class over the first couple of
days before reading the second
lecture.

Moving to correlation and regression opens up new insights into
our data sets, but still
lets us use what we have learned about Excel tools in setting up
and generating our results.
Regression lets us use relationships between and among our
variables to predict or explain
outcomes based upon inputs, factors we think might be related.
In our quest to understand what
impacts the compa-ratio and salary outcomes we see, we have
often been frustrated due to being
basically limited to examining only two variables at a time,
when we felt that we needed to
include many other factors. Regression, particularly multiple
regression, is the tool that allows
us to do this.
Regression
Regression takes us the next step in the journey. We move from
knowing which
variables are correlated to finding out which variables can be
used to actually predict outcomes
or explain the influence of different variables on a result. As
we might suspect, linear regression
involves a single dependent (outcome) and single independent
(input) variable. Linear
regression uses at least interval level data for both the
dependent and independent variables.

The form of a linear regression equation is:
Y = a + b*X; where Y is the output, X is the input, a is the
intercept (the value of y when
X = 0) on a graph, and b is the coefficient (showing the change
in Y for every 1 change in the
value of X.
Earlier, we found that the correlation between raise and
performance rating was 0.674
(rounded). While we did not make note of this in our
correlation discussion, it was part of the
correlation table. This correlation relates to a coefficient of
determination (CD) of 0.674^2 or
0.45 (rounded). As mentioned, this is not a particularly strong
correlation, and we would not
expect the graph of these values to show much of a straight line.
For purposes of understanding
linear regression, let’s look at a graph showing performance
rating as an input (an X variable)
predicting raise (Y). An example of a regression equation and
its graph is:
Raise (Y) vs Performance Rating (X)
y = 0.0512x + 0.5412
R² = 0.4538
0

1
2
3
4
5
6
7
0 20 40 60 80 100 120
This is a Scatter Diagram graph produced by Excel. The
regression line, equation, and R-
squared values have been added. Note that the Coefficient of
Determination (R2) is the 45% we
found earlier, and that the data points are not all that close to
the regression (AKA trend) line.
Note the format of the regression equation Y = 0.5412 +
0.0512X, this is the same as saying
Raise = 0.5412 + 0.0512* Performance Rating when we
substitute the variable names for the
algebraic letters.
Let us look at the equation. Since we know that the correlation
is significant (it is larger than our
0.278 cut-off discussed in lecture 2), the linear regression
equation is significant. The regression
says for every single point increase in the performance rating

(our X variable), the raise (The Y
variable) increases, on average by 0.0512%. If we extended the
line towards the y (vertical axis),
it would cross at Y = – 0.0512 and X = 0, this is an example
where looking at the origin points is
not particularly helpful as no one has a performance rating of 0.
This graph does tend to
reinforce our earlier comment that raise and performance rating,
even though the strongest
correlation, are not particularly good at predicting each other’s
value. We see too much
dispersion of data points around the best fit regression line
through the data points.
Most of us are probably not surprised, just as we feel compa-
ratio is not determined by a
single factor, we know raise is more complicated than simply
the performance rating. This is
where looking at multiple regression, the use of several factors,
might be more insightful.
Multiple Regression
Multiple Regression is probably the most powerful tool we will
look at in this course. It
allows us to examine the impact of multiple inputs (AKA
independent variables) on a single
output or result (AKA dependent variable). It also allows us to
include nominal and ordinal
variables in the results when they are used as dummy coded
variables.
Multiple regression has an interesting ability that we have not
been able to use before. It
can use nominal data variables as inputs to our outcomes, rather
than using them simply as

grouping labels. It does so by assigning either a 0 or 1 to the
variable value depending upon
whether some characteristic exists or not. For example, with
degree we essentially are looking to
see if a graduate degree has any impact, since everyone in the
sample has at least an
undergraduate degree. So, we code the existence of a graduate
degree with a 1, and the “non-
existence” with a 0. Similarly, with gender we are interested,
essentially, how females are being
treated, so we code them 1 (existence of being female). This
coding is called Dummy Coding,
and involves only using a 0 or 1 in specific situations where the
existence of a factor is
considered important. Note, other than some changes in the
value of the coefficients, the
outcomes would not differ if the codes were reversed. The
significance, or non-significance, of
degree or gender would remain the same regardless of the code
used. We will comment on this
more after we see our results.
Question 2
Question 2 for this week asks for a regression equation that
explains the impact of
various variables on our output of interest. Of course, in the
homework this is salary, while in
our lectures it is the compa-ratio.
Both linear and multiple regression are both set up in the same
fashion, so we will look at
only the multiple regression situation. For the data, put the
dependent variable, the output such

as salary or compa-ratio, in one column and then paste the
independent, input, variables in
sequential columns next to it. Make sure that none of the
columns contain letter characters. It is
also a good idea to include the variable labels for each data
column.
The Regression function is found in the Data | Analysis block
and is labeled Regression.
Here is a screen shot of a complete Regression set-up for a
regression equation for compa-ratio.
Note that unlike the correlation input, we have two ranges to
work with. The first is the output,
which for this example is compa-ratio (and would be salary for
the homework). The second is
for the inputs, which should include all of the numeric looking
variables, including the Degree
and Gender variables as shown below.
Data range entry for the Y (or outcome) and the X (or input)
variables are done separately
by either typing in the ranges or using dragging the cursor over
the data range after clicking on
the up arrow at the right end of the data entry boxes. The same
is done with the data entry box
after clicking the circle for Output range.
There are a number of options to consider. First, of course, is
the need to click the labels
box if your data ranges include labels. A second option is the
Constant is Zero equation. This
would force the regression equation to pass thru the X = 0 and
Y = 0 origin, even if this is not the
best fit. Use this with caution, even though it might make no
sense to have Y = 0 when all the X

variables are 0, using this option may not give us the equation
that best fits the data.
The residuals box provides a way to see how well each of the
plotted data points fits with
the predicted results. This will often allow us to see outliers –
cases that do not fit with the rest
of the data set. Outliers are sometimes indications of data entry
errors or, in the case of salary,
they may be paid using a different approach. One such example
would be a commission
salesperson being included with employees that are paid on a
straight salary, the basis of pay is
so different these two should not be analyzed in the same study.
Other options here allow for the
results to be turned into Z-scores (Standardized Residuals),
plotted on a graph, or have linear
plots made for the output and each separate input. Normal
Probability Plots are rather
complicated to discuss, and it is left to the student to explore
this if desired. You are encouraged
to play around with some of these options, even though they are
not required for the assignment.
Now that the data has been set up, let’s look at our hypothesis
testing process for the
question, first, of whether or not the regression equation is
helpful in explaining what impacts
compa-ratio outcomes.
Parts a and b. This part looks at the overall regression.
Step 1: Ha: The regression equation is not significant.

Ho: The regression equation is significant.
Step 2: Alpha = 0.05
Step 3: F stat and ANOVA-Regression, used to test regression
significance
Step 4: Decision Rule: Reject the null hypothesis if p-value <
0.05.
Step 5: Conduct the test.
After completing the set-up box, click on OK to produce the
result.
Here is a screen shot of a multiple regression analysis for the
question of what factors influence
compa-ratio. Note: we will split the discussion of the output
into two screen shots.
The first table in the output provides some summary statistics.
Two are important for us – the
multiple correlation, shown as R, which equals 0.655, a
moderate value; and, the R square or the
multiple coefficient of determination showing that about 43% of
the variation in compa-ratio
values can be explained by the shared variation in the variables
used in the analysis.
The second table shows the results of the actual statistical test
of the regression. Similar to the
ANOVA tables we looked at last week, it has two rows that are

used to generate our F statistic
(4.51) and the p-value (Significance F) of 0.0008.
Step 6: Conclusion and Interpretation.
What is the p-value? 0.0008
Decision: Rej or Not reject the null? Reject the null hypothesis.
Why? The p-value is less than (<) 0.05.
Conclusion about Compa-ratio factors? The input variables are
significantly related to compa-
ratio outcomes. Some of the compa-ratio outcomes can be
explained by the selected variables.
We used the phrase “some of” since the equation only explains
43% of the variance, less than
half.
Part c
Once we reject the null hypothesis, our attention changes to the
actual equation, the
variables and their corresponding coefficients. The third table
provides all the details we need to
reach our conclusions.
As with the correlations in question 1, we will use the
hypothesis testing process, but will
write it only once and use the p-values to make decisions on
each of the possible equation
variables.
Step 1: Ha: The variable coefficient is not significant (b = 0).
Ho: The variable coefficient is significant (b =/= 0).

Step 3: T stat and t-test for coefficients
Step 4: Decision Rule: Reject the null hypothesis if p-value <
0.05.
Step 5: Conduct the test. In this case, the test has already been
performed and is part of the
regression out. Here is a screen shot of the second half of the
Regression output.
Step 6: Conclusions and Interpretation
As with the correlations, we will use a single statement of the 6
steps to interpret the outcomes in
this part. Here is the completed table.
The Multiple Regression equation is similar to the linear
regression example given above
except it has more independent terms: Y = a + b1*X1 + b2*X2
+ B3*X3 + …. The b’s stand for
the coefficients that are multiplied by the value of each variable
(represented by the X’s).
In first column (L in the screen shot) are the possible regression
elements starting with
the intercept, which is always a part of the equation. The next
column (M) and the fifth column
are the really important columns. Column P, labeled p-value,
tells us which variables are

statistically significant. Just as with our previous tests, if the p-
value is less than (<) our chosen
alpha, we reject the related null hypothesis and accept the
alternate that the coefficient’s value is
different than 0, and the related variable should be included in
the final equation.
For our example, we find that only 3 variables are statistically
significant; the midpoint, the
performance rating, and the gender. With these 3 variables and
the intercept, the statistically
significant regression equation is:
Compa-ratio = 0.954 + 0.003*midpoint -0.002*performance
rating + 0.056*gender.
So, what does this equation mean? How do we interpret it? The
intercept (0.9545) is somewhat
of a place holder – it centers the line in the middle of the data
points, but has little other meaning
for us. The three variables, however tell us a lot. Changes in
each of them impact the compa-
ratio outcome independently of the others – it is as if we can
consider the other factors being held
constant as we examine each factor’s impact. So, all other
things the same, each dollar increase
in midpoint increases the compa-ratio value by 0.0034. This
relates to what we found last week
that compa-ratio is not independent of grade. At the same time,
and possibly surprisingly, every
increase in an employee’s performance rating causes the compa-
rating to decrease by .0024!
Finally, the equation says that gender is an important factor.
This factor alone means that the

company is violating the equal pay act. But, what might be
surprising is that for a change from
male (coded 0) to female (coded 1) the compa-ratio goes up by
0.0562! Females get a higher
compa-ratio (percent of midpoint) when all other things are
equal than males do, since the female
gender results in adding 0.056*1 to the compa-ratio while the
male gender has 0.056 * 0 (or 0)
added to their compa-ratio.
We did have one hint that this might be the case, when we
noticed in week 1 that the female
mean compa-ratio was higher than the male compa-ratio. But,
then some of the single factor
tests minimize this difference. This is one of multiple
regression’s greatest strengths, it will
show us the impact of a single variable by controlling for, or
keeping constant, the impact of all
other variables.
Parts d, e, and f
Gender is a significant element in the compa-ratio, as females
get a higher value when all
other variables are equal. We see this from the significant
positive coefficient to the variable
gender. Females are coded 1, so they get more added to their
result.
Here is a video on Regression: https://screencast-o-
matic.com/watch/cb6jfuIk8S
Question 3
This answer will depend on what other factors you would like to
see.

Question 4
As of this point, we have some strong evidence in the compa-
regression equation and the t-test
on average compa-ratios, that females get more pay for equal
work than males. The company is
violating the Equal Pay Act, in favor of women.
Question 5
What you say here describes your understanding of regression
analysis versus the power of
inferential tests of 2 variables at a time.
material.
couple of days.
https://screencast-o-matic.com/watch/cb6jfuIk8S
BUS 308 – Week 4 Lecture 2
As in many detective stories, we will often find that when one
thing changes, we see that
something else has changed as well. The correlation between
events is mirrored in data analysis
examinations with correlation analysis. This week’s focus

changes from detecting and
evaluating differences to looking at relationships. As students
often comment, finding
significant differences in gender based measures does not
explain why these differences exist.
Correlation, while not always explaining why things happen
gives detectives great clues on what
to examine more closely and helps move us towards
understanding why outcomes exist and what
impacts them. If we see correlations in the real world, we often
will spend time examining what
might underlie them; finding out if they are spurious or causal.
Linear Correlation
When two things seem to move in a somewhat predictable way,
we say they are
correlated. This correlation could be direct or positive, both
move in the same direction, or it
could be inverse or negative, where when one increases the
other decreases. The Law of Supply
and price in economics is a common example of an inverse
correlation, where the more supply
we have of something, the less we typically can charge for it;
the Law of Demand is an example
of a direct correlation as the more demand exists for something,
the more we can charge for it.
Height and weight in young children is another common
example of a direct correlation, as one
increases so does the other measure.
Probably the most commonly used correlation is the Pearson
Correlation Coefficient,
symbolized by r. It measures the strength of the association –
the extent to which measures
change together – between interval or ratio level measures as

well as the direction of the
relationship (inverse or direct). Several measures in our
company data set could use the Pearson
Correlation to show relationships; salary and midpoint, salary
and years of service, salary and
performance rating, etc. The Pearson Correlation runs from -
1.0 (perfect negative or inverse
correlation) thru 0 (no correlation) to +1.0 (perfect positive or
direct correlation).
A perfect correlation means that if we graphed the values, they
would fall exactly on a
straight line, either increase from bottom left to top right
(positive) or from top left to bottom
right (negative). The stronger the absolute value (ignoring the
sign), the stronger the correlation
and the more the data points would form a straight line when
plotted on a graph. The Excel Fx
function Correl, and the Data Analysis function Correlation
both produce Pearson Correlations.
Question 1
When we have a data set with multiple variables, we would
want to see what
relationships exist – a detective’s sort of “who works with
whom” around the result we are
looking for. Data set-up for a correlation is perhaps the
simplest of any we have seen. It
involves simply copying and pasting the variables from the Data
tab to the Week 4 worksheet.
Again, paste them to the right of the question area. The
screenshot below pasted them starting at
column V. The only issue is to only paste the variables we want
to use.

Question 1 asks for the correlation among the interval/ratio
level variables with salary,
and says to exclude compa-ratio. Setting you your data would
be fairly simple. Copy the salary
data to column Q (for example). Then copy the Midpoint thru
Service columns and paste them
next to salary. Finally copy the Raise column and paste it next
to the service column. Notice
that our data input range now excludes compa-ratio, Gender,
Degree, Gender1, and Grade that
are in the original data set. Except for compa-ratio (which was
excluded by the question), none
of these are at least interval level variables and therefore cannot
be used in a Pearson Correlation.
For our example, we will correlation compa-ratio with the other
interval/ratio level
variables with the exclusion of salary. Since compa-ratio =
salary/midpoint, it does not seem
reasonable to use salary in predicting compa-ratio or compa-
ratio in predicting salary.
Now that the data is set-up, we can proceed with the hypothesis
test on the question of
which variables are significantly correlated to each other.
However, we are going to proceed a
bit differently this week. Since you have all had practice in
setting up and performing these
steps, we are going to look at how to evaluate the correlations
in a different way.
The significance of the Pearson Correlation is tested with the t-
test, t = r * sqrt(n-
2)/sqrt(1-r^2), df = n-2; where n equals the number of data point

pairs used in the correlation.
So, we could set up the hypothesis testing steps for each of the
correlations (which we will see
shortly equals 15), or we can find the value of r that cuts-off the
significant and non-significant
correlation values. Having this critical value (which is
sometimes presented in correlation
tables), gives us a quick decision point (much like we use the p-
value).
The formal approach is:
Step 1: Ho: Correlation is not significant
Ha: Correlation is significant. (A two-tail test.)
Step 3: Spearman’s r, t, and the correlation t-test to test a
correlation
Step 4: Reject the null hypothesis if the correlation value is
larger than the critical value. (The
critical correlation value has a related t-statistic having a p-
value = 0.05. Larger correlations
result in smaller p-values. So, we are essentially saying reject
the null when the p-value is <
0.05; our usual standard.)
Statistical Significance
The issue is now, what is our critical r value?
Technical Point. If you are interested in how we obtain the
formula for determining the
minimum r value, the approach is shown below. If you are not

interested in the math, you can
safely skip this paragraph, and go to The Result paragraph
below.
We know that t = r* sqrt(n-2)/sqrt(1-r2)
Multiplying both sides by sqrt(1-r2) gives us t *sqrt (1- r2) =
r*sqrt(n-2)
Squaring both sides gives us: t2 * (1- r2) = r2* (n-2)
Multiplying each side out gives us: t2– t2* r2 = n r2-2* r2
Adding t2* r2 both sides gives us: t2= n* r2-2*r2+ t2 *r2
Factoring gives us: t2= r2 *(n -2+ t2)
Dividing both sides by *(n -2+ t2) gives us: t2 / (n -2+ t2) = r2
Taking the square root gives us: r = sqrt (t2 / (n -2+ t2)
The Result. The formula to use in finding the minimum
correlation value that is
statistically significant is: r = sqrt(t^2/(t^2 + n-2)), where t is
the 2-tail value.
We would find the appropriate t value by using the
t.inv.2T(alpha, df) function with alpha = 0.05
and df = n-2 or 48 (for our data set of 50 employees). Plugging
these values into the gives us a t-
value of 2.0106 or 2.011(rounded).
Putting 2.011 and 48 (n-2) into our formula gives us a r value of
0.278; therefore, in a

correlation table based on 50 pairs, any correlation greater or
equal to 0.278 would be
statistically significant.
So, what does all this mean? If we find a correlation based on
50 pairs of data (such as
what our data set will produce), any correlation value that
exceeds 0.278 would be found to be
statistically significant (p-value less than 0.05), and cause us to
reject the null hypothesis of not
significant.
So, when looking at a table of correlation values, we can
identify the significant
correlations immediately; these are any correlation above the
absolute value of 0.278 (that means
larger than + 0.278 (such as + .46) or less than -0.278 (such as -
0.53)). Knowing how to
interpret table results, we can proceed to creating the
correlations.
Step 5: Conduct the test.
Pearson correlations can be performed in two ways within
Excel. If we have a single pair
of variables we are interested in, for example compa-ratio and
performance rating, we could use
the fx (or Formulas) function CORREL(array1, array2) (note
array means the same as range) to
give us the correlation.
However, if we have several variables we want to correlate at
the same time, it is more
effective to use the Correlation function found in the Analysis
ToolPak in the Data Analysis tab.
Set up of the input data for Correlation is simple. Just ensure

that all of the variables to be
correlated are listed together, and only include interval or ratio
level data. For our data set, this
would mean we cannot include gender or degree; even though
they look like numerical data the 0
and 1 are merely labels as far as correlation is concerned.
In the Correlation data input box, list the entire data range and
indicate if your data has
labels or not (good idea to include these), select the output cell
and click OK. Here is a screen
shot of the input box and some of the data.
Here is a screen shot of the output table and part of the data.
Reading the Table. The table only shows correlations below the
diagonal (which has a
1.00 correlation of each variable with itself). Values above the
line would simply duplicate those
below it. The diagonal is a “pivot-point,” so to speak. In
reading the correlations we would start
with a row such as Age. The correlation of Age and Compa-
ratio is 0.195 (rounded), the
correlation of Age and Midpoint is 0.567. Then we get to the
diagonal. Instead of continuing
horizontally, we start going vertical (down the Age column).
So, the correlation of Age with
Performance rating is 0.139, with Service is 0.565, and with
raise -0.180. All correlations,
except for the first one (in this case Compa-ratio) would be read
this way in Correlation tables.

Step 6: Conclusion and Interpretation.
Normally, we would go thru our questions about the p-value for
each value. But since you are
familiar with the testing logic, for this question we are going to
“shortcut” the process. The first
question asked is what is the T value that cuts off the two tails
of the distribution with an alpha of
0.05? We calculated this above as 2.011. The second question
asks for the associated correlation
value for this t-value. Again, we found this above to be 0.278.
Part c asks us to use this information and identify the variables
significantly correlated to salary
in the homework and to compa-ratio for this example. Looking
at the output table above, only
Midpoint is significantly correlated to compa-ratio with a
correlation of .50 (rounded) which is
greater than our cut-off of 0.278.
Part d asks for any surprising results/correlations. This will
depend upon your table and what
you did or did not expect.
Part e asks if this information helps us answer our equal pay
question. The compa-ratio
correlations do not seem to be helpful as they do not shed any
insight on gender based issues.
While question 1 does not appear to rely upon the hypothesis
testing process, by showing the

logic behind finding the significant correlation cut-off value, we
can see that we are being
faithful to the logic even if not the actual step-by-step process
while making our decisions on
correlation significance.
Multiple Correlation
As interesting as linear correlation is, multiple correlation is
even more so. It correlates
several independent (input) variables with a single dependent
(output) variable. For example, it
would show the shared variation (multiple R squared, or
Multiple Coefficient of Determination)
for compa-ratio with the other variables in the data set at the
same time rather than in pairs as we
did in question 1.
While we can generate this value by itself, it is a bit
complicated and is rarely found
except in conjunction with a multiple regression equation. So,
having noted that this exists, let’s
move on to multiple regression.
material.
couple of days before reading the
third lecture for the week.

BUS308 Week 4 Lecture 1
Our investigation changes focus a bit this week. We started the
class by finding ways to
describe and summarize data sets – finding measures of the
center and dispersion of the data with
means, medians, standard deviations, ranges, etc. As interesting
as these clues were, they did not
tell us all we needed to know to solve our question about equal
work for equal pay. In fact, the
evidence was somewhat contradictory depending upon what
measure we focused on. In Weeks 2
and 3, we changed our focus to asking questions about
differences and how important different
sample outcomes were. We found that all differences were not
important, and that for many
relatively small result differences we could safely ignore them
for decision making purposes –
they were due to simple sampling (or chance) errors. We found
that this idea of sampling error
could extend into work and individual performance outcomes
observed over time; and that over-
reacting to such differences did not make much sense.
Now, in our continuing efforts to detect and uncover what the
data is hiding from us, we
want to start to find out why something happened, what caused
the data to act as it did? This
week we move from examining differences to looking at
relationships; that is, if some measure
changes does another measure change as well? And, if so, can
we use this information to make
predictions and/or understand what underlies this common
movement?
Our tools in doing this involve correlation, the measurement of

how closely two
variables move together; and regression, an equation showing
the impact of inputs on a final
output. A regression is similar to a recipe for a cake or other
food dish; take a bit of this and
some of that, put them together, and we get our result.
Correlation
We have seen correlations a lot, and probably have even used
them (formally or
informally). We know, for example, that all other things being
equal; the more we eat. the more
we weigh. Kids, up to the early teens, grow taller the older they
get. If we consistently speed,
we will get more speeding tickets than those who obey the
speed limit. The more efforts we put
into studying, the better grades we get. All of these are
examples of correlations.
Correlations exist in many forms. A somewhat specialized
correlation was the Chi
Square contingency test (for multi-row, multi-column tables) we
looked at last week, if we find
the distributions differ, then we say that the variables are
related/correlated. This correlation
would run from 0 (no correlation) thru positive values (the
larger the value the stronger the
relationship).
Probably the most commonly used correlation is the Pearson
Correlation Coefficient,
symbolized by r. It measures the strength of the association –
the extent to which measures change
together – between interval or ratio level measures. Excel’s Fx
Correl, and the Data Analysis

Correlation both produce Pearson Correlations.
Most correlations that we are familiar with show both the
direction (direct or inverse) as
well as the strength of the relationship, and run from -1.0 (a
strong and perfect inverse
correlation) through 0 (a weak and non-existent correlation) to
+1.0 (a strong an perfect direct
correlation). A direct correlation is positive; that is, both
variables move in the same direction,
such as weight and height for kids. An inverse, or negative,
correlation has variables moving in
different directions. For example, the number of hours you sleep
and how tired you feel; the
more hours, the less tired while the fewer hours, the more tired.
The strength of a correlation is shown by the value (regardless
of the sign). For example,
a correlation of +.78 is just as strong as a correlation of -.78;
the only difference is the direction
of the change. If we graphed a +.78 correlation the data points
would run from the lower left to
the upper right and somewhat cluster around a line we could
draw thru the middle of the data
points. A graph of a -.78 correlation would have the data points
starting in the upper left and run
down to the lower right. They would also cluster around a line.
Correlations below an absolute value of around .70 are
generally not considered to be
very strong. The reason for this is due to the coefficient of
determination(CD). This equals the
square of the correlation and shows the amount of shared

variation between the two variables.
Shared variation can be roughly considered the reason that both
variables move as they do when
one changes. The more the shared variation, the more one
variable can be used to predict the
other. If we square .70 we get .49, or about 50% of the
variation being shared. Anything less is
too weak of a relationship to be of much help.
Students often feel that a correlation shows a “cause-and-effect”
relationship; that is,
changes in one thing “cause” changes in the other variable. In
some cases, this is true – height
and weight for pre-teens, weight and food consumption, etc. are
all examples of possible cause-
and- effect relationships; but we can argue that even with these
there are other variables that
might interfere with the outcomes. And, in research, we cannot
say that one thing causes or
explains another without having a strong correlation present.
However, just as our favorite detectives find what they think is
a cause for someone to
have committed the crime, only to find that the motive did not
actually cause that person to
commit the crime; a correlation does not prove cause-and-
effect. An example of this is the
example the author heard in a statistics class of a perfect +1.00
correlation found between the
barrels of rum imported into the New England region of the
United States between the years of
1790 and 1820 and the number of churches built each year. If
this correlation showed a cause-
and-effect relationship, what does it mean? Does rum drinking
(the assumed result of importing
rum) cause churches to be built? Does the building of churches

cause the population to drink
more rum?
As tempting as each of these explanations is, neither is
reasonable – there is no theory or
justification to assume either is true. This is a spurious
correlation – one caused by some other,
often unknown, factor. In this case, the culprit is population
growth. During these years – many
years before Carrie Nation’s crusade against Demon Rum – rum
was the common drink for
everyone. It was even served on the naval ships of most
nations. And, as the population grew,
so did the need for more rum. At the same time, churches in the
region could only hold so many
bodies (this was before mega-churches that held multiple
services each Sunday); so, as the
population got too large to fit into the existing churches, new
ones were needed.
At times, when a correlation makes no sense we can find an
underlying variable fairly
easily with some thought. At other times, it is harder to figure
out, and some experimentation is
needed. The site http://www.tylervigen.com/spurious-
correlations is an interesting website
devoted to spurious correlations, take a look and see if you can
explain them. ��
Regression
Even if the correlation is spurious, we can often use the data in
making predictions until
we understand what the correlation is really showing us. This is

what regression is all about.
Earlier correlations between age, height, and even weight were
mentioned. In pediatrician
offices, doctors will often have charts showing typical weights
and heights for children of
different ages. These are the results of regressions, equations
showing relationships. For
example (and these values are made up for this example), a
child’s height might be his/her initial
height at birth plus and average growth of 3.5 inches per year.
If the average height of a
newborn child is about 19 inches, then the linear regression
would be:
Height = 19 inches plus 3.5 inches * age in years, or in math
symbols:
Y = a + b*x, where y stands for height, a is the intercept or
initial value at age 0
(immediate birth), b is the rate of growth per year, and x is the
age in years.
In both cases, we would read and interpret it the same way: the
expected height of a child is 19
inches plus 3.5 inches times its age. For a 12-year old, this
would be 19 + 3.5*12 = 19 + 42 = 61
inches or 5 feet 1 inch (assuming the made-up numbers are
accurate).
That was an example of a linear regression having one output
and a single, independent
variable as an input. A multiple regression equation is quite
similar but has several independent
input variables. It could be considered to be similar to a recipe
for a cake:

Cake = cake mix + 2* eggs + 1½ * cup milk + ½ * teaspoon
vanilla + 2 tablespoons* butter.
A regression equation, either linear or multiple, shows us how
“much” each factor is used in or
influences the outcome. The math format of the multiple
regression equation is quite similar to
that of the linear regression, it just includes more variables:
Y = a + b1*X1 + b2*X2 + b3*X3 + …; where a is the intercept
value when all the inputs
are 0, the various b’s are the coefficients that are multiplied by
each variable value, and
the x’s are the values of each input.
A note on how to read the math symbols in the equations. The
Y is considered the output or
result, and is often called the dependent variable as its value
depends on the other factors. The
different b’s (b1, b2, etc.) are coefficients and read b-sub-1, b-
sub-2, etc. The subscripts 1, 2, etc.
are used to indicate the different coefficient values that are
related to each of the input variables.
The X-sub-1, X-sub-2, etc., are the different variables used to
influence the output, and are called
independent variables. In the recipe example, Y would be the
quality of the cake, a would be the
cake mix (a constant as we use all of what is in the box), the
other ingredients would relate to the
bX terms. The 2*eggs would relate to b1*X1, where b1
would equal 2 and X1 stands for eggs,
the second input relates to the milk, etc.
http://www.tylervigen.com/spurious-correlations

If you have any questions on this material, please ask your
instructor.
After finishing with this lecture, please go to the first
discussion for the week, and engage
in a discussion with others in the class over the first couple of
days before reading the second
lecture.
This lecture focuses on the Chi Square, how to set-up the data
tables and how to conduct
the Chi Square tests on distributions. All the chi square related
functions are found in the fx or
Formulas list. None of these are found in the Data Analysis tab.
The chi square test compares the actual or observed count
distribution across groups
(such as how many in each grade) against an expected
distribution. We will see that different
ways exist to define what this expected distribution is.
Chi Square Tests
With the Chi Square tests, we are going to move from looking at
population parameters,
such as means and standard deviations, and move to looking at
patterns or distributions.
Generally, when looking at distributions and patterns we will

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