This chapter discusses hypothesis testing and its key concepts. It defines a hypothesis as a statement about a population parameter that is tested. The five steps of hypothesis testing are outlined as defining the null and alternative hypotheses, selecting a significance level, choosing a test statistic, determining a decision rule, and making a conclusion. The differences between one-tailed and two-tailed tests are explained. Examples are provided for conducting hypothesis tests on population means and proportions. The chapter also defines Type I and Type II errors and how to compute the probability of a Type II error.
Hypothesis Testing: Central Tendency – Normal (Compare 1:1)Matt Hansen
An extension on a series about hypothesis testing, this lesson reviews the 2 Sample T & Paired T tests as central tendency measurements for normal distributions.
Chapter 6 part2-Introduction to Inference-Tests of Significance, Stating Hyp...nszakir
Mathematics, Statistics, Introduction to Inference, Tests of Significance, The Reasoning of Tests of Significance, Stating Hypotheses, Test Statistics, P-values, Statistical Significance, Test for a Population Mean, Two-Sided Significance Tests and Confidence Intervals
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 8: Hypothesis Testing
8.1: Basics of Hypothesis Testing
Hypothesis Testing: Central Tendency – Normal (Compare 2+ Factors)Matt Hansen
An extension on a series about hypothesis testing, this lesson reviews the ANOVA test as a central tendency measurement for normal distributions. It also explains what residuals and boxplots are and how to use them with the ANOVA test.
A note and graphical illustration of type II errorGH Yeoh
In hypothesis testing, lab analysts usually have difficulty to understand the meaning of Type II error and its significance. Here we are trying to make it clearer.
Hypothesis Testing: Central Tendency – Normal (Compare 1:1)Matt Hansen
An extension on a series about hypothesis testing, this lesson reviews the 2 Sample T & Paired T tests as central tendency measurements for normal distributions.
Chapter 6 part2-Introduction to Inference-Tests of Significance, Stating Hyp...nszakir
Mathematics, Statistics, Introduction to Inference, Tests of Significance, The Reasoning of Tests of Significance, Stating Hypotheses, Test Statistics, P-values, Statistical Significance, Test for a Population Mean, Two-Sided Significance Tests and Confidence Intervals
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 8: Hypothesis Testing
8.1: Basics of Hypothesis Testing
Hypothesis Testing: Central Tendency – Normal (Compare 2+ Factors)Matt Hansen
An extension on a series about hypothesis testing, this lesson reviews the ANOVA test as a central tendency measurement for normal distributions. It also explains what residuals and boxplots are and how to use them with the ANOVA test.
A note and graphical illustration of type II errorGH Yeoh
In hypothesis testing, lab analysts usually have difficulty to understand the meaning of Type II error and its significance. Here we are trying to make it clearer.
The following calendar-year information is taken from the December.docxcherry686017
The following calendar-year information is taken from the December 31, 2011, adjusted trial balance and other records of Azalea Company.
1. Each team member is to be responsible for computing one of the following amounts. You are not to duplicate your teammates' work. Get any necessary amounts from teammates. Each member is to explain the computation to the team in preparation for reporting to class.
a. Materials used.
b. Factory overhead.
c. Total manufacturing costs.
d. Total cost of goods in process.
e. Cost of goods manufactured.
2. Check your cost of goods manufactured with the instructor. If it is correct, proceed to part (3).
3. Each team member is to be responsible for computing one of the following amounts. You are not to duplicate your teammates' work. Get any necessary amounts from teammates. Each member is to explain the computation to the team in preparation for reporting to class.
a. Net sales.
b. Cost of goods sold.
c. Gross profit.
d. Total operating expenses.
e. Net income or loss before taxes.
CALCULATE T TEST
Calculate the “t” value for independent groups for the following data using the formula provided in the attached word document. Using the raw measurement data presented, determine whether or not there exists a statistically significant difference between the salaries of female and male human resource managers using the appropriate t-test. Develop a testable hypothesis, confidence level, and degrees of freedom. Report the required “t” critical values based on the degrees of freedom. Show calculations.
Answer
The null hypothesis tested is
H0: There is no significant difference between the average salaries of female and male human resource managers. (µ1= µ2)
The alternative hypothesis is
H1: There is significant difference between the average salaries of female and male human resource managers. (µ1≠ µ2)
The test statistic used is
12
12
2
~
NN
DM
MM
tt
S
+-
-
=
Where
22
1122
1212
(1)(1)
11
2
DM
NsNs
S
NNNN
éùéù
-+-
=+
êúêú
+-
ëûëû
Here M1 = 62,200, M2 = 63,700
s1 = 9330.95, s2 = 6912.95
N1 = 10, N2 = 10 (See the excel sheet)
Then,
(
)
(
)
22
(101)9330.95(101)6912.95
11
101021010
DM
S
éù
-+-
éù
=+
êú
êú
+-
ëû
êú
ëû
= 3672.267768
Therefore test statistic,
62,20063,700
3672.267768
t
-
=
= -0.408466946
Degrees of freedom = N1 + N2 – 2 = 10 + 10 – 2 = 18
Let the significance level be 0.05.
Rejection criteria: Reject the null hypothesis, if the calculated value of t is greater than the critical value of t at 0.05 significance level.
The critical values can be obtained from the student’s t tables with 18 d.f. at 0.05 significance level.
Upper critical value = 2.1
Lower critical value = -2.1
0
.
4
0
.
3
0
.
2
0
.
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0
.
0
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i
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-
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0
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.
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,
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8
Conclusion: Fails to reject the null hypothesis. The sample does not provide enough evidence to support the claim that there is significant difference ...
Course Project, Part IIntroduction(REMOVE THIS LINE PRIOR CruzIbarra161
Course Project, Part I
Introduction
(REMOVE THIS LINE PRIOR TO SUBMITTING REPORT: Summarize what you have learned about confidence intervals. Discuss why it would be important to know the population mean of the data used for this term. Is this an important health measure?)
Sample Data
(REMOVE THIS LINE PRIOR TO SUBMITTING REPORT: List ALL of the sample data in the table below.)
Directions:
1. Use the table above to create an 80%, 95%, and 99% confidence interval.
2. Choose another confidence level (besides 80%, 95% or 99%) to create another confidence interval.
3. Provide a sentence for each confidence interval created above which explains what the confidence interval means in context of topic of your project.
Computations
(Round all values to TWO decimal places)
(REMOVE THIS LINE PRIOR TO SUBMITTING REPORT: Calculate each of the following.)
Sample Mean =
Sample Standard Deviation =
80% Confidence Interval:
80% Confidence Interval Margin of Error:
Sentence:
95% Confidence Interval:
95% Confidence Interval Margin of Error:
Sentence:
99% Confidence Interval:
99% Confidence Interval Margin of Error:
Sentence:
______% Confidence Interval:
______% Confidence Interval Margin of Error:
Sentence
Problem Analysis
(REMOVE THESE LINES PRIOR TO SUBMITTING REPORT: Write a half-page reflection. What trend do you see takes place to the confidence interval as the confidence level rises? Explain mathematically why that takes place. Explain how Part I of the project has helped you understand confidence intervals better? How did this project help you understand statistics better?)
Course Project, Part II
Preliminary Calculations
Round Preliminary Values to the nearest whole number.
Summary Table for:
Live Births
Mean
Median
Sample Standard Deviation
Minimum
Maximum
Sample Size
Summary Table for:
Deaths
Mean
Median
Sample Standard Deviation
Minimum
Maximum
Sample Size
Summary Table for:
Divorces
Mean
Median
Sample Standard Deviation
Minimum
Maximum
Sample Size
Summary Table for:
Marriages
Mean
Median
Sample Standard Deviation
Minimum
Maximum
Sample Size
Hypothesis Testing
With the information that you gather from the Summary Tables above, test the following (you can use excel when appropriate):
Hypothesis Test #1:
Determine if there is sufficient evidence to conclude the average amount of births is over 5000 in the United States and territories at the 0.05 level of significance.
Step 1: Clearly state a null and alternative hypothesis, identify the claim, and the type of test.
Ho: μ ≥ ≤ =
Ha: μ < > ≠
Circle One: Left Tailed Test Right Tailed Test Two-Tailed Test
Step 2: Determine the Rejection Region
Pick ONE multiple choice answer below and fill in the critical value. Round Critical Value to two decimal places.
a) ...
Testing of hypothesis - large sample testParag Shah
Different type of test which are used for large sample has been included in this presentation. Steps for each test and a case study is included for concept clarity and practice.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
Francesca Gottschalk - How can education support child empowerment.pptxEduSkills OECD
Francesca Gottschalk from the OECD’s Centre for Educational Research and Innovation presents at the Ask an Expert Webinar: How can education support child empowerment?
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
Biological screening of herbal drugs: Introduction and Need for
Phyto-Pharmacological Screening, New Strategies for evaluating
Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
Antifertility, Toxicity studies as per OECD guidelines
Embracing GenAI - A Strategic ImperativePeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
2. 2
GOALS
Define a hypothesis and hypothesis testing.
Describe the five-step hypothesis-testing procedure.
Distinguish between a one-tailed and a two-tailed
test of hypothesis.
Conduct a test of hypothesis about a population
mean.
Conduct a test of hypothesis about a population
proportion.
Define Type I and Type II errors.
Compute the probability of a Type II error.
3. 3
What is a Hypothesis?
A Hypothesis is a statement about the
value of a population parameter
developed for the purpose of testing.
Examples of hypotheses made about a
population parameter are:
– The mean monthly income for systems analysts is
$3,625.
– Twenty percent of all customers at Bovine’s Chop
House return for another meal within a month.
4. 4
What is Hypothesis Testing?
Hypothesis testing is a procedure, based
on sample evidence and probability
theory, used to determine whether the
hypothesis is a reasonable statement
and should not be rejected, or is
unreasonable and should be rejected.
6. 6
Important Things to Remember about H0 and H1
H0: null hypothesis and H1: alternate hypothesis
H0 and H1 are mutually exclusive and collectively exhaustive
H0 is always presumed to be true
H1 has the burden of proof
A random sample (n) is used to “reject H0”
If we conclude 'do not reject H0', this does not necessarily
mean that the null hypothesis is true, it only suggests that there
is not sufficient evidence to reject H0; rejecting the null
hypothesis then, suggests that the alternative hypothesis may
be true.
Equality is always part of H0 (e.g. “=” , “≥” , “≤”).
“≠” “<” and “>” always part of H1
7. 7
How to Set Up a Claim as Hypothesis
In actual practice, the status quo is set up as H0
If the claim is “boastful” the claim is set up as H1
(we apply the Missouri rule – “show me”).
Remember, H1 has the burden of proof
In problem solving, look for key words and
convert them into symbols. Some key words
include: “improved, better than, as effective as,
different from, has changed, etc.”
8. 8
Left-tail or Right-tail Test?
Keywords
Inequality
Symbol
Part of:
Larger (or more) than > H1
Smaller (or less) < H1
No more than ≤ H0
At least ≥ H0
Has increased > H1
Is there difference? ≠ H1
Has not changed = H0
Has “improved”, “is better
than”. “is more effective”
> H1
• The direction of the test involving
claims that use the words “has
improved”, “is better than”, and the like
will depend upon the variable being
measured.
• For instance, if the variable involves
time for a certain medication to take
effect, the words “better” “improve” or
more effective” are translated as “<”
(less than, i.e. faster relief).
• On the other hand, if the variable
refers to a test score, then the words
“better” “improve” or more effective”
are translated as “>” (greater than, i.e.
higher test scores)
14. 14
Testing for a Population Mean with a
Known Population Standard Deviation- Example
Jamestown Steel Company
manufactures and assembles desks
and other office equipment at
several plants in western New York
State. The weekly production of the
Model A325 desk at the Fredonia
Plant follows the normal probability
distribution with a mean of 200 and
a standard deviation of 16.
Recently, because of market
expansion, new production
methods have been introduced and
new employees hired. The vice
president of manufacturing would
like to investigate whether there has
been a change in the weekly
production of the Model A325 desk.
15. 15
Testing for a Population Mean with a
Known Population Standard Deviation- Example
Step 1: State the null hypothesis and the alternate
hypothesis.
H0: µ = 200
H1: µ ≠ 200
(note: keyword in the problem “has changed”)
Step 2: Select the level of significance.
α = 0.01 as stated in the problem
Step 3: Select the test statistic.
Use Z-distribution since σ is known and population
being sampled is normally distributed.
16. 16
Testing for a Population Mean with a
Known Population Standard Deviation- Example
Step 4: Formulate the decision rule.
Reject H0 if |Z| > Zα/2
58.2notis55.1
50/16
2005.203
/
2/01.
2/
2/
>
>
−
>
−
>
Z
Z
n
X
ZZ
α
α
σ
µ
Step 5: Make a decision and interpret the result.
Because 1.55 does not fall in the rejection region, H0 is not
rejected. We conclude that the population mean is not different from
200. So we would report to the vice president of manufacturing that the
sample evidence does not show that the production rate at the Fredonia
Plant has changed from 200 per week.
17. 17
Suppose in the previous problem the vice
president wants to know whether there has
been an increase in the number of units
assembled. To put it another way, can we
conclude, because of the improved
production methods, that the mean number
of desks assembled in the last 50 weeks was
more than 200?
Recall: σ=16, n=50, α=.01 (error for n in slide)
Testing for a Population Mean with a Known
Population Standard Deviation- Another Example
18. 18
Testing for a Population Mean with a Known
Population Standard Deviation- Example
Step 1: State the null hypothesis and the alternate
hypothesis.
H0: µ ≤ 200
H1: µ > 200
(note: keyword in the problem “an increase”)
Step 2: Select the level of significance.
α = 0.01 as stated in the problem
Step 3: Select the test statistic.
Use Z-distribution since σ is known
19. 19
Testing for a Population Mean with a Known
Population Standard Deviation- Example
Step 4: Formulate the decision rule.
Reject H0 if Z > Zα
Step 5: Make a decision and interpret the result.
Because 1.55 does not fall in the rejection region, H0 is not rejected.
We conclude that the average number of desks assembled in the last
50 weeks is not more than 200
20. 20
Type of Errors in Hypothesis Testing
Type I Error -
– Defined as the probability of rejecting the null
hypothesis when it is actually true.
– This is denoted by the Greek letter “α”
– Also known as the significance level of a test
Type II Error:
– Defined as the probability of “accepting” the null
hypothesis when it is actually false.
– This is denoted by the Greek letter “β”
21. 21
p-Value in Hypothesis Testing
p-VALUE is the probability of observing a sample
value as extreme as, or more extreme than, the
value observed, given that the null hypothesis is true.
In testing a hypothesis, we can also compare the p-
value to with the significance level (α).
If the p-value < significance level, H0 is rejected, else
H0 is not rejected.
22. 22
p-Value in Hypothesis Testing - Example
Recall the last problem where the
hypothesis and decision rules
were set up as:
H0: µ ≤ 200
H1: µ > 200
Reject H0 if Z > Zα
where Z = 1.55 and Zα =2.33
Reject H0 if p-value < α
0.0606 is not < 0.01
Conclude: Fail to reject H0
23. 23
What does it mean when p-value < α?
(a) .10, we have some evidence that H0 is not true.
(b) .05, we have strong evidence that H0 is not true.
(c) .01, we have very strong evidence that H0 is not true.
(d) .001, we have extremely strong evidence that H0 is not
true.
24. 24
Testing for the Population Mean: Population
Standard Deviation Unknown
When the population standard deviation (σ) is
unknown, the sample standard deviation (s) is used in
its place
The t-distribution is used as test statistic, which is
computed using the formula:
25. 25
Testing for the Population Mean: Population
Standard Deviation Unknown - Example
The McFarland Insurance Company Claims Department reports the mean
cost to process a claim is $60. An industry comparison showed this
amount to be larger than most other insurance companies, so the
company instituted cost-cutting measures. To evaluate the effect of
the cost-cutting measures, the Supervisor of the Claims Department
selected a random sample of 26 claims processed last month. The
sample information is reported below.
At the .01 significance level is it reasonable a claim is now less than $60?
26. 26
Testing for a Population Mean with a
Known Population Standard Deviation- Example
Step 1: State the null hypothesis and the alternate
hypothesis.
H0: µ ≥ $60
H1: µ < $60
(note: keyword in the problem “now less than”)
Step 2: Select the level of significance.
α = 0.01 as stated in the problem
Step 3: Select the test statistic.
Use t-distribution since σ is unknown
28. 28
Testing for the Population Mean: Population
Standard Deviation Unknown – Minitab Solution
29. 29
Testing for a Population Mean with a
Known Population Standard Deviation- Example
Step 5: Make a decision and interpret the result.
Because -1.818 does not fall in the rejection region, H0 is not rejected at
the .01 significance level. We have not demonstrated that the cost-cutting
measures reduced the mean cost per claim to less than $60. The difference
of $3.58 ($56.42 - $60) between the sample mean and the population mean
could be due to sampling error.
Step 4: Formulate the decision rule.
Reject H0 if t < -tα,n-1
30. 30
The current rate for producing 5 amp fuses at Neary
Electric Co. is 250 per hour. A new machine has
been purchased and installed that, according to the
supplier, will increase the production rate. A sample
of 10 randomly selected hours from last month
revealed the mean hourly production on the new
machine was 256 units, with a sample standard
deviation of 6 per hour.
At the .05 significance level can Neary conclude that
the new machine is faster?
Testing for a Population Mean with an Unknown
Population Standard Deviation- Example
31. 31
Testing for a Population Mean with a
Known Population Standard Deviation- Example continued
Step 1: State the null and the alternate hypothesis.
H0: µ ≤ 250; H1: µ > 250
Step 2: Select the level of significance.
It is .05.
Step 3: Find a test statistic. Use the t distribution
because the population standard deviation is not
known and the sample size is less than 30.
32. 32
Testing for a Population Mean with a
Known Population Standard Deviation- Example continued
Step 4: State the decision rule.
There are 10 – 1 = 9 degrees of freedom. The null
hypothesis is rejected if t > 1.833.
Step 5: Make a decision and interpret the results.
The null hypothesis is rejected. The mean number produced is
more than 250 per hour.
162.3
106
250256
=
−
=
−
=
ns
X
t
µ
33. 33
Tests Concerning A Proportion
A Proportion is the fraction or percentage that indicates the part of
the population or sample having a particular trait of interest.
The sample proportion is denoted by p and is estimated by x/n
The test statistic is computed as follows:
34. 34
Assumptions in Testing a Population Proportion
using the z-Distribution
A random sample is chosen from the population.
It is assumed that the binomial assumptions discussed in
Chapter 6 are met:
(1) the sample data collected are the result of counts;
(2) the outcome of an experiment is classified into one of two
mutually exclusive categories—a “success” or a “failure”;
(3) the probability of a success is the same for each trial; and
(4) the trials are independent
The test we will conduct shortly is appropriate when both nπ
and n(1- π ) are at least 5.
When the above conditions are met, the normal distribution can
be used as an approximation to the binomial distribution
35. 35
Test Statistic for Testing a Single
Population Proportion
n
p
z
)1( ππ
π
−
−
=
Sample proportion
Hypothesized
population proportion
Sample size
36. 36
Test Statistic for Testing a Single
Population Proportion - Example
Suppose prior elections in a certain state indicated
it is necessary for a candidate for governor to
receive at least 80 percent of the vote in the
northern section of the state to be elected. The
incumbent governor is interested in assessing
his chances of returning to office and plans to
conduct a survey of 2,000 registered voters in
the northern section of the state. Using the
hypothesis-testing procedure, assess the
governor’s chances of reelection.
37. 37
Test Statistic for Testing a Single
Population Proportion - Example
Step 1: State the null hypothesis and the alternate
hypothesis.
H0: π ≥ .80
H1: π < .80
(note: keyword in the problem “at least”)
Step 2: Select the level of significance.
α = 0.01 as stated in the problem
Step 3: Select the test statistic.
Use Z-distribution since the assumptions are met
and nπ and n(1-π) ≥ 5
38. 38
Testing for a Population Proportion - Example
Step 5: Make a decision and interpret the result.
The computed value of z (2.80) is in the rejection region, so the null hypothesis is rejected
at the .05 level. The difference of 2.5 percentage points between the sample percent (77.5
percent) and the hypothesized population percent (80) is statistically significant. The
evidence at this point does not support the claim that the incumbent governor will return to
the governor’s mansion for another four years.
Step 4: Formulate the decision rule.
Reject H0 if Z <-Zα
39. 39
Type II Error
Recall Type I Error, the level of significance,
denoted by the Greek letter “α”, is defined as
the probability of rejecting the null hypothesis
when it is actually true.
Type II Error, denoted by the Greek letter “β”,is
defined as the probability of “accepting” the null
hypothesis when it is actually false.
40. 40
Type II Error - Example
A manufacturer purchases steel bars to make cotter
pins. Past experience indicates that the mean tensile
strength of all incoming shipments is 10,000 psi and
that the standard deviation, σ, is 400 psi. In order to
make a decision about incoming shipments of steel
bars, the manufacturer set up this rule for the quality-
control inspector to follow: “Take a sample of 100
steel bars. At the .05 significance level if the sample
mean strength falls between 9,922 psi and 10,078
psi, accept the lot. Otherwise the lot is to be
rejected.”