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Section 7
Analyzing our Marketing Test, Survey Results
and Other Metrics Using Confidence Intervals
Rhonda Knehans Drake
Associate Professor, New York University
Data Analytics, Interpretation and Reporting
copyright © 2013
2
• When we estimate population averages or percentages based
on
samples, a certain amount of error is present.
• The amount of error present is a function mostly of your
sample size.
• The larger the sample, the less error in your estimates.
• Today we will learn how to place bounds around our estimates
obtained.
• With such an interval, we will then be able to say with 90%,
95%
or 99% confidence that the true population estimate will lie in
these bounds.
Introduction
3
The eight confidence interval formulas we will discuss are for
the following situations:
1. Average or mean based on large samples
2. Average or mean based on small samples
3. Response rate or survey percent based on large samples
4. Response rate or survey percent based on small samples
5. Difference between 2 averages for large samples
6. Difference between 2 averages for small samples
7. Difference between 2 response rates for large samples
8. Difference between 2 response rates for small sample
Confidence Intervals
A-B Split Tests
4
1. Confidence Interval for Averages or Means Based on
Large Samples (n ≥ 30)
To calculate a confidence interval around a mean, the following
information is required:
– The sample mean x obtained from the test.
– The sample standard deviation S obtained from the test.
Many software packages, including Microsoft ExcelTM, can
automatically calculate this value for you. (Review Section 3
for the
standard deviation formula.)
– The sample size n of the test.
This is the number of observations used to calculate your mean.
The
sample size must be greater than or equal to 30 in size.
– The desired confidence level: 90%, 95% or 99%.
A confidence interval constructed around the mean will
guarantee, with
your specified level of confidence, the true mean will fall
within those
bounds.
Sample Means
(Large Sample)
5
• Once all information is known, you construct the confidence
interval around the mean by adding and subtracting from the
mean a multiple of your standard deviation associated with the
sample mean. The “multiple” depends on your desired
confidence level.
• The formula for a confidence interval around the mean is
calculated as follows:
Where:
• S is the standard deviation associated with the sample.
• n is the sample size.
• Z is equal to 1.645, 1.96 or 2.575 for a 90%, 95% or 99%
confidence level
Sample Means
(Large Sample)
6
Example:
Money magazine conducts a survey of 100 retirees across the
US
and asks them how much they have in their retirement fund.
You obtain an average of $84.75 with a standard deviation of
$18.75, both in thousands of dollars.
You are about to write an article based on this average but
realize
that the true average is something more or less than this in
reality.
Construct a 95% confidence interval around this average.
Sample Means
(Large Sample)
7
Sample Means
(Large Sample)
84.75 1.96 (18.75 / √100)
84.75 1.96 (1.875)
84.75 3.675
($81,075 , $88,425)
1.
2.
3.
4.
8
Later we will discuss how to chose the confidence level and
address if 95% was the appropriate level for this example.
Sample Means
(Large Sample)
9
• Let’s do the previous
example again but using
the Plan-alyzer.
1.
2.
3.
Sample Means
(Large Sample)
Select the tab “Table of
Calculators”
Select “Confidence
Interval Calculators for
Averages, Large Samples”
Select “One Sample”
10
Sample Means
(Large Sample)
Input the required info.
11
Sample Means
(Large Sample)
See the answer.
12
2) Confidence Interval for Averages or Means Based on
Small Samples (n < 30)
To calculate a confidence interval around a mean, the following
information is required:
– The sample mean x obtained from the test.
– The sample standard deviation S obtained from the test.
Many software packages, including Microsoft ExcelTM, can
automatically calculate this value for you. (Review Section 3
for the
standard deviation formula.)
– The sample size n of the test.
This is the number of observations used to calculate your mean.
The
sample size must be greater than or equal to 30 in size.
– The desired confidence level: 90%, 95% or 99%.
A confidence interval constructed around the mean will
guarantee,
with your specified level of confidence, the true mean will fall
within
those bounds.
Sample Means
(Small Sample)
13
• Once all information is known, you construct the confidence
interval around the mean by adding and subtracting from the
mean a multiple of your standard deviation associated with the
sample mean. The “multiple” depends on your desired
confidence level.
• The formula for a confidence interval around the mean is
the same as the prior formula except we use a value from the
“t-distribution” which is for approximate normally distributed
data:
Where:
• S is the standard deviation associated with the sample.
• n is the sample size.
• t is obtained by using the excel function TINV as will be seen
shortly.
Sample Means
(Small Sample)
14
Example:
Suppose Money Magazine only conducted the survey to a
sample
of 10 retirees instead of 100 as in our prior example, all else the
same.
Construct a 95% confidence interval around this average.
Sample Means
(Small Sample)
15
We construct the confidence interval as before except we will
use
The t-distribution.
84.75 ± (t) (18.75/√10 )
Where the value of t = TINV(.05,9)
= 2.262
84.75 ± (2.262) (18.75/3.1623)
84.75 ± (2.262) (5.93)
84.75 ± 13.41
($71.34, $98.16)
(Small Sample)
(1-conf
level)
(n-1)
Sample Means
16
Note our confidence interval is wider for two reasons:
1. The smaller sample size
1. Our sample is less than 30 we cannot assume it is normal but
only approximately normal so our multiplier is larger (2.262
versus 1.96).
Sample Means
(Small Sample)
17
• Let’s do the previous
example again but using
the Plan-alyzer.
1.
2.
3.
Sample Means
(Small Sample)
Select the tab “Table of
Calculators”
Select “Confidence
Interval Calculators for
Averages, Small Samples”
Select “One Sample”
18
Sample Means
(Small Sample)
Input the required info.
19
Sample Means
(Small Sample)
See the answer.
20
3. Confidence Intervals for Response Rates or Survey
Percentages Based on Large Samples (where n*p and
n*(1 - p) are both ≥ 5)
To calculate a confidence interval around a sample proportion,
the
following information is required.
– The sample proportion p obtained from the test.
– The sample size n of the test.
This is the number of observations used to calculate your
proportion.
The sample size, when multiplied by the sample proportion and
when
multiplied by one minus the sample proportion, must both be
greater
than or equal to 5.
– The desired confidence level: 90%, 95% or 99%.
A confidence interval constructed around the sample proportion
will
guarantee, with your specified level of confidence, the true
population
proportion will fall within those bounds.
Sample Proportions
(Large Sample)
21
• Once all information is known, you construct the confidence
interval around the sample proportion by adding and subtracting
from the sample proportion a multiple of the standard deviation
associated with the sample proportion. The “multiple” depends
on your desired confidence level.
• The formula for a confidence interval around the sample
proportion is calculated as follows:
• Where n is your sample size and Z is 1.645, 1.96, 2.575 for a
90%, 95% or 99% confidence interval
This is the standard deviation for a
proportion or response rate
Sample Proportions
(Large Sample)
22
Example:
AT&T samples a new prospect list and sends them an offer to
order their new wireless cellular service.
They sample 10,000 prospects and receive a 0.89% response
rate.
What is the margin of error at 90% confidence?
Sample Proportions
(Large Sample)
23
• So, for our example we have the confidence interval is:
= .0089 ± (1.645)·√ (.0089)(1- .0089 )/10,000
= .0089 ± (1.645)·√ (.0089)(.9911 )/10,000
= .0089 ± (1.645)·√ 10000008
= .0089 ± (1.645)·(.0008944)
= .0089 ± .0015
(.0074, .0104) or (.74%, 1.04%)
Sample Proportions
(Large Sample)
0.0089 1.645*(√[(0.0089)*(0.9911)/10,000]
0.0089 1.645*(√0.0000008)
0.0089 1.645*(0.0008944)
0.0089 .0015
(0.0074 , 0.0104) or (0.74% , 1.04%)
1.
2.
3.
4.
5.
24
• Let’s do the previous
example again but using
the Plan-alyzer.
1.
2.
3.
Sample Proportions
(Large Sample)
Select the tab “Table of
Calculators”
Select “Confidence
Interval Calculators for
Percentages, Large
Samples”
Select “One Sample”
25
Sample Proportion
(Large Sample)
Input the required info.
26
Sample Proportion
(Large Sample)
See the answer.
27
4. Confidence Intervals for Response Rates or Survey
Percentages Based on Small Samples (where either n*p or
n*(1 - p) are < 5)
To calculate a confidence interval around a sample proportion,
the
following information is required:
– The sample proportion p obtained from the test.
– The sample size n of the test.
This is the number of observations used to calculate your
proportion.
The sample size, when multiplied by the sample proportion and
when
multiplied by one minus the sample proportion, must both be
greater
than or equal to 5.
– The desired confidence level: 90%, 95% or 99%.
A confidence interval constructed around the sample proportion
will
guarantee, with your specified level of confidence, the true
population
proportion will fall within those bounds.
Sample Proportions
(Small Sample)
28
• Once all information is known, you construct the confidence
interval around the sample proportion by adding and subtracting
from the sample proportion a multiple of the standard deviation
associated with the sample proportion. The “multiple” depends
on your desired confidence level.
• The formula for a confidence interval around the sample
proportion is the same as the prior formula except we use a
value from the “t-distribution” which is for approximate
normally
distributed data:
• Where n is your sample size and t is obtained by using the
Excel function TINV as will be seen shortly.
This is the standard deviation for a
proportion or response rate
Sample Proportions
(Small Sample)
29
Example:
Suppose AT&T only sampled 100 prospects instead of 10,000 as
in our previous example, all else the same.
What is the margin of error at 90% confidence?
Sample Proportions
(Small Sample)
30
We construct the confidence interval as before except we will
use
the t-distribution.
- .0089 )/100
Where the value of t = TINV(.10,99)
= 1.66
= .0089 ± (1.66)·√(.0089)(.9911 )/100
= .0089 ± (1.66)·√ .0000882
= .0089 ± (1.66)·(.0093914)
= (0.00, 0.0245) or (0.00%, 2.45%)
The lower bound here cannot be negative, so we change it to
zero.
Sample Proportions
(Small Sample)
(1-conf
level)
(n-1)
31
• Let’s do the previous
example again but using
the Plan-alyzer.
1.
2.
3.
Sample Proportions
(Small Sample)
Select the tab “Table of
Calculators”
Select “Confidence
Interval Calculators for
Percentages, Small
Samples”
Select “One Sample”
32
Sample Proportions
(Small Sample)
Input the required info.
33
See the answer.
Sample Proportions
(Small Sample)
34
5. Confidence Interval for the Difference between 2 Means or
Averages for Large Samples (n1 ≥ 30 and n2 ≥ 30 )
To calculate a confidence interval around the difference
between
two sample means, the following information is required:
– The means of both samples (x1 and x2).
– The standard deviation of both samples (S1 and S2).
Many software packages, including Microsoft ExcelTM, can
automatically calculate these values.
– The size of both samples (n1 and n2).
These are the number of observations that went into calculating
each
of your means. Both sample sizes n1 and n2 must be greater
than or
equal to 30 in size.
– The desired confidence level: 90%, 95% or 99%.
A confidence interval constructed around the sample proportion
will
guarantee, with your specified level of confidence, the true
population
proportion will fall within those bounds.
•
Confidence Interval Estimation (A-B Split Testing)
(Difference Between Two Samples Means – Large Samples)
35
• Once all information is known, you construct the confidence
interval around the difference between two mean by adding and
subtracting from the difference in means a multiple of the
standard deviation associated with the difference. The
“multiple” depends on your desired confidence level.
• The formula for a confidence interval around the difference
between means is calculated as follows:
• Where Z is 1.645, 1.96, 2.575 for a 90%, 95% or 99%
confidence interval.
This is the standard deviation for the
difference in averages.
Confidence Interval Estimation (A-B Split Testing)
(Difference Between Two Sample Means – Large Samples)
(X1 – X2) ± (Z)(√(S1
2 /n1) + (S2
2 /n2))
36
Example:
You sample 100 home sales in San Francisco and 100 home
sales
in NYC for 2010 with the following results:
NSF = 100 XSF = $745.25 SSF = $40
NNYC = 100 XNYC = $775.10 SNYC = $45
Is there any difference in home prices between NYC and San
Francisco? Base your answer on the 95% confidence level.
Confidence Interval Estimation (A-B Split Testing)
(Difference Between Two Sample Means – Large Samples)
37
• Assume SSF = $40 and SNYC = $45
(775.10 – 745.25) ± (1.96)·√ [(402/100)+(452/100)]
29.85 ± (1.96)·√ 16 + 20.25
29.85 ± (1.96)·√ 36.25
29.85 ± (1.96)·(6.021)
29.85 ± 11.80
($18,050, $41,650)
Confidence Interval Estimation (A-B Split Testing)
(Difference Between Two Sample Means – Large Samples)
1.
2.
3.
4.
5.
(775.10-745.25) 1.96*(√[(402/100)+(452/100 )])
29.85 1.96*(√(16+20.25))
29.85 1.96*(6.021)
29.85 11.80
($18.050 , $41.650)
38
How do we interpret?
• What if the interval was -$18,050 to $41,540. How would you
interpret and is it okay in this case to have a negative value?
Confidence Interval Estimation (A-B Split Testing)
(Difference Between Two Sample Means – Large Samples)
– With 95% confidence, we can say that NYC home prices in
2010 are higher than SF home prices by anywhere from
$18,050 to $41,540.
– If Zero were in the interval then you would say no
difference between the two (a hypothesis test!!).
– It means there's no statistical evidence to conclude that
NYC prices are different from those of SF.
– Its ok to have a negative value.
39
• Let’s do the previous
example again but using
the Plan-alyzer.
1.
2.
3.
Select the tab “Table of
Calculators”
Select “Confidence
Interval Calculators for
Averages, Large Samples”
Select “Test vs. Control”
Confidence Interval Estimation (A-B Split Testing)
(Difference Between Two Sample Means – Large Samples)
40
Input the required info.
Confidence Interval Estimation (A-B Split Testing)
(Difference Between Two Sample Means – Large Samples)
41
See the answer.
Confidence Interval Estimation (A-B Split Testing)
(Difference Between Two Sample Means – Large Samples)
42
6. Confidence Interval for the Difference between 2 Means or
Averages for Small Samples (n1 < 30 or n2 < 30).
– If one or both samples is less than 30 in size then you will
replace the Z value with the t value.
– You will again use the TINV function in excel.
– Your parameters are TINV(1- confidence level, n1+n2 - 2).
– All else the same.
– This will be used for small market research problems.
Confidence Interval Estimation (A-B Split Testing)
(Difference Between Two Sample Means – Small Samples)
43
7. Confidence Intervals for the Difference Between 2
Percentages for Large Samples (n1*p1, n1*(1-p1), n2*p2,
n2*(1-p2), all ≥ 5)
To calculate a confidence interval around the difference
between
two sample proportions, the following information is required:
– The proportions (p1 and p2) for both samples.
– The size of both samples (n1 and n2).
These are the number of observations used in calculating each
of the
sample proportions. Both sample sizes, when multiplied by
their
respective sample proportions and when multiplied by one
minus their
respective sample proportions, must all be greater than or equal
to 5.
– The desired confidence level: 90%, 95% or 99%.
A confidence interval constructed around the sample proportion
will
guarantee, with your specified level of confidence, the true
population
proportion will fall within those bounds.
Confidence Interval Estimation (A-B Split Testing)
(Difference Between Two Sample Proportions – Large Samples)
44
• Once all information is known, you construct the confidence
interval around the difference between two proportions by
adding
and subtracting from the difference in proportions a multiple of
the
standard deviation associated with the difference. The
“multiple”
depends on your desired confidence level.
• The formula for a confidence interval around the difference
between proportions is calculated as follows:
• Where Z is 1.645, 1.96, 2.575 for a 90%, 95% or 99%
confidence
interval.
This is the standard deviation for the
difference in proportions.
Confidence Interval Estimation (A-B Split Testing)
(Difference Between Two Sample Proportions – Large Samples)
45
Example:
You are in charge of new card acquisitions at American
Express.
You conduct a new offer test for the green card versus your
control offer with the following results
Did the test beat the control with 95% confidence? Do you have
a
winner?
Sample Size Response Rate
Control Offer with 10,000 Bonus Miles 10,000 1.10%
Test Offer with 25,000 Bonus Miles 10,000 1.38%
Confidence Interval Estimation (A-B Split Testing)
(Difference Between Two Sample Proportions – Large Samples)
1.
2.
3.
4.
46
Confidence Interval Estimation (A-B Split Testing)
(Difference Between Two Sample Proportions – Large Samples)
0.00297
0.000001 .0028
.0028
47
So how do we interpret?
• With 95% confidence the test can do worse than the control by
-.017%
OR do better than the control by .577%.
• As such we say the test and the control are not significantly
different
since the confidence interval wrapped around the difference in
response rates contains zero.
• Had the lower bound been above zero then we would say the
test has
beaten the control.
• But let’s be real here. For all purposes, the test is a winner.
The lower
bound is soooo close to zero. So worst case the test is the same
as
the control with much upside potential.
Confidence Interval Estimation (A-B Split Testing)
(Difference Between Two Sample Proportions – Large Samples)
48
So how do we interpret (continuation)?
• But remember just because the test beaten the control from a
statistical
point of view, that does not mean that it won from a marketing
point of
view.
• In this example we were giving away additional sky miles. So
the test
will need to beat the control by some minimum most likely
greater than
zero or else we will not generate the same revenue.
• Suppose I told you that based on the cost of the additional sky
miles
the test needs to beat the control by at least .025% to break-
even.
Would you consider it a winner?
• What if I told you the test needs to beat the control by .25% to
break
even. Would you now consider the test a winner?
Confidence Interval Estimation (A-B Split Testing)
(Difference Between Two Sample Proportions – Large Samples)
49
• Let’s do the previous
example again but using
the Plan-alyzer.
1.
2.
3.
Select the tab “Table of
Calculators”
Select “Confidence
Interval Calculators for
Proportions, Large
Samples”
Select “Test vs. Control”
Confidence Interval Estimation (A-B Split Testing)
(Difference Between Two Sample Proportions – Large Samples)
50
Input the required info.
Confidence Interval Estimation (A-B Split Testing)
(Difference Between Two Sample Proportions – Large Samples)
51
See the answer.
Confidence Interval Estimation (A-B Split Testing)
(Difference Between Two Sample Proportions – Large Samples)
52
8. Confidence Intervals for the Difference Between 2
Percentages for Small Samples (where n1*p1 or n1*(1-p1)
or n2*p2 or n2*(1-p2) < 5)
• You will replace the Z value with the t value.
• You will again use the TINV function in excel.
• Your parameters are TINV(1- confidence level, n1+n2 - 2).
• All else the same.
• This will be used for small market research problems.
Confidence Interval Estimation (A-B Split Testing)
(Difference Between Two Sample Proportions – Small Samples)
53
• Remember, interpretation of confidence intervals is not that
simple.
• They will not tell you what to do.
• They simply give you valid best and worst case scenarios to
take into consideration.
• They give you additional information upon which to help you
base your marketing decisions:
• Worst case, are the results meeting your criteria?
• How does the upside potential compare to the downside
potential?
Interpretation of Confidence Intervals
54
• No direct marketer should ever consider evaluating their test
results with a confidence level lower than 90%. To do so
assumes
way to much risk.
• And, fishing for a confidence level that yields significance
should
never be practiced.
• The rules that any good direct marketer should follow
regarding
significance are shown on the next slide.
Setting the Confidence Level
55
Evaluate your test response
rate at 95% confidence level
Significant?
Yes No
Is it significant at the 99%
confidence level?
Not that we want to go that
low but is it significant at the
90% confidence level?
Yes No Yes No
A no brainer.
Let’s roll!
That’s okay…at
a minimum let’s
consider a
partial to full
roll out.
Okay, so we
have something
here. Let’s
either retest or
go for a partial
rollout.
Not good. We
should scrape
this from further
consideration.
Setting the Confidence Level
56
It is important to keep in mind the following facts regarding the
creation of a confidence interval.
• If you want more confidence in your estimates, the resulting
interval
will widen.
• If you increase your sample size, the resulting interval will
become
tighter.
• The more accuracy you need in your test estimate, the higher
you
should set your confidence level.
• The confidence level you set should depend on the risk you
are
willing to take in making an incorrect decision.
Setting the Confidence Level
57
• If our sample represents a large percent (> 10%) of the
population in total,
then we typically apply a correction factor to our margin of
error estimates.
• The larger our sample is as a percent of the total population
the more valid
our estimates.
• Of the following two samples, which would you think would
yield a better
parameter estimate?
• A sample of size 5,000 from a population of size 10,000 in
total
• A sample of size 5,000 from a population of size 10,000,000
in total.
• If our sample represents 10% or more of the total population,
you
multiple the margin of error for the first four formulas by:
Finite Population Correction Factor
58
Example:
Going back to our first exercise, suppose the survey of 100
people
was conduct to the 800 residents of the Happy Retirement
Village and
not Money Magazine subscribers.
What is our correction factor and what is the new interval?
Finite Population Correction Factor
3.
1.
2.
3.4398
59
7.1 Briefly explain how the width of a confidence interval
decreases
with an increase in the sample size. Give me an example.
7.2 Briefly explain how the width of a confidence interval
decreases
with a decrease in the confidence level. Give me an example.
7.3 According to a study done by Dr. Martha S. Linet and
others, the mean
duration of the recent headache was 8.2 hours for a sample of
5055
females aged 12 through 29. Assume that this sample
represents the
current population of all headaches for all females aged 12
through 29
and that the standard deviation for this sample is 2.4 hours.
Make a 95%
confidence interval for the mean duration of all headaches for
all 12-to-29-
year-old females. Do by hand and using The Plan-alyzer.
7.4 A sample of 12 observations was drawn from a population
of size 100.
Calculate a 95% confidence interval around the average for this
sample
by hand. HINT: You will want to use the finite population
correction factor for this
problem as found on slides 57 and 58.
13 15 9 11 8 16 14 9 10 14
16 12
Section 7 Exercises
60
7.5 A company wants to estimate the mean net weight of its
“Big Top Circus”
cereal boxes. A sample of 16 such boxes produced the mean net
weight of
31.98 ounces with a standard deviation of .26 ounces. Make a
95%
confidence interval for the mean net weight of all boxes. Do by
hand and
using The Plan-alyzer.
7.6 Crate and Barrel Cataloger promises its customers that the
products
ordered will be mailed within 72 hours after an order is placed.
The quality
control department at the company checks from time to time to
see if this
promise is fulfilled. Recently, the quality control department
took a sample
of 50 orders and found that 42 of them were mailed within 72
hours of the
placement of the orders.
a) Construct a 95% confidence interval for the percentage of all
orders
that are mailed within 72 hours of their placement. Do by hand
and
using The Plan-alyzer.
b) Suppose the confidence interval obtained in part a is too
wide. How
can the width of this interval be reduced? Discuss all possible
alternatives. Which of these alternatives is the best?
Section 7 Exercises
61
7.7 In virtual reality a person views a computer-generated scene
that changes
as if the viewer’s body were in motion. Some individual
experience
unpleasant side effects from virtual reality, such as nausea,
dizziness, or
disorientation. In a recent study by Clare Tegan of Britain’s
Defense
Research Agency, each of the 150 people included in the study
spent 20
minutes wearing a head-mounted virtual reality system through
which he
or she explored a virtual environment consisting of a series of
rooms.
Either during their time in the virtual environment or in the 10
minutes
immediately afterward, 61% of these 150 persons suffered side
effects.
Find the 95% confidence interval for the proportion of all
virtual reality
users who would suffer side effects. Do by hand and using The
Plan-
alyzer.
Section 7 Exercises
62
7.8 One of the major problems faced by department stores is a
high
percentage of returns. The manager of Macy’s wanted to
estimate the
percentage of all sales that result in returns. A sample of 40
sales showed
that 8 of them had products returned within the time allowed for
returns.
a) Construct a 99% confidence interval for the percentage of all
sales
that result in returns. Do by hand and using The Plan-alyzer.
b) Do you think 99% confidence is appropriate in this case and
if not
what would be a more appropriate level of confidence to use?
7.9 According to a survey, the mean price of gasoline in the
U.S. was
$1.20 per gallon in 1995 and $1.10 per gallon in 1994 (Wow,
don’t you
wish!) Suppose these means were based on random samples of
100
gas station for 1995 and 120 gas station for 1994. Also, assume
that
the sample standard deviations were $.11 for 1995 and $.10 for
1994.
Find a 90% confidence interval for the difference between the
mean
gasoline prices for 1995 and 1994. Do by hand and using The
Plan-
alyzer.
Section 7 Exercises
63
7.10 An insurance company wants to know if the average speed
at which men
drive cars is higher than that of women drivers. The company
took a
random sample of 27 cars driven by men on a highway and
found the
mean speed to be 68 miles per hour with a standard deviation of
2.2 miles.
Another sample of 18 cars driven by women on the same
highway gave a
mean speed of 65 miles per hour with a standard deviation of
2.5 miles.
Assume that the speeds at which all men and all women drive
cars on this
highway are both known to be normally distributed. Construct a
99%
confidence interval for the difference between the mean speeds
of cars
driven by all men and all women drivers on this highway. Do
by hand and
using The Plan-alyer.
Section 7 Exercises
64
7.11 Removed.
7.12 In a Prevention magazine survey released in 2008,
Princeton Survey
Research Association examined the weight of children aged 3 to
17.
According to this study, 24% of children in this age group were
overweight in 2000, and 31% were considered overweight in
2008.
Suppose that these percentages are based on random samples of
400
and 500 children in the given age group in 2000 and 2008,
respectively.
Conduct a 95% confidence interval for the difference between
the
portions of the overweight 3-to-17-year-olds in 2000 and 2008.
Do by
hand and using The Plan-alyzer.
Section 7 Exercises
Number of sources: 7
Topic: Should all states require motorcyclist and passengers to
wear helmets?
Type of document: Research Paper: THIS IS A RESEARCH
PAPER!!
Academic Level:Undergraduate
Number of Pages: 9 (Double Spaced)
Category: English
Language Style: English (U.S.)
Writing Style: APA
Order Instructions:
Should all states require motorcyclist and passengers to wear
helmets?
• Using your thesis statement and research, present the problem
that needs to be addressed with your proposed solution. Note:
Your solution, advantages, and challenges, will be in Parts 2
and 3.
Write a three to four (3-4) page paper in which you: 3 PAGES
NEEDED
1. Provide an appropriate title and an interesting opening
paragraph to appeal to your stated audience (appeal with logic,
ethics, or emotion).
2. Include a defensible, relevant thesis statement in the first
paragraph. (Revised from Assignment 2; Check below)
3. Describe the history and status of the issue and provide an
overview of the problem(s) that need to be addressed. This
should be one or two (1or 2) paragraphs.
4. Explain the first problem (economic, social, political,
environmental, complexity, inequity, ethical/moral, etc.) and
provide support for your claims. This should be one or two (1 or
2) paragraphs.
5. Explain the second problem (economic, social, political,
environmental, complexity, inequity, ethical/moral, etc.). and
provide support for your claims. This should be one or two (1 or
2) paragraphs.
6. Explain the third problem (economic, social, political,
environmental, complexity, inequity, ethical/moral, etc.) and
provide support for your claims. This should be one or two (1 or
2) paragraphs.
7. Provide a concluding paragraph that summarizes the stated
problems and promises a solution.
8. Develop a coherently structured paper with an introduction,
body, and conclusion.
9. Use effective transitional words, phrases, and sentences
throughout the paper.
10. Support claims with at least three (3) quality, relevant
references. Use credible, academic sources available through
Strayer University’s Resource Center.
Your assignment must follow these formatting guidelines:
• Be typed, double spaced, using Times New Roman font (size
12), with one-inch margins on all sides; citations and references
must follow APA or school-specific format. Check with your
professor for any additional instructions.
• Include a cover page containing the title of the assignment, the
student’s name, the professor’s name, the course title, and the
date. The cover page and the reference page are not included in
the required assignment page length.
• Note: Submit your assignment to Connect Composition Plus
and to the designated plagiarism program so that you can make
revisions before submitting your paper to your professor.
The specific course learning outcomes associated with this
assignment are:
• Recognize the elements and correct use of a thesis statement.
• Recognize the use of summary, paraphrasing, and quotation to
communicate the main points of a text.
• Analyze the rhetorical strategies of ethos, pathos, logos in
writing samples and for incorporation into essays or
presentations.
• Prepare a research project that supports an argument with
structure and format appropriate to the genre.
• Recognize how to organize ideas with transitional words,
phrases, and sentences.
• Incorporate relevant, properly documented sources to
substantiate ideas.
• Write clearly and concisely about selected topics using proper
writing mechanics.
• Use technology and information resources to research selected
issues for this course.
• Write a six to eight (6-8) page paper in which you: 6 PAGES
NEEDED
Provide Part I: Revision of A Problem Exists (3-4 pages)
1. Revise, using feedback from the professor and classmates,
your Persuasive Paper Part I: A Problem Exists.
Develop Part 2:
Solution
to Problem and Advantages (3-4 pages for 6-8 pages total)
2. Include a defensible, relevant thesis statement clearly in the
first paragraph. (The thesis statement may need to be modified
to reflect added information and purpose of this part.)
3. Explain a detailed, viable solution that supports your thesis.
This should be one or two (1-2) paragraphs.
4. State, explain, and support the first advantage (economic,
social, political, environmental, social, equitable, ethical/moral,
etc.) to your solution. This should be one or two (1-2)
paragraphs.
5. State, explain, and support the second advantage (economic,
social, political, environmental, social, equitable, ethical/moral,
etc.) to your solution. This should be one or two (1-2)
paragraphs.
6. State, explain, and support the third (and fourth if desired)
advantage (economic, social, political, environmental, social,
equitable, ethical/moral, etc.) to your solution. This should be
one or two (1-2) paragraphs.
7. Use effective transitional words, phrases, and sentences.
8. Provide a concluding paragraph / transitional paragraph that
summarizes the proposed solution and its advantages.
9. Develop a coherently structured paper with an introduction,
body, and conclusion.
10. Use one (1) or more rhetorical strategies (ethos, logos,
pathos) to explain advantages.
11. Support advantage claims with at least three (3) additional
quality relevant references. Use at least six (6) total for Parts 1
and 2.
Your assignment must follow these formatting guidelines:
• Be typed, double spaced, using Times New Roman font (size
12), with one-inch margins on all sides; citations and references
must follow APA or school-specific format. Check with your
professor for any additional instructions.
• Include a cover page containing the title of the assignment, the
student’s name, the professor’s name, the course title, and the
date. The cover page and the reference page are not included in
the required assignment page length.
• Note: Submit your assignment to Connect Composition Plus
and to the designated plagiarism program so that you can make
revisions before submitting your paper to your professor.
ASSIGNMENT TWO OUTLINE:
Assignment 2: Research Proposal – Thesis, Major Points, and
Plan
Due Week 3 and worth 80 points
Select a topic on which your persuasive writing paper will be
focused.
Write a one to two (1-2) page research proposal in which you:
Identify the topic you selected and explain two (2) reasons for
using it.
Include a defensible, relevant thesis statement in the first
paragraph.
Describe three (3) major characteristics of your audience
(official position, decision-making power, current view on
topic, other important characteristic).
Describe the paper’s scope and outline the major sections.
Identify and explain the questions to be answered.
Explain your research plan, including the methods of
researching and organizing research.
Develop a coherently structured paper with an introduction,
body, and conclusion.
Document at least three (3) primary sources and three (3)
secondary sources. Use credible, academic sources available
through Strayer University’s Resource Center.
Your assignment must follow these formatting guidelines:
Be typed, double spaced, using Times New Roman font (size
12), with one-inch margins on all sides; references must follow
APA or school-specific format. Check with your professor for
any additional instructions.
Include a cover page containing the title of the assignment, the
student’s name, the professor’s name, the course title, and the
date. The cover page and the reference page are not included in
the required page length.
The specific course learning outcomes associated with this
assignment are:
Recognize the elements and correct use of a thesis statement.
Write a research proposal that states the claim and scope of the
research project.
Outline the main sections of the research project.
Devise an ordered research plan to obtain appropriate resources.
Write clearly and concisely about selected topics using proper
writing mechanics.
Use technology and information resources to research selected
issues for this course.

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Section 7 Analyzing our Marketing Test, Survey Results .docx

  • 1. Section 7 Analyzing our Marketing Test, Survey Results and Other Metrics Using Confidence Intervals Rhonda Knehans Drake Associate Professor, New York University Data Analytics, Interpretation and Reporting copyright © 2013 2 • When we estimate population averages or percentages based on samples, a certain amount of error is present. • The amount of error present is a function mostly of your sample size. • The larger the sample, the less error in your estimates. • Today we will learn how to place bounds around our estimates obtained.
  • 2. • With such an interval, we will then be able to say with 90%, 95% or 99% confidence that the true population estimate will lie in these bounds. Introduction 3 The eight confidence interval formulas we will discuss are for the following situations: 1. Average or mean based on large samples 2. Average or mean based on small samples 3. Response rate or survey percent based on large samples 4. Response rate or survey percent based on small samples 5. Difference between 2 averages for large samples 6. Difference between 2 averages for small samples 7. Difference between 2 response rates for large samples 8. Difference between 2 response rates for small sample Confidence Intervals
  • 3. A-B Split Tests 4 1. Confidence Interval for Averages or Means Based on Large Samples (n ≥ 30) To calculate a confidence interval around a mean, the following information is required: – The sample mean x obtained from the test. – The sample standard deviation S obtained from the test. Many software packages, including Microsoft ExcelTM, can automatically calculate this value for you. (Review Section 3 for the standard deviation formula.) – The sample size n of the test. This is the number of observations used to calculate your mean. The sample size must be greater than or equal to 30 in size. – The desired confidence level: 90%, 95% or 99%. A confidence interval constructed around the mean will guarantee, with your specified level of confidence, the true mean will fall within those
  • 4. bounds. Sample Means (Large Sample) 5 • Once all information is known, you construct the confidence interval around the mean by adding and subtracting from the mean a multiple of your standard deviation associated with the sample mean. The “multiple” depends on your desired confidence level. • The formula for a confidence interval around the mean is calculated as follows: Where: • S is the standard deviation associated with the sample. • n is the sample size. • Z is equal to 1.645, 1.96 or 2.575 for a 90%, 95% or 99% confidence level Sample Means
  • 5. (Large Sample) 6 Example: Money magazine conducts a survey of 100 retirees across the US and asks them how much they have in their retirement fund. You obtain an average of $84.75 with a standard deviation of $18.75, both in thousands of dollars. You are about to write an article based on this average but realize that the true average is something more or less than this in reality. Construct a 95% confidence interval around this average. Sample Means (Large Sample) 7
  • 6. Sample Means (Large Sample) 84.75 1.96 (18.75 / √100) 84.75 1.96 (1.875) 84.75 3.675 ($81,075 , $88,425) 1. 2. 3. 4.
  • 7. 8 Later we will discuss how to chose the confidence level and address if 95% was the appropriate level for this example. Sample Means (Large Sample) 9 • Let’s do the previous example again but using the Plan-alyzer. 1. 2. 3. Sample Means
  • 8. (Large Sample) Select the tab “Table of Calculators” Select “Confidence Interval Calculators for Averages, Large Samples” Select “One Sample” 10 Sample Means (Large Sample) Input the required info. 11 Sample Means (Large Sample) See the answer.
  • 9. 12 2) Confidence Interval for Averages or Means Based on Small Samples (n < 30) To calculate a confidence interval around a mean, the following information is required: – The sample mean x obtained from the test. – The sample standard deviation S obtained from the test. Many software packages, including Microsoft ExcelTM, can automatically calculate this value for you. (Review Section 3 for the standard deviation formula.) – The sample size n of the test. This is the number of observations used to calculate your mean. The sample size must be greater than or equal to 30 in size. – The desired confidence level: 90%, 95% or 99%. A confidence interval constructed around the mean will guarantee, with your specified level of confidence, the true mean will fall within those bounds. Sample Means
  • 10. (Small Sample) 13 • Once all information is known, you construct the confidence interval around the mean by adding and subtracting from the mean a multiple of your standard deviation associated with the sample mean. The “multiple” depends on your desired confidence level. • The formula for a confidence interval around the mean is the same as the prior formula except we use a value from the “t-distribution” which is for approximate normally distributed data: Where: • S is the standard deviation associated with the sample. • n is the sample size. • t is obtained by using the excel function TINV as will be seen shortly. Sample Means (Small Sample) 14
  • 11. Example: Suppose Money Magazine only conducted the survey to a sample of 10 retirees instead of 100 as in our prior example, all else the same. Construct a 95% confidence interval around this average. Sample Means (Small Sample) 15 We construct the confidence interval as before except we will use The t-distribution. 84.75 ± (t) (18.75/√10 ) Where the value of t = TINV(.05,9) = 2.262 84.75 ± (2.262) (18.75/3.1623)
  • 12. 84.75 ± (2.262) (5.93) 84.75 ± 13.41 ($71.34, $98.16) (Small Sample) (1-conf level) (n-1) Sample Means 16 Note our confidence interval is wider for two reasons: 1. The smaller sample size 1. Our sample is less than 30 we cannot assume it is normal but only approximately normal so our multiplier is larger (2.262 versus 1.96). Sample Means (Small Sample)
  • 13. 17 • Let’s do the previous example again but using the Plan-alyzer. 1. 2. 3. Sample Means (Small Sample) Select the tab “Table of Calculators” Select “Confidence Interval Calculators for Averages, Small Samples” Select “One Sample”
  • 14. 18 Sample Means (Small Sample) Input the required info. 19 Sample Means (Small Sample) See the answer. 20 3. Confidence Intervals for Response Rates or Survey Percentages Based on Large Samples (where n*p and n*(1 - p) are both ≥ 5) To calculate a confidence interval around a sample proportion, the following information is required. – The sample proportion p obtained from the test.
  • 15. – The sample size n of the test. This is the number of observations used to calculate your proportion. The sample size, when multiplied by the sample proportion and when multiplied by one minus the sample proportion, must both be greater than or equal to 5. – The desired confidence level: 90%, 95% or 99%. A confidence interval constructed around the sample proportion will guarantee, with your specified level of confidence, the true population proportion will fall within those bounds. Sample Proportions (Large Sample) 21 • Once all information is known, you construct the confidence interval around the sample proportion by adding and subtracting from the sample proportion a multiple of the standard deviation associated with the sample proportion. The “multiple” depends on your desired confidence level. • The formula for a confidence interval around the sample proportion is calculated as follows:
  • 16. • Where n is your sample size and Z is 1.645, 1.96, 2.575 for a 90%, 95% or 99% confidence interval This is the standard deviation for a proportion or response rate Sample Proportions (Large Sample) 22 Example: AT&T samples a new prospect list and sends them an offer to order their new wireless cellular service. They sample 10,000 prospects and receive a 0.89% response rate. What is the margin of error at 90% confidence? Sample Proportions (Large Sample)
  • 17. 23 • So, for our example we have the confidence interval is: = .0089 ± (1.645)·√ (.0089)(1- .0089 )/10,000 = .0089 ± (1.645)·√ (.0089)(.9911 )/10,000 = .0089 ± (1.645)·√ 10000008 = .0089 ± (1.645)·(.0008944) = .0089 ± .0015 (.0074, .0104) or (.74%, 1.04%) Sample Proportions (Large Sample) 0.0089 1.645*(√[(0.0089)*(0.9911)/10,000] 0.0089 1.645*(√0.0000008) 0.0089 1.645*(0.0008944) 0.0089 .0015
  • 18. (0.0074 , 0.0104) or (0.74% , 1.04%) 1. 2. 3. 4. 5. 24 • Let’s do the previous
  • 19. example again but using the Plan-alyzer. 1. 2. 3. Sample Proportions (Large Sample) Select the tab “Table of Calculators” Select “Confidence Interval Calculators for Percentages, Large Samples” Select “One Sample” 25 Sample Proportion
  • 20. (Large Sample) Input the required info. 26 Sample Proportion (Large Sample) See the answer. 27 4. Confidence Intervals for Response Rates or Survey Percentages Based on Small Samples (where either n*p or n*(1 - p) are < 5) To calculate a confidence interval around a sample proportion, the following information is required: – The sample proportion p obtained from the test. – The sample size n of the test. This is the number of observations used to calculate your proportion.
  • 21. The sample size, when multiplied by the sample proportion and when multiplied by one minus the sample proportion, must both be greater than or equal to 5. – The desired confidence level: 90%, 95% or 99%. A confidence interval constructed around the sample proportion will guarantee, with your specified level of confidence, the true population proportion will fall within those bounds. Sample Proportions (Small Sample) 28 • Once all information is known, you construct the confidence interval around the sample proportion by adding and subtracting from the sample proportion a multiple of the standard deviation associated with the sample proportion. The “multiple” depends on your desired confidence level. • The formula for a confidence interval around the sample proportion is the same as the prior formula except we use a value from the “t-distribution” which is for approximate normally distributed data:
  • 22. • Where n is your sample size and t is obtained by using the Excel function TINV as will be seen shortly. This is the standard deviation for a proportion or response rate Sample Proportions (Small Sample) 29 Example: Suppose AT&T only sampled 100 prospects instead of 10,000 as in our previous example, all else the same. What is the margin of error at 90% confidence? Sample Proportions (Small Sample) 30
  • 23. We construct the confidence interval as before except we will use the t-distribution. - .0089 )/100 Where the value of t = TINV(.10,99) = 1.66 = .0089 ± (1.66)·√(.0089)(.9911 )/100 = .0089 ± (1.66)·√ .0000882 = .0089 ± (1.66)·(.0093914) = (0.00, 0.0245) or (0.00%, 2.45%) The lower bound here cannot be negative, so we change it to zero. Sample Proportions (Small Sample) (1-conf level) (n-1)
  • 24. 31 • Let’s do the previous example again but using the Plan-alyzer. 1. 2. 3. Sample Proportions (Small Sample) Select the tab “Table of Calculators” Select “Confidence Interval Calculators for Percentages, Small Samples” Select “One Sample”
  • 25. 32 Sample Proportions (Small Sample) Input the required info. 33 See the answer. Sample Proportions (Small Sample) 34 5. Confidence Interval for the Difference between 2 Means or Averages for Large Samples (n1 ≥ 30 and n2 ≥ 30 ) To calculate a confidence interval around the difference between two sample means, the following information is required: – The means of both samples (x1 and x2).
  • 26. – The standard deviation of both samples (S1 and S2). Many software packages, including Microsoft ExcelTM, can automatically calculate these values. – The size of both samples (n1 and n2). These are the number of observations that went into calculating each of your means. Both sample sizes n1 and n2 must be greater than or equal to 30 in size. – The desired confidence level: 90%, 95% or 99%. A confidence interval constructed around the sample proportion will guarantee, with your specified level of confidence, the true population proportion will fall within those bounds. • Confidence Interval Estimation (A-B Split Testing) (Difference Between Two Samples Means – Large Samples) 35 • Once all information is known, you construct the confidence interval around the difference between two mean by adding and subtracting from the difference in means a multiple of the
  • 27. standard deviation associated with the difference. The “multiple” depends on your desired confidence level. • The formula for a confidence interval around the difference between means is calculated as follows: • Where Z is 1.645, 1.96, 2.575 for a 90%, 95% or 99% confidence interval. This is the standard deviation for the difference in averages. Confidence Interval Estimation (A-B Split Testing) (Difference Between Two Sample Means – Large Samples) (X1 – X2) ± (Z)(√(S1 2 /n1) + (S2 2 /n2)) 36 Example:
  • 28. You sample 100 home sales in San Francisco and 100 home sales in NYC for 2010 with the following results: NSF = 100 XSF = $745.25 SSF = $40 NNYC = 100 XNYC = $775.10 SNYC = $45 Is there any difference in home prices between NYC and San Francisco? Base your answer on the 95% confidence level. Confidence Interval Estimation (A-B Split Testing) (Difference Between Two Sample Means – Large Samples) 37 • Assume SSF = $40 and SNYC = $45 (775.10 – 745.25) ± (1.96)·√ [(402/100)+(452/100)] 29.85 ± (1.96)·√ 16 + 20.25 29.85 ± (1.96)·√ 36.25 29.85 ± (1.96)·(6.021) 29.85 ± 11.80
  • 29. ($18,050, $41,650) Confidence Interval Estimation (A-B Split Testing) (Difference Between Two Sample Means – Large Samples) 1. 2. 3. 4. 5. (775.10-745.25) 1.96*(√[(402/100)+(452/100 )]) 29.85 1.96*(√(16+20.25)) 29.85 1.96*(6.021)
  • 30. 29.85 11.80 ($18.050 , $41.650) 38 How do we interpret? • What if the interval was -$18,050 to $41,540. How would you interpret and is it okay in this case to have a negative value? Confidence Interval Estimation (A-B Split Testing) (Difference Between Two Sample Means – Large Samples) – With 95% confidence, we can say that NYC home prices in 2010 are higher than SF home prices by anywhere from $18,050 to $41,540. – If Zero were in the interval then you would say no
  • 31. difference between the two (a hypothesis test!!). – It means there's no statistical evidence to conclude that NYC prices are different from those of SF. – Its ok to have a negative value. 39 • Let’s do the previous example again but using the Plan-alyzer. 1. 2. 3. Select the tab “Table of Calculators” Select “Confidence Interval Calculators for Averages, Large Samples”
  • 32. Select “Test vs. Control” Confidence Interval Estimation (A-B Split Testing) (Difference Between Two Sample Means – Large Samples) 40 Input the required info. Confidence Interval Estimation (A-B Split Testing) (Difference Between Two Sample Means – Large Samples) 41 See the answer. Confidence Interval Estimation (A-B Split Testing) (Difference Between Two Sample Means – Large Samples) 42 6. Confidence Interval for the Difference between 2 Means or Averages for Small Samples (n1 < 30 or n2 < 30). – If one or both samples is less than 30 in size then you will
  • 33. replace the Z value with the t value. – You will again use the TINV function in excel. – Your parameters are TINV(1- confidence level, n1+n2 - 2). – All else the same. – This will be used for small market research problems. Confidence Interval Estimation (A-B Split Testing) (Difference Between Two Sample Means – Small Samples) 43 7. Confidence Intervals for the Difference Between 2 Percentages for Large Samples (n1*p1, n1*(1-p1), n2*p2, n2*(1-p2), all ≥ 5) To calculate a confidence interval around the difference between two sample proportions, the following information is required: – The proportions (p1 and p2) for both samples. – The size of both samples (n1 and n2). These are the number of observations used in calculating each of the
  • 34. sample proportions. Both sample sizes, when multiplied by their respective sample proportions and when multiplied by one minus their respective sample proportions, must all be greater than or equal to 5. – The desired confidence level: 90%, 95% or 99%. A confidence interval constructed around the sample proportion will guarantee, with your specified level of confidence, the true population proportion will fall within those bounds. Confidence Interval Estimation (A-B Split Testing) (Difference Between Two Sample Proportions – Large Samples) 44 • Once all information is known, you construct the confidence interval around the difference between two proportions by adding and subtracting from the difference in proportions a multiple of the standard deviation associated with the difference. The “multiple” depends on your desired confidence level. • The formula for a confidence interval around the difference between proportions is calculated as follows:
  • 35. • Where Z is 1.645, 1.96, 2.575 for a 90%, 95% or 99% confidence interval. This is the standard deviation for the difference in proportions. Confidence Interval Estimation (A-B Split Testing) (Difference Between Two Sample Proportions – Large Samples) 45 Example: You are in charge of new card acquisitions at American Express. You conduct a new offer test for the green card versus your control offer with the following results Did the test beat the control with 95% confidence? Do you have a winner?
  • 36. Sample Size Response Rate Control Offer with 10,000 Bonus Miles 10,000 1.10% Test Offer with 25,000 Bonus Miles 10,000 1.38% Confidence Interval Estimation (A-B Split Testing) (Difference Between Two Sample Proportions – Large Samples) 1. 2. 3. 4. 46
  • 37. Confidence Interval Estimation (A-B Split Testing) (Difference Between Two Sample Proportions – Large Samples) 0.00297 0.000001 .0028 .0028 47 So how do we interpret? • With 95% confidence the test can do worse than the control by -.017% OR do better than the control by .577%. • As such we say the test and the control are not significantly different since the confidence interval wrapped around the difference in response rates contains zero. • Had the lower bound been above zero then we would say the test has beaten the control. • But let’s be real here. For all purposes, the test is a winner. The lower bound is soooo close to zero. So worst case the test is the same as
  • 38. the control with much upside potential. Confidence Interval Estimation (A-B Split Testing) (Difference Between Two Sample Proportions – Large Samples) 48 So how do we interpret (continuation)? • But remember just because the test beaten the control from a statistical point of view, that does not mean that it won from a marketing point of view. • In this example we were giving away additional sky miles. So the test will need to beat the control by some minimum most likely greater than zero or else we will not generate the same revenue. • Suppose I told you that based on the cost of the additional sky miles the test needs to beat the control by at least .025% to break- even. Would you consider it a winner? • What if I told you the test needs to beat the control by .25% to break even. Would you now consider the test a winner?
  • 39. Confidence Interval Estimation (A-B Split Testing) (Difference Between Two Sample Proportions – Large Samples) 49 • Let’s do the previous example again but using the Plan-alyzer. 1. 2. 3. Select the tab “Table of Calculators” Select “Confidence Interval Calculators for Proportions, Large Samples” Select “Test vs. Control”
  • 40. Confidence Interval Estimation (A-B Split Testing) (Difference Between Two Sample Proportions – Large Samples) 50 Input the required info. Confidence Interval Estimation (A-B Split Testing) (Difference Between Two Sample Proportions – Large Samples) 51 See the answer. Confidence Interval Estimation (A-B Split Testing) (Difference Between Two Sample Proportions – Large Samples) 52 8. Confidence Intervals for the Difference Between 2 Percentages for Small Samples (where n1*p1 or n1*(1-p1) or n2*p2 or n2*(1-p2) < 5) • You will replace the Z value with the t value.
  • 41. • You will again use the TINV function in excel. • Your parameters are TINV(1- confidence level, n1+n2 - 2). • All else the same. • This will be used for small market research problems. Confidence Interval Estimation (A-B Split Testing) (Difference Between Two Sample Proportions – Small Samples) 53 • Remember, interpretation of confidence intervals is not that simple. • They will not tell you what to do. • They simply give you valid best and worst case scenarios to take into consideration. • They give you additional information upon which to help you base your marketing decisions: • Worst case, are the results meeting your criteria? • How does the upside potential compare to the downside
  • 42. potential? Interpretation of Confidence Intervals 54 • No direct marketer should ever consider evaluating their test results with a confidence level lower than 90%. To do so assumes way to much risk. • And, fishing for a confidence level that yields significance should never be practiced. • The rules that any good direct marketer should follow regarding significance are shown on the next slide. Setting the Confidence Level 55 Evaluate your test response rate at 95% confidence level Significant?
  • 43. Yes No Is it significant at the 99% confidence level? Not that we want to go that low but is it significant at the 90% confidence level? Yes No Yes No A no brainer. Let’s roll! That’s okay…at a minimum let’s consider a partial to full roll out. Okay, so we have something here. Let’s either retest or
  • 44. go for a partial rollout. Not good. We should scrape this from further consideration. Setting the Confidence Level 56 It is important to keep in mind the following facts regarding the creation of a confidence interval. • If you want more confidence in your estimates, the resulting interval will widen. • If you increase your sample size, the resulting interval will become tighter. • The more accuracy you need in your test estimate, the higher you should set your confidence level.
  • 45. • The confidence level you set should depend on the risk you are willing to take in making an incorrect decision. Setting the Confidence Level 57 • If our sample represents a large percent (> 10%) of the population in total, then we typically apply a correction factor to our margin of error estimates. • The larger our sample is as a percent of the total population the more valid our estimates. • Of the following two samples, which would you think would yield a better parameter estimate? • A sample of size 5,000 from a population of size 10,000 in total • A sample of size 5,000 from a population of size 10,000,000 in total. • If our sample represents 10% or more of the total population, you multiple the margin of error for the first four formulas by:
  • 46. Finite Population Correction Factor 58 Example: Going back to our first exercise, suppose the survey of 100 people was conduct to the 800 residents of the Happy Retirement Village and not Money Magazine subscribers. What is our correction factor and what is the new interval? Finite Population Correction Factor 3. 1. 2. 3.4398
  • 47. 59 7.1 Briefly explain how the width of a confidence interval decreases with an increase in the sample size. Give me an example. 7.2 Briefly explain how the width of a confidence interval decreases with a decrease in the confidence level. Give me an example. 7.3 According to a study done by Dr. Martha S. Linet and others, the mean duration of the recent headache was 8.2 hours for a sample of 5055 females aged 12 through 29. Assume that this sample represents the current population of all headaches for all females aged 12 through 29 and that the standard deviation for this sample is 2.4 hours. Make a 95% confidence interval for the mean duration of all headaches for all 12-to-29- year-old females. Do by hand and using The Plan-alyzer. 7.4 A sample of 12 observations was drawn from a population of size 100. Calculate a 95% confidence interval around the average for this sample by hand. HINT: You will want to use the finite population correction factor for this problem as found on slides 57 and 58.
  • 48. 13 15 9 11 8 16 14 9 10 14 16 12 Section 7 Exercises 60 7.5 A company wants to estimate the mean net weight of its “Big Top Circus” cereal boxes. A sample of 16 such boxes produced the mean net weight of 31.98 ounces with a standard deviation of .26 ounces. Make a 95% confidence interval for the mean net weight of all boxes. Do by hand and using The Plan-alyzer. 7.6 Crate and Barrel Cataloger promises its customers that the products ordered will be mailed within 72 hours after an order is placed. The quality control department at the company checks from time to time to see if this promise is fulfilled. Recently, the quality control department took a sample
  • 49. of 50 orders and found that 42 of them were mailed within 72 hours of the placement of the orders. a) Construct a 95% confidence interval for the percentage of all orders that are mailed within 72 hours of their placement. Do by hand and using The Plan-alyzer. b) Suppose the confidence interval obtained in part a is too wide. How can the width of this interval be reduced? Discuss all possible alternatives. Which of these alternatives is the best? Section 7 Exercises 61 7.7 In virtual reality a person views a computer-generated scene that changes as if the viewer’s body were in motion. Some individual experience unpleasant side effects from virtual reality, such as nausea, dizziness, or
  • 50. disorientation. In a recent study by Clare Tegan of Britain’s Defense Research Agency, each of the 150 people included in the study spent 20 minutes wearing a head-mounted virtual reality system through which he or she explored a virtual environment consisting of a series of rooms. Either during their time in the virtual environment or in the 10 minutes immediately afterward, 61% of these 150 persons suffered side effects. Find the 95% confidence interval for the proportion of all virtual reality users who would suffer side effects. Do by hand and using The Plan- alyzer. Section 7 Exercises 62 7.8 One of the major problems faced by department stores is a high
  • 51. percentage of returns. The manager of Macy’s wanted to estimate the percentage of all sales that result in returns. A sample of 40 sales showed that 8 of them had products returned within the time allowed for returns. a) Construct a 99% confidence interval for the percentage of all sales that result in returns. Do by hand and using The Plan-alyzer. b) Do you think 99% confidence is appropriate in this case and if not what would be a more appropriate level of confidence to use? 7.9 According to a survey, the mean price of gasoline in the U.S. was $1.20 per gallon in 1995 and $1.10 per gallon in 1994 (Wow, don’t you wish!) Suppose these means were based on random samples of 100 gas station for 1995 and 120 gas station for 1994. Also, assume that the sample standard deviations were $.11 for 1995 and $.10 for 1994. Find a 90% confidence interval for the difference between the
  • 52. mean gasoline prices for 1995 and 1994. Do by hand and using The Plan- alyzer. Section 7 Exercises 63 7.10 An insurance company wants to know if the average speed at which men drive cars is higher than that of women drivers. The company took a random sample of 27 cars driven by men on a highway and found the mean speed to be 68 miles per hour with a standard deviation of 2.2 miles. Another sample of 18 cars driven by women on the same highway gave a mean speed of 65 miles per hour with a standard deviation of 2.5 miles. Assume that the speeds at which all men and all women drive cars on this highway are both known to be normally distributed. Construct a 99%
  • 53. confidence interval for the difference between the mean speeds of cars driven by all men and all women drivers on this highway. Do by hand and using The Plan-alyer. Section 7 Exercises 64 7.11 Removed. 7.12 In a Prevention magazine survey released in 2008, Princeton Survey Research Association examined the weight of children aged 3 to 17. According to this study, 24% of children in this age group were overweight in 2000, and 31% were considered overweight in 2008. Suppose that these percentages are based on random samples of 400
  • 54. and 500 children in the given age group in 2000 and 2008, respectively. Conduct a 95% confidence interval for the difference between the portions of the overweight 3-to-17-year-olds in 2000 and 2008. Do by hand and using The Plan-alyzer. Section 7 Exercises Number of sources: 7 Topic: Should all states require motorcyclist and passengers to wear helmets? Type of document: Research Paper: THIS IS A RESEARCH PAPER!! Academic Level:Undergraduate Number of Pages: 9 (Double Spaced) Category: English Language Style: English (U.S.) Writing Style: APA Order Instructions: Should all states require motorcyclist and passengers to wear helmets? • Using your thesis statement and research, present the problem that needs to be addressed with your proposed solution. Note: Your solution, advantages, and challenges, will be in Parts 2 and 3. Write a three to four (3-4) page paper in which you: 3 PAGES NEEDED
  • 55. 1. Provide an appropriate title and an interesting opening paragraph to appeal to your stated audience (appeal with logic, ethics, or emotion). 2. Include a defensible, relevant thesis statement in the first paragraph. (Revised from Assignment 2; Check below) 3. Describe the history and status of the issue and provide an overview of the problem(s) that need to be addressed. This should be one or two (1or 2) paragraphs. 4. Explain the first problem (economic, social, political, environmental, complexity, inequity, ethical/moral, etc.) and provide support for your claims. This should be one or two (1 or 2) paragraphs. 5. Explain the second problem (economic, social, political, environmental, complexity, inequity, ethical/moral, etc.). and provide support for your claims. This should be one or two (1 or 2) paragraphs. 6. Explain the third problem (economic, social, political, environmental, complexity, inequity, ethical/moral, etc.) and provide support for your claims. This should be one or two (1 or 2) paragraphs. 7. Provide a concluding paragraph that summarizes the stated problems and promises a solution. 8. Develop a coherently structured paper with an introduction, body, and conclusion. 9. Use effective transitional words, phrases, and sentences throughout the paper. 10. Support claims with at least three (3) quality, relevant references. Use credible, academic sources available through Strayer University’s Resource Center. Your assignment must follow these formatting guidelines: • Be typed, double spaced, using Times New Roman font (size 12), with one-inch margins on all sides; citations and references must follow APA or school-specific format. Check with your professor for any additional instructions. • Include a cover page containing the title of the assignment, the student’s name, the professor’s name, the course title, and the
  • 56. date. The cover page and the reference page are not included in the required assignment page length. • Note: Submit your assignment to Connect Composition Plus and to the designated plagiarism program so that you can make revisions before submitting your paper to your professor. The specific course learning outcomes associated with this assignment are: • Recognize the elements and correct use of a thesis statement. • Recognize the use of summary, paraphrasing, and quotation to communicate the main points of a text. • Analyze the rhetorical strategies of ethos, pathos, logos in writing samples and for incorporation into essays or presentations. • Prepare a research project that supports an argument with structure and format appropriate to the genre. • Recognize how to organize ideas with transitional words, phrases, and sentences. • Incorporate relevant, properly documented sources to substantiate ideas. • Write clearly and concisely about selected topics using proper writing mechanics. • Use technology and information resources to research selected issues for this course. • Write a six to eight (6-8) page paper in which you: 6 PAGES NEEDED Provide Part I: Revision of A Problem Exists (3-4 pages) 1. Revise, using feedback from the professor and classmates, your Persuasive Paper Part I: A Problem Exists. Develop Part 2: Solution
  • 57. to Problem and Advantages (3-4 pages for 6-8 pages total) 2. Include a defensible, relevant thesis statement clearly in the first paragraph. (The thesis statement may need to be modified to reflect added information and purpose of this part.) 3. Explain a detailed, viable solution that supports your thesis. This should be one or two (1-2) paragraphs. 4. State, explain, and support the first advantage (economic, social, political, environmental, social, equitable, ethical/moral, etc.) to your solution. This should be one or two (1-2) paragraphs. 5. State, explain, and support the second advantage (economic, social, political, environmental, social, equitable, ethical/moral, etc.) to your solution. This should be one or two (1-2) paragraphs. 6. State, explain, and support the third (and fourth if desired) advantage (economic, social, political, environmental, social, equitable, ethical/moral, etc.) to your solution. This should be one or two (1-2) paragraphs. 7. Use effective transitional words, phrases, and sentences. 8. Provide a concluding paragraph / transitional paragraph that summarizes the proposed solution and its advantages. 9. Develop a coherently structured paper with an introduction, body, and conclusion. 10. Use one (1) or more rhetorical strategies (ethos, logos, pathos) to explain advantages.
  • 58. 11. Support advantage claims with at least three (3) additional quality relevant references. Use at least six (6) total for Parts 1 and 2. Your assignment must follow these formatting guidelines: • Be typed, double spaced, using Times New Roman font (size 12), with one-inch margins on all sides; citations and references must follow APA or school-specific format. Check with your professor for any additional instructions. • Include a cover page containing the title of the assignment, the student’s name, the professor’s name, the course title, and the date. The cover page and the reference page are not included in the required assignment page length. • Note: Submit your assignment to Connect Composition Plus and to the designated plagiarism program so that you can make revisions before submitting your paper to your professor. ASSIGNMENT TWO OUTLINE: Assignment 2: Research Proposal – Thesis, Major Points, and Plan Due Week 3 and worth 80 points Select a topic on which your persuasive writing paper will be focused.
  • 59. Write a one to two (1-2) page research proposal in which you: Identify the topic you selected and explain two (2) reasons for using it. Include a defensible, relevant thesis statement in the first paragraph. Describe three (3) major characteristics of your audience (official position, decision-making power, current view on topic, other important characteristic). Describe the paper’s scope and outline the major sections. Identify and explain the questions to be answered. Explain your research plan, including the methods of researching and organizing research. Develop a coherently structured paper with an introduction, body, and conclusion. Document at least three (3) primary sources and three (3) secondary sources. Use credible, academic sources available through Strayer University’s Resource Center. Your assignment must follow these formatting guidelines: Be typed, double spaced, using Times New Roman font (size 12), with one-inch margins on all sides; references must follow APA or school-specific format. Check with your professor for any additional instructions.
  • 60. Include a cover page containing the title of the assignment, the student’s name, the professor’s name, the course title, and the date. The cover page and the reference page are not included in the required page length. The specific course learning outcomes associated with this assignment are: Recognize the elements and correct use of a thesis statement. Write a research proposal that states the claim and scope of the research project. Outline the main sections of the research project. Devise an ordered research plan to obtain appropriate resources. Write clearly and concisely about selected topics using proper writing mechanics. Use technology and information resources to research selected issues for this course.