Quiz 4
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Table of Contents
Part 1 of 1 - Multiple Choice
Question 1 of 20
Of type I and type II error, which is traditionally regarded as more serious?
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A. Type I is considered to be more serious.
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B. Type II is considered to be more serious.
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C. Type I and Type II are equally serious.
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D. Neither Type I or Type II is serious and both can be avoided.
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Question 2 of 20
If you are constructing a confidence interval for a single mean, the confidence interval will _____ with an
increase in the sample size.
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A. decrease
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B. increase
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C. stay the same
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D. increase or decrease, depending on the sample data
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Question 3 of 20
The chi-square distribution for developing a confidence interval for a standard deviation has degrees of
freedom equal to:
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A. n + 2
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B. n +1
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C. n
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D. n – 1
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E. n – 2
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Question 4 of 20
The null hypothesis usually represents the:
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A. theory the researcher would like to prove
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B. preconceived ideas of the researcher
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C. perceptions of the sample population
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D. status quo of the situation being studied
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Question 5 of 20
Which of the following statements is true regarding the chi-square goodness-of-fit test for normality?
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A. The test does depend on which and how many categories we use for the
histogram.
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B. The test is not very effective unless the sample size is large, say, at least 80 or 100.
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C. The test tends to be too sensitive if the sample size is really large.
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D. None of these choices is true.
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E. Choices a, b, and c are all true.
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Question 6 of 20
A type I error occurs when the:
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A. null hypothesis is incorrectly accepted when it is false
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B. null hypothesis is incorrectly rejected when it is true
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C. sample mean differs from the population mean
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D. test procedure itself is fundamentally biased
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Question 7 of 20
The rejection region is the set of sample data that leads to the rejection of the alternative hypothesis.
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A. True
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B. False
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Question 8 of 20
In general, increasing the confidence level will narrow the confidence interval, and decreasing the
confidence level widens the interval.
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A. True
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B. False
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Question 9 of 20
Which of the following statements are true of the null and alternative hypotheses?
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A. Exactly one hypothesis must be true.
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B. Both hypotheses must be true.
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C. It is possible for both hypotheses to be true.
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D. It is possible for neither hypothesis to be true.
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Question 10 of 20
A low p–value provides evidence for accepting the null hypothesis and rejecting the alternative.
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A. True
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B. False
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Question 11 of 20
If you increase the confidence level, the confidence interval .
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Question 12 of 20
A teacher who is trying to prove that a new method of teaching economics is more effective than a
traditional one will conduct a:
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A. one-tailed test
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B. two-tailed test
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C. point estimate of the population parameter
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D. confidence interval
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Question 13 of 20
The degrees of freedom for the t and chi-square distributions is a numerical parameter of the
distribution that defines the precise shape of the distribution.
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A. True
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B. False
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Question 14 of 20
The p-value of a test is the probability of observing a test statistic at least as extreme as the one
computed given that the null hypothesis is true.
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A. True
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B. False
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Question 15 of 20
As the sample size increases, the t-distribution becomes more similar to the ____ distribution.
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A. normal
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B. exponential
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C. multinominal
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D. chi-square
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E. binomial
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Question 16 of 20
Which sign is possible in an alternative hypothesis?
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A. >
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B. <
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C. ≠
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D. All of these signs are possible
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Question 17 of 20
The p–value of a sample is the probability of seeing a sample with
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A. at most as much evidence in favor of the null hypothesis as the sample actually
observed.
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B. at most as much evidence in favor of the alternative hypothesis as the sample
actually observed.
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C. at least as much evidence in favor of the null hypothesis as the sample actually
observed.
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D. at least as much evidence in favor of the alternative hypothesis as the sample
actually observed.
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Question 18 of 20
Confidence intervals are a function of the:
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A. population, the sample, and the standard deviation
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B. sample, the variable of interest, and the degrees of freedom
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C. data in the sample, the confidence level, and the sample size
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D. sampling distribution, the confidence level, and the degrees of freedom
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E. mean, median, and mode
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Question 19 of 20
If two samples contain the same number of observations, then the data must be paired.
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A. True
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B. False
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Question 20 of 20
A type II error occurs when:
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A. the null hypothesis is incorrectly accepted when it is false
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B. the null hypothesis is incorrectly rejected when it is true
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C. the sample mean differs from the population mean
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D. the test procedure itself is fundamentally biased
PUAD 630
ANALYTICAL TECHNIQUES IN PUBLIC ADMINISTRATION
F ERZANA H AVEWALA
C ONFIDENCE I NTERVAL
E STIMATION
Confidence Intervals
1
Statistical inferences are always based on an underlying probability model, which
means that some type of random mechanism must generate the data.
Two random mechanisms are generally used:
Random sampling from a larger population
Randomized experiments
Generally, statistical inferences are of two types:
Confidence interval estimation uses the data to obtain a point estimate and a confidence
interval around this point estimate.
Hypothesis testing determines whether the observed data provide support for a particular
hypothesis.
PUAD 630: Analytical Techniques in Public Administration
Margin of Error and the Interval Estimate
2
Inferences about population parameters are based on the sampling distribution of a
point estimate, such as the sample mean.
An interval estimate can be computed by adding and subtracting a margin of error to
the point estimate.
Point estimate ± Margin of error
Most confidence intervals are of the form:
Point estimate ± Multiple × Standard Error
PUAD 630: Analytical Techniques in Public Administration
3
Confidence Interval for a Mean
PUAD 630: Analytical Techniques in Public Administration
Confidence Interval for a Mean
4
To obtain a confidence interval for μ, first specify a confidence level
usually 90%, 95%, or 99%
Then use the sampling distribution of the point estimate to determine the multiple of
the standard error (SE) to go out on either side of the point estimate to achieve the
given confidence level.
If the confidence level is 95%, the value used most frequently in applications, the multiple is
approximately 2. More precisely, it is a t-value.
A typical confidence interval for μ is of the form: X t-multiple SE ( X )
where SE ( X ) s
n
PUAD 630: Analytical Techniques in Public Administration
Confidence Interval for a Mean
5
To obtain the correct t-multiple, let α be 1 minus the confidence level (expressed as a
decimal).
For example, if the confidence level is 90%, then α = 0.10.
Then the appropriate t-multiple is the value that cuts off probability α/2 in each tail
of the t distribution with n−1 degrees of freedom.
As the confidence level increases, the length of the confidence interval also increases.
As n increases, the standard error s/√n decreases, so the length of the confidence
interval tends to decrease for any confidence level.
PUAD 630: Analytical Techniques in Public Administration
Example 1. Customer Response to a New Sandwich
6
Objective: To obtain a 95% confidence interval for the mean satisfaction rating of the
new sandwich.
Data:
A random sample of 40 customers who ordered a new sandwich were surveyed.
Each was asked to rate the sandwich on a scale of 1 to 10. (Rating results appear in column B)
Solution:
First method: using only Excel, find the 95% confidence interval for the mean
satisfaction rating.
PUAD 630: Analytical Techniques in Public Administration
Example 1.
PUAD 630: Analytical Techniques in Public Administration
Example 1. Customer Response to a New Sandwich
8
Solution:
An alternative method uses StatTools’s One-Sample Analysis on the Satisfaction
variable.
StatTools Toolbar: >Statistical Inference > Confidence Interval >Mean/Std. Deviation
PUAD 630: Analytical Techniques in Public Administration
Example 1.
(continued)
PUAD 630: Analytical Techniques in Public Administration
Example 1. Customer Response to a New Sandwich
10
In this example, two assumptions lead to the confidence interval:
First, you might question whether the sample is really a random sample. It is likely a
convenience sample, not really a random sample.
However, unless there is some reason to believe that this sample differs in some relevant
aspect from the entire population, it is probably safe to treat it as a random sample.
A second assumption is that the population distribution is normal, even though the
population distribution cannot be exactly normal.
This is probably not a problem because confidence intervals based on the t distribution
are robust to violations of normality, and the normal population assumption is less crucial
for larger sample sizes because of the central limit theorem.
PUAD 630: Analytical Techniques in Public Administration
11
Confidence Interval for a Total
PUAD 630: Analytical Techniques in Public Administration
Confidence Interval for a Total
12
be a point estimate of T
Let T be a population total we want to estimate and let 𝑇
based on a simple random sample of size n from a population of size N.
First, we need a point estimate of T.
For the population total T, it is reasonable to sum all of the values in the sample, denoted Ts
and then “project” this total to the population with this equation:
are given as:
The mean and standard deviation of the sampling distribution of 𝑇
Because σ is usually unknown, s is used instead of σ to obtain the approximate
standard error of 𝑇 ∶
The point estimate of T is the point estimate of the mean multiplied by N, and the
standard error of this point estimate is the standard error of the sample mean
multiplied by N.
PUAD 630: Analytical Techniques in Public Administration
Example 2. Estimating Total Tax Refunds
13
Objective: To find a 95% confidence interval for the total (net) amount the IRS must
pay out to a set of 1,000,000 taxpayers.
Data:
Data set is the refunds from a random sample of 500 taxpayers.
Solution:
Calculate the confidence interval with StatTools.
PUAD 630: Analytical Techniques in Public Administration
Example 2.
PUAD 630: Analytical Techniques in Public Administration
15
Confidence Interval
for a Proportion
PUAD 630: Analytical Techniques in Public Administration
Confidence Interval for a Proportion
16
To form a confidence interval for any population proportion p, we require a point
estimate, the standard error of this point estimate, and a multiple that depends on
the confidence level:
It can be shown that for sufficiently large n, the sampling distribution of 𝑝Ƹ is
approximately normal with mean p
Standard error of sample proportion:
Confidence interval for a proportion:
PUAD 630: Analytical Techniques in Public Administration
Example 3. Estimating the Response to a New Sandwich
17
Objective: To illustrate the procedure for finding a 95% confidence interval for the
proportion of customers who rate the new sandwich at least 6 on a 10-point scale.
Data:
A sample of 40 customers who ordered a new sandwich were surveyed. Each was
asked to rate the sandwich on a scale of 1 to 10. (The results are shown in column B)
Solution:
The sample proportion is calculated directly from the sample data with a COUNTIF
function.
Use StatTools three different ways:
One-Sample Analysis from Population Sample
One-Sample Analysis from Summary Table with Counts
One-Sample Analysis from Summary Table with Proportions
PUAD 630: Analytical Techniques in Public Administration
Confidence Interval for a Proportion
Example 3.
PUAD 630: Analytical Techniques in Public Administration
19
Confidence Interval for the
Difference Between Means
PUAD 630: Analytical Techniques in Public Administration
Confidence Interval for the Difference Between Means
20
One of the most important applications of statistical inference is the comparison of
two population means.
For statistical reasons, independent samples must be distinguished from paired
samples.
PUAD 630: Analytical Techniques in Public Administration
Independent Samples
21
The appropriate sampling distribution of the difference between sample means is the
t distribution with n1 + n2 – 2 degrees of freedom.
Confidence interval for difference between means:
Pooled estimate of common standard deviation:
Standard error of difference between sample means:
PUAD 630: Analytical Techniques in Public Administration
Example 4. Reliability of Treadmill Motors at SureStep
22
Objective: To find a 95% confidence interval for the difference between mean
lifetimes of motors, and to see how this confidence interval can help SureStep choose
the better supplier.
Data:
SureStep Company installs motors from supplier A on 30 of its treadmills and
motors from supplier B on another 30 of its treadmills.
It then runs these treadmills and records the number of hours until the motor fails.
PUAD 630: Analytical Techniques in Public Administration
Example 4. Reliability of Treadmill Motors at SureStep
23
Solution:
Use StatTools’s Two-Sample Confidence Interval procedure to find a confidence
interval for the difference between mean lifetimes of the motors of the two
suppliers.
PUAD 630: Analytical Techniques in Public Administration
Example 4.
PUAD 630: Analytical Techniques in Public Administration
Equal-Variance Assumption
25
The two-sample analysis in the previous example makes the assumption that the
standard deviations of the two populations are equal.
How can you tell if they are equal, and what do you do if they are clearly not equal?
A statistical test for equality of two population variances is automatically shown at the
bottom of the StatTools Two-Sample output.
If there is reason to believe that the population variances are unequal, a slightly different
procedure can be used to calculate a confidence interval for the difference between the
means.
The appropriate standard error of
is now:
PUAD 630: Analytical Techniques in Public Administration
Paired Samples
26
When the samples to be compared are paired in some natural way, such as a pretest
and posttest for each person, or husband-wife pairs, there is a more appropriate form
of analysis than the two-sample procedure.
The paired procedure itself is very straightforward:
It does not directly analyze two separate variables (e.g., pretest scores and posttest scores); it
analyzes their differences.
For each pair in the sample, calculate the difference between the two scores for the pair.
Then perform a one-sample analysis on these differences.
PUAD 630: Analytical Techniques in Public Administration
Example 5. Husband and Wife Reactions to Presentations
27
Objective: To use a paired-sample procedure to find a 95% confidence interval for the
mean difference between husbands’ and wives’ ratings of sales presentations.
Data:
A random sample of husbands and wives are asked (separately) to rate the sales
presentation at Stevens Honda-Buick automobile dealership on a scale of 1 to 10.
Solution:
Use StatTools three different ways:
One-Sample Analysis
A new variable for the difference is created.
We then calculate a confidence interval for the mean of the differences, exactly as before
Paired Sample Analysis
Two Sample Analysis
PUAD 630: Analytical Techniques in Public Administration
Example 5.
PUAD 630: Analytical Techniques in Public Administration
29
Confidence Interval for the
Difference between Proportions
PUAD 630: Analytical Techniques in Public Administration
Confidence Interval for the Difference between Proportions
30
The basic form of analysis is the same as in the two-sample analysis for differences
between means.
However, instead of comparing two means, we now compare proportions.
Confidence interval for difference between proportions:
Standard error of difference between sample proportions:
PUAD 630: Analytical Techniques in Public Administration
Example 6. Sales Response to Coupons for Discounts on Appliances
31
Objective: To find a 95% confidence interval for the difference between proportions
of customers purchasing appliances with and without 5% discount coupons.
Data:
An appliance store selects 300 of its best customers and randomly divides them into
two sets of 150 customers each.
It then mails a notice about a sale to all 300 but includes a coupon for an extra 5%
off the sale price to the second set of customers only.
As the sale progresses, the store keeps track of which of these customers purchase
appliances.
PUAD 630: Analytical Techniques in Public Administration
Example 6. Sales Response to Coupons for Discounts on Appliances
32
The sample difference is 13.33%, and the objective is to find a confidence interval
for this difference for the entire population.
Solution:
Use StatTools to find a confidence interval
for the difference between two proportions.
Click on the Format button in the dialog
box and check the Stacked option. This is
because the Yes and No values are stacked
on top of one another in column B.
There are two categorical variables.
The subpopulations you are comparing:
whether they received a coupon or didn't.
Response = whether they purchased or not.
PUAD 630: Analytical Techniques in Public Administration
Example 6.
PUAD 630: Analytical Techniques in Public Administration
34
Sample Size Selection
PUAD 630: Analytical Techniques in Public Administration
Sample Size Selection
35
Confidence intervals are a function of three things:
Data in the sample directly affect the length of a confidence interval through their sample
standard deviation(s).
There are random sampling plans that can reduce the amount of variability in the sample
and hence reduce confidence interval length.
Variance reduction is also possible in randomized experiments.
As confidence level increases, the length of the confidence interval increases as well.
However, the confidence level is rarely used to control the length of the confidence
interval.
Instead, confidence level choice is usually based on convention, and 95% is by far the
most commonly used value.
Sample size(s) is/are the most obvious way to control confidence interval length is to choose
the sample size(s) appropriately.
PUAD 630: Analytical Techniques in Public Administration
Sample Size Selection
36
The goal is to make the length of a confidence interval sufficiently narrow.
Each confidence interval discussed so far (with the exception of the confidence
interval for a standard deviation) is a point estimate plus or minus some quantity.
The “plus or minus” part is called the half-length of the interval.
The usual approach is to specify the half-length B you would like to obtain. Then you find
the sample size(s) necessary to achieve this half-length.
PUAD 630: Analytical Techniques in Public Administration
Sample Size Selection for Estimation of the Mean
37
The appropriate sample size for estimation of the mean can be calculated from the
formula for the confidence interval for the mean,
by setting
and solving for n:
Unfortunately, sample size selection must be done before a sample is observed, and
value s is not yet available.
The usual solution is to replace s by some reasonable estimate σest of the population
standard deviation, and to replace the t-multiple with the corresponding z-multiple from the
standard normal distribution.
The resulting sample size formula is:
PUAD 630: Analytical Techniques in Public Administration
Example 7. Sample Size Selection for Estimating Reaction
38
Objective: To find the sample size of customers required to achieve a sufficiently
narrow confidence interval for the mean rating of the new sandwich.
Data:
The fast-food manager in Example 1 surveyed 40 customers, each of whom rated a
new sandwich on a scale of 1 to 10.
The sample standard deviation was 1.597
Based on the data, a 95% confidence interval for the mean rating of all potential
customers extended from 5.739 to 6.761, with a half-length of 0.511.
How large a sample would be needed to reduce this half-length to approximately 0.3?
Solution:
Using the formula:
we can calculate:
These calculations can also be performed using StatTools
StatTools Toolbar > Statistical Inference>Sample Size Selection
PUAD 630: Analytical Techniques in Public Administration
Example 7.
PUAD 630: Analytical Techniques in Public Administration
Sample Size Selection for Estimation of Other Parameters
40
The sample-size analysis for the mean carries over with very few changes to other
parameters.
Sample size formula for estimating a proportion:
Sample size formula for estimating the difference between means:
Sample size formula for estimating the difference between proportions:
PUAD 630: Analytical Techniques in Public Administration
Example 8. Sample Size Selection for Estimating the Proportion
41
Objective: To find the sample size of customers required to achieve a sufficiently
narrow confidence interval for the proportion of customers who have tried the new
sandwich.
Data:
The data set is the same as in Example 1.
Now the fast-food manager wants to look at the proportion of customers who have
tried the new sandwich.
If she is fairly sure that 30% of customers have tried the new sandwich, she can
estimate the proportion to be = 0.3.
She wants to estimate a 90% and 95% confidence interval for this proportion to have
half-length 0.05.
PUAD 630: Analytical Techniques in Public Administration
Example 8.
PUAD 630: Analytical Techniques in Public Administration
Example 9. Sample Size Selection for Difference of Means
43
Objective: To see how many employees in each experimental group must be sampled
to achieve a sufficiently narrow confidence interval for the difference between the
mean numbers of complaints.
Data:
A customer service center has two types of employees: those who have had a recent
course in dealing with customers (but little actual experience) and those with a lot of
experience dealing with customers (but no formal course).
The company wants to estimate the difference between the two types of employees
in terms of the average number of customer complaints regarding poor service in
the last six months.
The company plans to obtain information on a randomly selected sample of each
type of employee, using equal sample sizes.
The company estimates that the common standard deviation for each group of
employees is about 5 complaints.
How many employees should be in each sample to achieve a 95% confidence interval
with approximate half-length 2?
PUAD 630: Analytical Techniques in Public Administration
Example 9.
PUAD 630: Analytical Techniques in Public Administration
Example 10. Sample Size Selection for Analyzing Difference of
Proportions
45
Objective: To see how many products in each plant must be sampled to achieve a
sufficiently narrow confidence interval for the difference between the proportions of
out-of-spec products.
Data:
A supervisor at a company with two plants wants to know how much the proportion
of out-of-spec products differs across the two plants.
He suspects the proportion of out-of-spec products in each plant is in the range of
about 5%, and he wants a 99% confidence interval to have approximate half-length
0.005.
What is the sample size required from each of the two samples?
If his initial calculations yield a sample size that is prohibitive, he decreases the
confidence level to 95% and increases the desired half-length to 0.025.
What is the sample size required now?
PUAD 630: Analytical Techniques in Public Administration
Example 10.
PUAD 630: Analytical Techniques in Public Administration
PUAD 630
ANALYTICAL TECHNIQUES IN PUBLIC ADMINISTRATION
F ERZANA H AVEWALA
Hypothesis Testing
Introduction
1
In hypothesis testing, an analyst collects sample data and checks whether the data
provide enough evidence to support a theory, or hypothesis.
The hypothesis that an analyst is attempting to prove is called the alternative
hypothesis.
It is also frequently called the research hypothesis.
The opposite of the alternative hypothesis is called the null hypothesis.
It usually represents the current thinking or status quo.
That is, it is usually the accepted theory that the analyst is trying to disprove.
The burden of proof is on the alternative hypothesis.
PUAD 630: Analytical Techniques in Public Administration
The basic research situation for hypothesis testing
2
It is assumed that the parameter μ is known for the population before treatment.
The purpose of the study is to determine whether the treatment has an effect on the
population mean.
PUAD 630: Analytical Techniques in Public Administration
A Research Study
3
From the point of view of the hypothesis test, the entire population receives the treatment
and then a sample is selected from the treated population.
In the actual research study, however, a sample is selected from the original population and
the treatment is administered to the sample.
From either perspective, the result is a treated sample that represents the treated population.
PUAD 630: Analytical Techniques in Public Administration
Hypothesis Testing
4
The purpose of the hypothesis test is to decide between two explanations:
The difference between the sample and the population can be explained by sampling error
(there does not appear to be a treatment effect).
The difference between the sample and the population is too large to be explained by
sampling error (there does appear to be a treatment effect).
PUAD 630: Analytical Techniques in Public Administration
The Hypothesis Test: Step 1
5
State the hypothesis about the unknown population.
The null hypothesis, H0, states that there is no change in the general population before
and after an intervention.
In the context of an experiment, H0 predicts that the independent variable had no effect on
the dependent variable.
The alternative hypothesis, H1, states that there is a change in the general population
following an intervention.
In the context of an experiment, predicts that the independent variable did have an effect
on the dependent variable.
PUAD 630: Analytical Techniques in Public Administration
Developing
Hypotheses
DevelopingNull
Nulland
and Alternative
Alternative Hypotheses
6
It is not always obvious how the null and alternative hypotheses should be
formulated.
Care must be taken to structure the hypotheses appropriately so that the test
conclusion provides the information the researcher wants.
The context of the situation is very important in determining how the hypotheses
should be stated.
In some cases it is easier to identify the alternative hypothesis first. In other cases
the null is easier.
Correct hypothesis formulation will take practice.
PUAD 630: Analytical Techniques in Public Administration
Alternative Hypothesis as a Research Hypothesis
7
Many applications of hypothesis testing involve an attempt to gather evidence in
support of a research hypothesis.
In such cases, it is often best to begin with the alternative hypothesis and make it the
conclusion that the researcher hopes to support.
The conclusion that the research hypothesis is true is made if the sample data
provide sufficient evidence to show that the null hypothesis can be rejected.
PUAD 630: Analytical Techniques in Public Administration
Developing Null and Alternative Hypotheses
8
Alternative Hypothesis as a Research Hypothesis
Example:
A new teaching method is developed that is believed to be better than the current
method.
Alternative
Null
Hypothesis: The new teaching method is better.
Hypothesis: The new method is no better than the old method.
PUAD 630: Analytical Techniques in Public Administration
Developing Null and Alternative Hypotheses
9
Alternative Hypothesis as a Research Hypothesis
Example:
A new sales force bonus plan is developed in an attempt to increase sales.
Alternative
Null
Hypothesis: The new bonus plan increase sales.
Hypothesis: The new bonus plan does not increase sales.
PUAD 630: Analytical Techniques in Public Administration
Developing Null and Alternative Hypotheses
10
Alternative Hypothesis as a Research Hypothesis
Example:
A new drug is developed with the goal of lowering blood pressure more than the
existing drug.
Alternative
Hypothesis: The new drug lowers blood pressure more than the
existing drug.
Null
Hypothesis: The new drug does not lower blood pressure more than the
existing drug.
PUAD 630: Analytical Techniques in Public Administration
Null Hypothesis as an Assumption to be Challenged
11
We might begin with a belief or assumption that a statement about the value of a
population parameter is true.
We then using a hypothesis test to challenge the assumption and determine if there is
statistical evidence to conclude that the assumption is incorrect.
In these situations, it is helpful to develop the null hypothesis first.
PUAD 630: Analytical Techniques in Public Administration
Developing
Hypotheses
DevelopingNull
Nulland
and Alternative
Alternative Hypotheses
12
Null Hypothesis as an Assumption to be Challenged
Example:
The label on a soft drink bottle states that it contains 67.6 fluid ounces.
Null
Hypothesis: The label is correct.
Alternative
µ ≥ 67.6 ounces.
Hypothesis: The label is incorrect. µ < 67.6 ounces
PUAD 630: Analytical Techniques in Public Administration
The Set of Potential Samples
13
The set of potential samples is divided into those that are likely to be obtained and those that
are very unlikely to be obtained if the null hypothesis is true.
PUAD 630: Analytical Techniques in Public Administration
Summary
of
Forms
for
Null
and
Alternative
Developing Null and Alternative Hypotheses
Hypotheses about a Population
Mean
14
The equality part of the hypotheses always appears in the null hypothesis.
In general, a hypothesis test about the value of a population mean
µ must take one
of the following three forms (where µ0 is the hypothesized value of the population
mean).
𝐻0 : 𝜇 ≥ 𝜇0
𝐻𝑎 : 𝜇 < 𝜇0
𝐻0 : 𝜇 ≤ 𝜇0
𝐻0 : 𝜇 = 𝜇0
𝐻𝑎 : 𝜇 > 𝜇0
𝐻𝑎 : 𝜇 ≠ 𝜇0
One-tailed
(lower-tail)
One-tailed
(upper-tail)
Two-tailed
PUAD 630: Analytical Techniques in Public Administration
Uncertainty and Errors in Hypothesis Testing
15
Hypothesis testing is an inferential process, which means that it uses limited
information as the basis for reaching a general conclusion.
Specifically, a sample provides only limited or incomplete information about the whole
population, and yet a hypothesis test uses a sample to draw a conclusion about the
population.
In this situation, there is always the possibility that an incorrect conclusion will be made.
PUAD 630: Analytical Techniques in Public Administration
Type I Errors
16
A Type I error occurs when a researcher rejects a null hypothesis that is actually
true.
In a typical research situation, a Type I error means the researcher concludes that a
treatment does have an effect when in fact it has no effect.
A Type I error occurs when a researcher unknowingly obtains an extreme,
nonrepresentative sample.
Fortunately, the hypothesis test is structured to minimize the risk that this will occur.
The alpha level for a hypothesis test is the probability that the test will lead to a
Type I error.
That is, the alpha level determines the probability of obtaining sample data in the critical
region even though the null hypothesis is true.
PUAD 630: Analytical Techniques in Public Administration
Type II Errors
17
A Type II error occurs when a researcher fails to reject a null hypothesis that is really
false.
In a typical research situation, a Type II error means that the hypothesis test has failed to detect a
real treatment effect.
A type II error occurs when the alternative hypothesis is true but there isn’t enough evidence in
the sample to reject the null hypothesis.
A Type II error occurs when the sample mean is not in the critical region even though the
treatment has an effect on the sample.
Often this happens when the effect of the treatment is relatively small.
The consequences of a Type II error are usually not as serious as those of a Type I error.
This type of error is traditionally considered less important than a type I error, but it can lead to
serious consequences in real situations.
In general terms, a Type II error means that the research data do not show the results that
the researcher had hoped to obtain.
The researcher can accept this outcome and conclude that the treatment either has no effect or
has only a small effect that is not worth pursuing, or the researcher can repeat the experiment
(usually with some improvement) and try to demonstrate that the treatment really does work.
PUAD 630: Analytical Techniques in Public Administration
Type II Errors
18
Unlike a Type I error, it is impossible to determine a single, exact probability for a
Type II error.
Instead, the probability of a Type II error depends on a variety of factors and therefore is a
function, rather than a specific number.
Nonetheless, the probability of a Type II error is represented by the symbol β, the Greek
letter beta.
The power of a test is 1 minus the probability of a type II error.
It is the probability of rejecting the null hypothesis when the alternative hypothesis is true.
There are several ways to achieve high power, the most obvious of which is to increase
sample size.
PUAD 630: Analytical Techniques in Public Administration
Possible Outcomes of a Statistical Decision
19
PUAD 630: Analytical Techniques in Public Administration
Significance Level
20
To decide how strong the evidence in favor of the alternative hypothesis must be to
reject the null hypothesis, one approach is to prescribe the probability of a type I
error that you are willing to tolerate.
This type I error probability is usually denoted by α
The value of α is called the significance level of the test.
The α level establishes a criterion, or "cut-off", for making a decision about the null
hypothesis.
The α level also determines the risk of a Type I error.
α = .01, α = .05 (most used), α = .001
PUAD 630: Analytical Techniques in Public Administration
Rejection Region
21
The rejection region is the set of sample data that leads to the rejection of the null
hypothesis.
The significance level, α, determines the size of the rejection region.
Sample results in the rejection region are called statistically significant at the α level.
This critical region consists of outcomes that are very unlikely to occur if the null
hypothesis is true. That is, the critical region is defined by sample means that are
almost impossible to obtain if the treatment has no effect.
PUAD 630: Analytical Techniques in Public Administration
Selecting an Alpha Level
22
The primary concern when selecting an alpha level is to minimize the risk of a Type
I error.
Thus, alpha levels tend to be very small probability values.
By convention, the largest permissible value is α = .10
However, as the alpha level is lowered, the hypothesis test demands more evidence
from the research results.
It is important to understand the effect of varying α:
If α is small, such as 0.01, the probability of a type I error is small, and a lot of sample
evidence in favor of the alternative hypothesis is required before the null hypothesis can be
rejected
When α is larger, such as 0.10, the rejection region is larger, and it is easier to reject the null
hypothesis.
PUAD 630: Analytical Techniques in Public Administration
The Critical Region Boundaries
23
The locations of the critical region boundaries
for three different levels of significance: α = .05, α = .01, and α = .001.
PUAD 630: Analytical Techniques in Public Administration
Significance
24
A result is said to be statistically significant if it is very unlikely to occur when the
null hypothesis is true.
That is, the result is sufficient to reject the null hypothesis.
Thus, a treatment has a significant effect if the decision from the hypothesis test is to reject
H0
If the test statistic results are in the critical region, we conclude that the difference is
significant or that the treatment has a significant effect.
In this case, we reject the null hypothesis.
If the mean difference is not in the critical region, we conclude that the evidence
from the sample is not sufficient.
The decision is fail to reject the null hypothesis.
PUAD 630: Analytical Techniques in Public Administration
Sample Means in the Critical Region
25
Sample means that fall in the critical region (shaded area) have a probability < α
--- In this case, the null hypothesis should be rejected.
Sample means that do not fall in the critical region have a probability > α
--- In this case, we fail to reject the null hypothesis.
PUAD 630: Analytical Techniques in Public Administration
Significance from p-values
26
A second approach is to avoid the use of a significance level and instead simply
report how significant the sample evidence is.
This approach is currently more popular.
It is done by means of a p-value.
The
p-value is the probability of seeing a random sample at least as extreme as the
observed sample, given that the null hypothesis is true.
The smaller the p-value, the more evidence there is in favor of the alternative
hypothesis.
Sample evidence is statistically significant at the α level only if the p-value is less than α.
The
advantage of the p-value approach is that you don’t have to choose a
significance value α ahead of time, and p-values are included in virtually all
statistical software output.
PUAD 630: Analytical Techniques in Public Administration
Suggested
Guidelines
for Interpreting
p-Values
Suggested
Guidelines
for Interpreting
p-Values
27
Less than .01: Overwhelming evidence to conclude Ha is true.
Between .01 and .05 : Strong evidence to conclude Ha is true.
Between .05 and .10 : Weak evidence to conclude Ha is true.
Greater than .10: Insufficient evidence to conclude Ha is true.
PUAD 630: Analytical Techniques in Public Administration
Hypothesis Tests and Confidence Intervals
28
The results of hypothesis tests are often accompanied by confidence intervals.
This provides two complementary ways to interpret the data.
There is also a more formal connection between the two, at least for two-tailed tests.
When
using a confidence interval to perform a two-tailed hypothesis test, reject
the null hypothesis if and only if the hypothesized value does not lie inside a
confidence interval for the parameter.
PUAD 630: Analytical Techniques in Public Administration
Practical versus Statistical Significance
29
Statistically significant results are those that produce sufficiently small p-values.
In other words, statistically significant results are those that provide strong evidence in
support of the alternative hypothesis.
Such results are not necessarily significant in terms of importance. They might be
significant only in the statistical sense.
There is always a possibility of statistical significance but not practical significance
with large sample sizes.
By contrast, with small samples, results may not be statistically significant even if they
would be of practical significance.
PUAD 630: Analytical Techniques in Public Administration
Factors That Influence a Hypothesis Test
30
The final decision in a hypothesis test is determined by the value obtained for the z-
score statistic.
Two factors help determine whether the z-score will be large enough to reject H0.
In
a hypothesis test, higher variability can reduce the chances of finding a
significant treatment effect.
Increasing the number of scores in the sample produces a smaller standard error
and a larger value for the z-score.
PUAD 630: Analytical Techniques in Public Administration
Assumptions for Hypothesis Tests with z-Scores
31
It is assumed that the participants used in the study were selected randomly.
The values in the sample must consist of independent observations.
Two events (or observations) are independent if the occurrence of the first event has no
effect on the probability of the second event.
The standard deviation for the unknown population (after treatment) is assumed to
be the same as it was for the population before treatment.
To evaluate hypotheses with z-scores, we have used the unit normal table to identify
the critical region.
This table can be used only if the distribution of sample means is normal.
PUAD 630: Analytical Techniques in Public Administration
Directional Tests
32
In a directional hypothesis test, or a one-tailed test, the statistical hypotheses (H0 and
HA) specify either an increase or a decrease in the population mean.
That is, they make a statement about the direction of the effect.
When a specific direction is expected for the treatment effect, it is possible for the
researcher to perform a directional test.
The first step (and the most critical step) is to state the statistical hypotheses.
The null hypothesis states that there is no treatment effect and that the alternative hypothesis
says that there is an effect.
The two hypotheses are mutually exclusive and cover all of the possibilities.
A one-tailed alternative is one that is supported only by evidence in a single direction.
A two-tailed alternative is one that is supported by evidence in either of two directions.
Once hypotheses are set up, it is easy to detect whether the test is one-tailed or two-tailed.
One-tailed alternatives are phrased in terms of “”.
Two-tailed alternatives are phrased in terms of “≠” .
PUAD 630: Analytical Techniques in Public Administration
Critical Region and Directional Tests
33
The critical region is defined by sample outcomes that are very unlikely to occur if
the null hypothesis is true (that is, if the treatment has no effect).
Because the critical region is contained in one tail of the distribution, a directional test is
commonly called a one-tailed test.
Also
note that the proportion specified by the alpha level is not divided between
two tails, but rather is contained entirely in one tail.
PUAD 630: Analytical Techniques in Public Administration
Comparison of One-Tailed vs. Two-Tailed Tests
34
The major distinction between one-tailed and two-tailed tests is in the criteria they
use for rejecting H0.
A one-tailed test allows you to reject the null hypothesis when the difference between the
sample and the population is relatively small, provided the difference is in the specified
direction.
A two-tailed test requires a relatively large difference independent of direction.
PUAD 630: Analytical Techniques in Public Administration
Hypothesis Tests Procedures
35
Just as we developed confidence intervals for a variety of parameters, we can develop
hypothesis tests for other parameters.
In each case, the sample data are used to calculate a test statistic that has a wellknown sampling distribution.
Then a corresponding p-value measures the support for the alternative hypothesis.
PUAD 630: Analytical Techniques in Public Administration
Hypothesis Tests for a
Population Mean
PUAD 630: Analytical Techniques in Public Administration
Hypothesis Tests for a Population Mean
1
As with confidence intervals, the key to the analysis is the sampling distribution of
the sample mean.
If you subtract the true mean from the sample mean and divide the difference by the
standard error, the result has a t distribution with n – 1 degrees of freedom.
In a hypothesis-testing context, the true mean to use is the null hypothesis value, specifically,
the borderline value between the null and alternative hypotheses.
value is usually labeled μ0.
To run the test, referred to as the t test for a population mean, you calculate the
test statistic as shown below:
This
PUAD 630: Analytical Techniques in Public Administration
Example 1. A New Pizza Style at Pepperoni Pizza Restaurant
2
The manager of Pepperoni Pizza Restaurant has recently begun experimenting with a
new method of baking pizzas.
He would like to base the decision whether to switch from the old method to the
new method on customer reactions, so he performs an experiment.
For 100 randomly selected customers who order a pepperoni pizza for home
delivery, he includes both an old-style and a free new-style pizza.
He asks the customers to rate the difference between the pizzas on a -10 to +10
scale, where -10 means that they strongly favor the old style, +10 means they strongly
favor the new style, and 0 means they are indifferent between the two styles.
How might he proceed by using hypothesis testing?
PUAD 630: Analytical Techniques in Public Administration
Null and Alternative Hypotheses
3
The manager would like to prove that the new method provides better-tasting pizza,
so this becomes the alternative hypothesis.
The pizza manager’s alternative hypothesis is one-tailed because he is trying to prove that
the new-style pizza is better than the old-style pizza.
The opposite, that the old-style pizzas are at least as good as the new-style pizzas,
becomes the null hypothesis.
He judges which of these are true on the basis of the mean rating over the entire
customer population, labeled μ.
If it turns out that μ≤ 0, the null hypothesis is true.
If μ> 0, the alternative hypothesis is true.
Usually, the null hypothesis is labeled H0 and the alternative hypothesis is labeled Ha
In our example, they can be specified as H0:μ≤ 0 and Ha:μ> 0.
The null and alternative hypotheses divide all possibilities into two nonoverlapping sets,
exactly one of which must be true.
PUAD 630: Analytical Techniques in Public Administration
Types of Errors
4
Regardless of whether the manager decides to accept or reject the null hypothesis, it
might be the wrong decision.
A type I error occurs when he incorrectly rejects a null hypothesis that is true.
A type II error occurs when he incorrectly accepts a null hypothesis that is false.
The traditional hypothesis-testing procedure favors caution in terms of rejecting the
null hypothesis. Given this rather conservative way of thinking, you are inclined to
accept the null hypothesis unless the sample evidence provides strong support for the
alternative hypothesis.
PUAD 630: Analytical Techniques in Public Administration
Example 1. A New Pizza Style at Pepperoni Pizza Restaurant
5
Objective: To use a one-sample t test to see whether consumers prefer the new-style
pizza to the old style.
Data: The ratings for the 40 randomly selected customers are shown.
Solution:
To run the test, calculate the test statistic
using the borderline null hypothesis
value = 0, and report how much
probability is beyond it in the right tail
of the appropriate t distribution.
The right tail is appropriate because
the alternative is one-tailed of the
“greater than” variety.
StatTools>Statistical Inference> Hypothesis
Test>Mean/Std. Deviation
PUAD 630: Analytical Techniques in Public Administration
Example 1.
PUAD 630: Analytical Techniques in Public Administration
Example 1. A New Pizza Style at Pepperoni Pizza Restaurant
6
A two-tailed test would be relevant if the pizza manager were deciding whether to
use the old method or switch to the new method.
Calculate the same t-statistic, but
because of the two-tailed nature
of the test, the previous p-value
is doubled.
PUAD 630: Analytical Techniques in Public Administration
Example 1.
(continued)
PUAD 630: Analytical Techniques in Public Administration
Hypothesis Tests for a
Population Proportion
PUAD 630: Analytical Techniques in Public Administration
Hypothesis Tests for a Population Proportion
1
To test a population proportion p, recall that the sample proportion has a sampling
distribution that is approximately normal when the sample size is reasonably large.
Specifically, the distribution of the standardized value
is approximately normal with mean 0 and standard deviation 1.
This leads to the following
z test for a population proportion.
Let p0
be the borderline value of p between the null and alternative hypotheses.
Then p0 is substituted for p to obtain the test statistic below:
PUAD 630: Analytical Techniques in Public Administration
Example 2. Customer Complaints
2
Objective: To use a test for a proportion to see whether a new process of responding
to complaint letters results in an acceptably low proportion of unsatisfied customers.
Data:
With the new process in place, the
manager has tracked 400 letter writers
and found that 23 of them are
“unsatisfied” after 30 days.
The manager’s goal is to reduce the
proportion of unsatisfied customers
after 30 days to 7.5% or less
StatTools > Statistical Inference > Hypothesis
Test > Proportion
PUAD 630: Analytical Techniques in Public Administration
48
Hypothesis Tests for a Population Proportion
Example 2.
PUAD 630: Analytical Techniques in Public Administration
Hypothesis Tests for
Differences between
Population Means
PUAD 630: Analytical Techniques in Public Administration
Hypothesis Tests for Differences between Population Means
1
The comparison problem, where the difference between two population means is
tested, is one of the most important problems analyzed with statistical methods.
The form of the analysis depends on whether the two samples are independent or paired.
If the samples are paired, then the test is referred to as the t test for difference
between means from paired samples.
If the samples are independent, the test is referred to as the t test for difference
between means from independent samples.
Test statistic for independent samples test of difference between means:
where
PUAD 630: Analytical Techniques in Public Administration
Example 3. Measuring the Effects of Soft-Drink Cans
2
Objective: To use paired-sample t tests for differences between means to see whether
consumers rate the attractiveness, and their likelihood to purchase, higher for a newstyle can than for the traditional-style can.
Data:
180 randomly selected customers are asked to rate the attractiveness and the likelihood
of purchasing both the traditional-style and the new-style cans on a scale of 1 to 7:
Attractiveness of the traditional-style can (AO)
Attractiveness of the new-style can (AN)
Likelihood of purchasing a product with the traditional-style can (WBO)
Likelihood of purchasing a product with the new-style can (WBN)
PUAD 630: Analytical Techniques in Public Administration
Example 3. Measuring the Effects of Soft-Drink Cans
3
A paired-sample procedure makes the most efficient use of the data.
There are several differences of interest.
The two most obvious are the difference between the attractiveness ratings of the two styles
and the difference between the likelihoods of buying the two styles
—that is, AO minus AN and WBO and WBN.
A third difference of interest is the difference between the attractiveness ratings of the new
style and the likelihoods of buying the new can—that is, AN minus WBN. This difference
indicates whether perceptions of the new-style can are likely to translate into actual sales.
Finally, a fourth difference that might be of interest is the difference between the third
difference (AN minus WBN) and the similar difference for the old style (AO minus WBO).
This checks whether the translation of perceptions into sales is any different for the two
styles of cans
PUAD 630: Analytical Techniques in Public Administration
Example 3. Measuring the Effects of Soft-Drink Cans
4
Run the one-sample procedure on each of the difference variables
PUAD 630: Analytical Techniques in Public Administration
54
Example 3.
PUAD 630: Analytical Techniques in Public Administration
Example 4. Productivity Due to Exercise
5
Objective: To use a two-sample t test for the difference between means to see whether
regular exercise increases worker productivity.
Data:
Informatrix Software Company installed exercise equipment on site a year ago and
wants to know if it has had an effect on productivity.
The company gathered data on a sample of 80 randomly chosen employees:
23 used the exercise facility regularly, 6 exercised regularly elsewhere, and 51 admitted
to being nonexercisers.
The 51 nonexercisers were compared to the 29 exercisers based on the employees’
productivity over the year, as rated by their supervisors on a scale of 1 to 25, with 25
being the best.
Is there a difference in productivity ratings between employees who exercise regularly
and employees who don’t exercise?
PUAD 630: Analytical Techniques in Public Administration
Example 4.
PUAD 630: Analytical Techniques in Public Administration
Hypothesis Tests for
Differences between
Population Proportions
PUAD 630: Analytical Techniques in Public Administration
Hypothesis Tests for Differences between Population Proportions
1
One of the most common uses of hypothesis testing is to test whether two
population proportions are equal.
The following z test for difference between proportions can then be used.
As usual, the test on the difference between the two values requires a standard error.
Standard error for difference between sample proportions:
Resulting test statistic for difference between proportions:
PUAD 630: Analytical Techniques in Public Administration
Example 5. Employee Empowerment
2
Objective: To use a test for the difference between proportions to see whether a
program of accepting employee suggestions is appreciated by employees.
Data:
ArmCo Company initiated a number of policies to respond to employee suggestions
at its Midwest plant, but no such initiatives were taken at its other plants.
To check whether the initiatives had a lasting effect, 100 randomly selected
employees at the Midwest plant and 300 employees from the other plants were asked
to fill out a questionnaire six months after implementation of the new policies at the
Midwest plant.
Two specific items on the questionnaire were:
Management at this plant is generally responsive to employee suggestions for
improvements in the manufacturing process.
Management at this plant is more responsive to employee suggestions now than it
used to be.
PUAD 630: Analytical Techniques in Public Administration
Example 5. Employee Empowerment
3
Use a test for the difference between proportions to test the following hypotheses:
Management at the Midwest plant is
more responsive than at the other plants
(as perceived by employees)
Things have improved more at the
Midwest plant than at the other plants.
PUAD 630: Analytical Techniques in Public Administration
.
61
Example 5.
PUAD 630: Analytical Techniques in Public Administration
Tests for Normality
PUAD 630: Analytical Techniques in Public Administration
Tests for Normality
1
Many statistical procedures are based on the assumption that population data are
normally distributed.
The tests that allow you to test this assumption are called tests for normality.
The first test is called a chi-square goodness-of-fit test.
A histogram of the sample data is compared to the expected bell-shaped histogram that
would be observed if the data were normally distributed with the same mean and standard
deviation as in the sample.
If the two histograms are sufficiently similar, the null hypothesis of normality is accepted.
The goodness-of-fit measure in the equation below is used as a test statistic.
PUAD 630: Analytical Techniques in Public Administration
Example 6. Metal Strips
2
Objective: To use the chi-square goodness-of-fit test to see whether a normal
distribution of the metal strip widths is reasonable.
Data:
A company manufactures strips of metal that are supposed to have width of 10
centimeters.
For purposes of quality control, the manager plans to run some statistical tests on
these strips.
Realizing that these statistical procedures generally assume normally distributed
widths, he needs to first test this normality assumption on 90 randomly sampled
strips.
Does the distribution of metal strip widths appear to follow a normal distribution?
PUAD 630: Analytical Techniques in Public Administration
Example 6.
PUAD 630: Analytical Techniques in Public Administration
Example 6. Metal Strip Widths
3
A more powerful test than the chi-square test of normality is the Lilliefors test.
This test is based on the cumulative distribution function (cdf), which shows the probability
of being less than or equal to any particular value.
Specifically, the Lilliefors test compares two cdfs: the cdf from a normal distribution and the
cdf corresponding to the given data.
This latter cdf, called the empirical cdf, shows the fraction of observations less than or
equal to any particular value.
If the maximum vertical distance between the two cdfs is sufficiently large, the null
hypothesis of normality can be rejected.
PUAD 630: Analytical Techniques in Public Administration
Example 6.
(continued)
PUAD 630: Analytical Techniques in Public Administration
Example 6. Metal Strip Widths
4
A popular, but informal, test of normality is the quantile-quantile (Q-Q) plot.
It is basically a scatterplot of the standardized values from the data set versus the values that
would be expected if the data were perfectly normally distributed.
The Q-Q plot for the Width data appears below.
PUAD 630: Analytical Techniques in Public Administration
Chi-Square Test for
Independence
PUAD 630: Analytical Techniques in Public Administration
Chi-Square Test for Independence
1
The chi-square test for independence is used in situations where a population is
categorized in two different ways.
For example, people might be characterized by their smoking habits and their drinking
habits. The question then is whether these two attributes are independent in a probabilistic
sense.
They are independent if information on a person’s drinking habits is of no use in predicting
the person’s smoking habits (and vice versa).
The null hypothesis for this test is that the two attributes are independent.
This test is based on the counts in a contingency (or cross-tabs) table.
It tests whether the row variable is probabilistically independent of the column variable.
PUAD 630: Analytical Techniques in Public Administration
Example 7. Relationship between Demands for Laptops
2
Objective: To use the chi-square test of independence to test whether demand for
Windows laptops is independent of demand for Mac laptops.
Data:
Big Office wants to know whether the demands for Windows and Mac laptops are
related in any way.
Big Office has daily information on categories of demand for 250 days, with each
day’s demand for each type of computer categorized as Low, Medium Low, Medium
High, or High.
PUAD 630: Analytical Techniques in Public Administration
Example 7. Relationship Between Demands for Laptops
3
Find the test statistic for chi-square test for independence:
Find the expected counts assuming row and column independence:
Perform the calculations for the test by selecting Chi-Square Independence Test
from the StatTools Statistical Inference dropdown list.
PUAD 630: Analytical Techniques in Public Administration
73
Example 7.
PUAD 630: Analytical Techniques in Public Administration

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