QUESTIONS FOR REVIEW AND CRITICAL THINKING/ANSWERS
1. What is the purpose of a statistical hypothesis?
Hypotheses were defined in Chapter 3 as formal statements of explanations stated in a testable form.
Generally, hypotheses should be stated in concrete fashion so that the method of empirical testing seems
almost obvious. In statistics classes, students are exposed to hypothesis testing as a contrast between a
2. What is a significance level? How does a researcher choose a significance level?
A significance level is a critical probability associated with a statistical hypothesis test that indicates how
likely an inference supporting a difference between an observed value and some statistical expectation is
3. What is the difference between a significance level and a p-value?
4. How is a p-value used to test a hypothesis?
5. Distinguish between Type I and Type II error.
A Type I error occurs when a condition that is true in the population is rejected based on statistical
observations. When a researcher sets an acceptable significance level a-priori (), he/she is determining
If the alternative condition is in fact true but we conclude that we should not reject the null hypothesis, we
make a Type II error. This type of error is the probability of failing to reject a false null hypothesis. This
6. What are the factors that determine the choice of the appropriate statistical technique?
Making the correct choice can be determined by considering:
1. The type of question to be answered.
7. A researcher is asked to determine whether or not a sales objective (in dollars) of better than $75,000
per salesperson is possible. A market test is done involving 20 salespeople. What conclusion would
you reach? Their sales results are as follows:
a. 28,000 105,000 58,000 93,000 96,000
One-Sample Statistics
N Mean Std. Deviation
Std. Error
Mean
V1 19 78000.00 15742.723 3611.628
One-Sample Test
Test Value = 75000
t
Mean
95% Confidence
Interval of the
Difference
Lower
8. Assume you have the following data: H1:
m
≠ 200, S = 30, n = 64, and
X
= 218. Conduct a
two-tailed hypothesis test at the 0.05 significance level.
H1:
m
≠ 200
n = 64
X
= 218
Z
c . 1 .
= 1.96 at 0.05 significance level
S
X
=
S n = 30 8 = 3 . 75
m
=
X ± Z
c . 1 .
S
X
m
= 218 ± 1.96 (3.75)
m
= 218 ± 7.35
m
= 210.65 — 225.35
The confidence interval does not include 200, therefore the null hypothesis is not rejected.
9. If the data in Question 8 had been generated with a sample of 25 (n = 25), what statistical test would
have been appropriate?
10. The answers to a researcher’s question will be nominally scaled. What statistical test is appropriate
to compare the sample data with hypothesized population data?
11. A researcher plans to ask employees whether they favor, oppose, or are indifferent about a change
in the company retirement program. Formulate a hypothesis for a chi square test and the way the
variable would be created.
A researcher has considerable flexibility to determine the expected distribution of answers for chi square
12. Give an example in which a Type I error may be more serious than a Type II error.
Students’ responses will vary. Here’s one example: If the null hypothesis is that there has been no change
13. Refer to the pizza store location
2
data on page 523. What statistical decisions could be made if
the 0.01 significance level were selected rather than the 0.05 level?
The critical
2
value with one degree of freedom at the .01 significance level is 6.635. If the appropriate
2
14. Determine a hypothesis that the data below may address and perform a 2 test on the survey data.
a. The X Factor should be broadcast before 9:00 p.m.
Agree 40
Neutral 35
Disagree 25
100
15. A researcher hypothesizes that 15 percent of people in a test market will recall seeing a particular
advertisement. In a sample of 1,200, 20 percent of the people say they recall the ad. Perform a
hypothesis test.
H0:
p
= .15
H1:
p
¹
.15
Sample:
Sp=
pq
n
. 2
( )
. 8
( )
1200
Sp= [
. 16
1200
Sp= .000133
Z =
p – p
S
p
Z =
. 20 – . 15
. 0115
Z =
. 05
. 0115
Z = 4.348
RESEARCH ACTIVITIES
1. [Internet Question] What is the ideal climate? Fill in the following blanks: The lowest temperature in
January should be no lower than ______ degrees. At least ______ days should be sunny in January.
a. List at least 15 places where you would like to live. Using the Internet, find the average
low temperature in January for each place. This information is available through various
b. Using the same website, record how many days in January are typically sunny. Test
c. For each location, record whether or not there was measurable precipitation yesterday.
Students’ analyses will differ depending on the benchmark and locations selected.
2. [Ethics Question] Examine the statistical choices under Analyze in SPSS. Click on Compare Means.
To compare an observed mean to some benchmark or hypothesized population mean, the available
a. What is the p-value? Is the hypothesis supported?
b. Write the 95% confidence interval which corresponds to an of .05.
c. Technically, since the sample size is greater than 30, a Z-test might be more appropriate.
However, since the t-test result is readily available with SPSS, the researcher presents
this result. Is there an ethical problem with using the one-sample t-test?
When sample size (n) is larger than 30, the t-distribution and Z-distribution are almost identical.
CASE 21.1 PREMIER MOTORS
Objective: The objective is for the student to form a statistical hypothesis test, to calculate sample
variance and a standard deviation, and to conduct a hypothesis test to measure whether or not the
hypothesis should be accepted or rejected.
Summary: Premier Motors is an automobile dealership that regularly advertises in its local market area.
Premier has claimed that a certain make and model of car averages 30 miles to a gallon of gasoline. In its
advertising, Premier Motors mentions that the miles per gallon may vary with driving conditions. A local
consumer group challenges the advertising claim.
Questions
1. Formulate a statistical hypothesis appropriate for the consumer group’s purpose.
The question should be stated as a two-tail hypothesis test. H0:
m
= 30, H1:
m ¹
30. The consumer
2. Calculate the mean average miles per gallon. Compute the sample variance and sample standard
deviation.
Mean 28.172
Median 28.7
3. Construct the appropriate statistical test for your hypothesis, using a 0.05 significance level.
H0:
m
= 30
H1:
m ¹
30
Using a hand calculation the results are as follows:
m = X ± Z
c . 1 .
S
X
Upper Limit
m
=
X + Z S n
( )
m
= 28.172 + 1.96 [3.03/
25
]
m
m
= 29.359
Lower Limit
m
=
X – Z S n
( )
m
= 28.172 – 1.96 [3.03/
25
]
m
= 28.172 – 1.877
m
= 26.98
Thus, the confidence interval for the hypothesis test would be:
m
= 26.98 — 29.359
This suggests the researchers accept the null hypothesis at the 95 percent level of significance.
Descriptive Statistics
25 21.90 35.10 28.1720 3.0339
25
Miles per gallon
Valid N (listwise)
N Minimum Maximum Mean Std. Deviation
One-Sample Statistics
25 28.1720 3.0339 .6068
Miles per gallon
N Mean Std. Deviation
Std. Error
Mean
One-Sample Test
46.429 24 .000 28.1720 26.9197 29.4243
Miles per gallon
t df Sig. (2-tailed)
Mean
Difference Lower Upper
95% Confidence
Interval of the
Difference
Test Value = 0
One-Sample Test
41.484 24 .000 25.1720 23.9197 26.4243
Miles per gallon
t df Sig. (2-tailed)
Mean
Difference Lower Upper
95% Confidence
Interval of the
Difference
Test Value = 3.0