Inference for Distributions
Moore IPS Chapter 7
© 2012 W.H. Freeman and Company
7.1-1
It has been claimed that women live longer than men; however, men tend to
be older than their wives.
Ages of 16 husbands and wives from England were obtained. These data
should be analyzed with a:
a. two-sample t test.
b. paired samples t test.
c. two-sample z test.
7.1 Inference for the Mean of a Population
7.1-2
An SRS of 20 third-grade children is selected and each is given a test to
measure their reading ability. We are interested in a 90% confidence interval for
the population mean score. In the sample, the mean score is 64 points and the
standard deviation is 12 points.
The margin of error associated with the confidence interval is:
a. 2.68 points.
b. 4.64 points.
s
12
t19 ×
= 1.729 ×
= 4.64
n
64
c. 5.62 points
7.1 Inference for the Mean of a Population
7.1-2c
C
A SRS of 20 third-grade children is selected and each is given a test to measure
their reading ability. We are interested in a 90% confidence interval for the
population mean score. In the sample, the mean score is 64 points and the
standard deviation is 12 points.
The margin of error associated with the confidence interval is
a. 2.68 points.
b. 4.64 points.
c. 5.62 points
t19 ×
s
12
= 1.729 ×
= 4.64
n
64
7.1 Inference for the Mean of a Population
7.1-3
You are thinking of using a t procedure to construct a 95% confidence
interval for the mean of a population. You suspect the distribution of the
population is not normal and may be skewed. Which of the following
statements is correct?
a. You should not use the t procedure since the population does not
have a normal distribution.
b. You may use the t procedure provided your sample size is large, say
at least 50.
c. You may use the t procedure since it is robust to nonnormality.
Large samples : The t procedures can be used even for clearly skewed
distributions when the sample is large, roughly n ≥ 40 or 50.
7.1 Inference for the Mean of a Population
7.1-4
I took a sample of the grade point averages for students in my class.
For 25 students, the standard deviation of grade points was 0.65 and
the mean was 2.89. The standard error for the sample was:
a. 0.578.
b. 0.026.
c. 0.130.
s
0.65
=
= 0.13
n
25
7.1 Inference for the Mean of a Population
7.1-5
The one-sample t statistic from a sample of n = 19 observations for the
two-sided test of
H 0: m = 6
H a: m ≠ 6
has the value t = 1.93. Based on this information:
a. we would reject the null hypothesis at a = 0.10.
b. 0.025 < P-value < 0.05.
c. both (a) and (b) are correct.
7.1 Inference for the Mean of a Population
7.1-6
To construct a 92% confidence interval, the correct z* to use is:
a. 1.75.
b. 1.41.
c. 1.645.
7.1 Inference for the Mean of a Population
7.1-6c
C
To construct a 92% confidence interval, the correct z* to use is
a. 1.75
b. 1.41
c. 1.645
7.1 Inference for the Mean of a Population
7.1-7
Below are normal quantile plots for two data sets. For which set(s)
is(are) inference using the t distribution valid?
a.
b.
c. Both a and b
d. Neither a nor b
This normal quantile plot shows an approximately straight line, meaning the data came
from an approximately normal population.
7.1 Inference for the Mean of a Population
7.1-8
The manager of an automobile dealership is considering a new bonus
plan to increase sales. Currently, the mean sales rate per salesperson
is five automobiles per month. The correct set of hypotheses to test the
effect of the bonus plan is
a. H0: m = 5, Ha: m > 5
b. H0: m > 5, Ha: m = 5
c. H0:
d. H0:
x = 5, Ha: x > 5
x = 5, Ha: x= 5
7.1 Inference for the Mean of a Population
7.1-9
As the degrees of freedom become larger, the difference between the t
and z distributions:
a. becomes smaller.
b. stays the same.
c. becomes larger.
7.1 Inference for the Mean of a Population
7.1-10
The heights (in inches) of adult males in the U.S. are believed to be
normally distributed with mean µ. The average height of a random
sample of 25 American adult males is found to be x= 69.72 inches,
and the standard deviation of the 25 heights is found to be s = 4.15.
A 90% confidence interval for µ is:
a. 69.72 ± 1.09.
x
±
t24
×
b. 69.72 ± 1.37.
c. 69.72 ± 1.42.
s
n
4.15
69.72 ± 1.711 ×
25
7.1 Inference for the Mean of a Population
7.1-10
C
The heights (in inches) of adult males in the U.S. are believed to be
normally distributed with mean µ. The average height of a random
sample of 25 American adult males is found to be x= 69.72 inches
and the standard deviation of the 25 heights is found to be s = 4.15.
A 90% confidence interval for µ is
a. 69.72 ± 1.09
b. 69.72 ± 1.37
c. 69.72 ± 1.42
71.14 - 69.72 = 1.42
7.1 Inference for the Mean of a Population
7.2-1
A researcher wants to know if calcium is an effective treatment for
lowering blood pressure. He assigns one randomly chosen group of
subjects to take calcium supplements; the other group will get placebo.
At the end of the treatment period, he measures the difference in blood
pressure. The 50 members of the calcium group had blood pressure
lowered by an average of 25 points with standard deviation 10
points. The 50 members of the placebo group had blood pressure
lowered by an average of 15 points with standard deviation 8
points.
To analyze this information we will use a:
a. two-sample t procedure.
b. two-sample z procedure.
c. matched pairs procedure.
7.2 Comparing Two Means
7.2-2
It has been claimed that women live longer than men; however, men
tend to be older than their wives. Ages of 16 husbands and wives
from England were obtained. The null hypothesis of equality of means
is rejected. What conclusion can be made from this study?
a. English husbands are older than their wives.
b. Husbands are older than their wives.
c. The sample was too small. No conclusions can be made.
7.2 Comparing Two Means
7.2-3
The math department at a university is hiring a new professor.
Each of two candidates for the job will be interviewed by five current
department members. Each member will score candidates on a
scale of 1 to 10 according to several important criteria. From the
candidates’ applications, the department head suspects that
Candidate 1 is more qualified for the job than Candidate 2. This
data is best analyzed using:
a. one sample t procedure.
b. two sample t procedures.
c. matched pairs procedure.
7.2 Comparing Two Means
7.2-4
A local farmer is interested in comparing the yields of two varieties of tomatoes.
In an experimental field, she selects 40 locations and assigns 20 plants from
each variety at random to the locations. She determines the average per plant
(in pounds). She computes a 95% confidence interval for the difference in
mean yields between the two varieties using the two-sample t procedures with
the resulting interval (2.13, 6.41). For testing H 0 : m1 = m2 H a : m1 ¹ m2 using the
two-sample t procedures we can say that:
a. the P-value could be greater than 0.05.
b. the P-value must be less than 0.05.
c. no information about the P-value can be obtained without the test statistic.
7.2 Comparing Two Means
7.2-5
A local farmer is interested in comparing the yields of two varieties of
tomatoes. In an experimental field, he selects 20 locations and assigns
10 plants from each variety at random to the locations. He determines
the yield per plant (in pounds).
The mean yield for plants of variety 1 was 16.3 pounds with a standard
deviation of 3 pounds. The mean yield for plants of variety 2 was 18.4
pounds with a standard deviation of 4 pounds. The standard error of
the difference in sample means is:
a. 2.10 pounds.
b. 2.50 pounds.
c. 1.58 pounds.
7.2 Comparing Two Means
7.2-6
The water diet requires you to drink two cups of water every half hour
from when you get up until you go to bed, but eat anything you want.
Four adult volunteers agreed to test this diet. They are weighed prior to
beginning the diet and six weeks after. Their weights in pounds are:
Person
1
2
3
4
mean
Weight before
180
125
240
150
173.75
49.56
Weight after
170
130
215
152
166.75
36.09
Difference
10
-5
25
-2
??
s.d.
13.64
What is the mean of the differences?
a. 170.25
b. 7
10+(-5)+25+(-2) / 4 = 7
c. 13.64
7.2 Comparing Two Means
7.2-7
The water diet requires you to drink two cups of water every half hour from
when you get up until you go to bed, but eat anything you want. Four adult
volunteers agreed to test this diet. They are weighed prior to beginning the
diet and six weeks after. Their weights in pounds are:
Person
1
2
3
4
mean
Weight before
180
125
240
150
173.75
49.56
Weight after
170
130
215
152
166.75
36.09
Difference
10
-5
25
-2
??
s.d.
13.64
What is the appropriate alternative hypothesis, to see if the diets leads to
weight loss?
a. μd = 0
Because this is a before and after experiment for weight loss, a
b. μd > 0
positive value indicates you have lost weight.
c. μd < 0
7.2 Comparing Two Means
7.2-8
The water diet requires you to drink two cups of water every half hour
from when you get up until you go to bed, but eat anything you want.
Four adult volunteers agreed to test this diet. They are weighed prior to
beginning the diet and six weeks after. Their weights in pounds are:
Person
1
2
3
Weight before
180
125
240
150
173.75
49.56
Weight after
170
130
215
152
166.75
36.09
Difference
10
-5
25
-2
What is the test statistic?
a. 0.06
b. 1.03
4
7
t=
13.64
mean
??
s.d.
13.64
= 1.03
4
c. 2.35
7.2 Comparing Two Means
7.2-9
The water diet requires you to drink two cups of water every half hour
from when you get up until you go to bed, but eat anything you want.
Four adult volunteers agreed to test this diet. They are weighed prior to
beginning the diet and six weeks after. Their weights in pounds are:
Person
1
2
3
4
mean
Weight before
180
125
240
150
173.75
49.56
Weight after
170
130
215
152
166.75
36.09
Difference
10
-5
25
-2
??
s.d.
13.64
What is the rejection region at the 5% level of significance?
a. t > 2.353
b. t > 3.182
from Table C with 3df and 5% level of significance
for a one sided test
c. t > 1.943
7.2 Comparing Two Means
7.3-1
An American political party analyst was curious whether younger
adults were becoming more conservative. He wanted to see if the
mean age of registered Republicans was lower than that of
registered Conservatives. He selected an SRS of 15 Republicans
and found the sample mean to be 39 with standard deviation of 4
years, while an SRS of 27 Democrats had a mean age of 44.5 and a
standard deviation of 6. Let μ1 and μ2 represent the mean ages of
the populations of all registered Republicans and Democrats,
respectively.
The critical value (t*) for a 90% confidence interval for μ1 – μ2 is:
a. 1.645
b. 1.684
Since max S/ min S < 2, we use the pooled method of
analysis. This will yield a critical value with n1+n2-2
degrees of freedom.
c. 1.761
d. 1.753
7.3 Optional Topics in Comparing Distributions
7.3-2
Like the t procedures for means, the F test and other procedures for
standard deviations are robust against non-normal population
distributions  especially for larger samples.
a. True
b. False
7.3 Optional Topics in Comparing Distributions
7.3-3
Inference procedures for comparing the standard deviations of two
normal populations are based on what distribution?
a. Chi-square
b. F
c. t
d. z
7.3 Optional Topics in Comparing Distributions
7.3-4
The degrees of freedom associated with the F distribution are:
a. n-1.
b. n1-1 (numerator) and n2-1 (denominator).
c. n1-1 (denominator) and n2-1 (numerator).
d. There are no degrees of freedom for the F distribution.
7.3 Optional Topics in Comparing Distributions
7.3-5
When computing the power of a pooled two-sample t test, if the sample
size is increased, what is the effect on the power of the test?
a. Increases
b. Decreases
c. Remains the same
7.3 Optional Topics in Comparing Distributions