Inference for Distributions

Moore IPS Chapter 7

© 2012 W.H. Freeman and Company

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

7.1-1

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.

c. 5.62 points

7.1-2

7.1 Inference for the Mean of a Population

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

7.1-2c

7.1 Inference for the Mean of a Population

C

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.

7.1-3

7.1 Inference for the Mean of a Population

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.

7.1-4

7.1 Inference for the Mean of a Population

The one-sample t statistic from a sample of n = 19 observations for the

two-sided test of

H0: = 6

Ha: ≠ 6 has the value t = 1.93. Based on this information:

a. we would reject the null hypothesis at = 0.10.

b. 0.025 < P-value < 0.05.

c. both (a) and (b) are correct.

7.1-5

7.1 Inference for the Mean of a Population

To construct a 92% confidence interval, the correct z* to use is:

a. 1.75.

b. 1.41.

c. 1.645.

7.1-6

7.1 Inference for the Mean of a Population

To construct a 92% confidence interval, the correct z* to use is

a. 1.75

b. 1.41

c. 1.645

7.1-6c

7.1 Inference for the Mean of a Population

C

Below are normal quantile plots for two data sets. For which set(s)

is(are) inference using the t distribution valid?

a. c. Both a and b

7.1-7

7.1 Inference for the Mean of a Population

This normal quantile plot shows an approximately straight line, meaning the data came from an approximately normal population.

d. Neither a nor bb.

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: = 5, Ha: > 5

b. H0: > 5, Ha: = 5

c. H0: = 5, Ha: > 5

d. H0: = 5, Ha: = 5

7.1-8

7.1 Inference for the Mean of a Population

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.

7.1-9

7.1 Inference for the Mean of a Population

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 = 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.

7.1-10

7.1 Inference for the Mean of a Population

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 = 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

7.1-10

7.1 Inference for the Mean of a Population

C

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.

7.2-1

7.2 Comparing Two Means

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.

7.2-2

7.2 Comparing Two Means

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.

7.2-3

7.2 Comparing Two Means

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 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.

7.2-4

7.2 Comparing Two Means

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.

7.2-5

7.2 Comparing Two Means

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 s.d.

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 ?? 13.64

What is the mean of the differences?

a. 170.25

b. 7

c. 13.64

7.2-6

7.2 Comparing Two Means

10+(-5)+25+(-2) / 4 = 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 s.d.

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 ?? 13.64

What is the appropriate alternative hypothesis, to see if the diets leads to weight loss?

a. μd = 0

b. μd > 0

c. μd < 0

7.2-7

7.2 Comparing Two Means

Because this is a before and after experiment for weight loss, a positive value indicates you have lost weight.

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 s.d.

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 ?? 13.64

What is the test statistic?

a. 0.06

b. 1.03

c. 2.35

7.2-8

7.2 Comparing Two Means

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 s.d.

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 ?? 13.64

What is the rejection region at the 5% level of significance?

a. t > 2.353

b. t > 3.182

c. t > 1.943

7.2-9

7.2 Comparing Two Means

from Table C with 3df and 5% level of significance for a one sided test

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

c. 1.761

d. 1.753

7.3-1

7.3 Optional Topics in Comparing Distributions

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.

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

7.3-2

7.3 Optional Topics in Comparing Distributions

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

7.3-3

7.3 Optional Topics in Comparing Distributions

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.

7.3-4

7.3 Optional Topics in Comparing Distributions

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

7.3-5

7.3 Optional Topics in Comparing Distributions