Deck 11: Statistical Inferences Based on Two Samples

Question 1

In an experiment involving matched pairs, a sample of 12 pairs of observations is collected. The degrees of freedom for the t statistic is 10.

Answer 1

False

Question 2

In testing the difference between two population variances, it is a common practice to compute the F statistic so that its value is always greater than or equal to one.

Answer 2

True

Question 3

There are two types of machines, called type A and type B. Both type A and type B can be used to produce a certain product. The production manager wants to compare efficiency of the two machines. He assigns each of the 15 workers to both types of machines to compare their hourly production rate. In other words, each worker operates machine A and machine B for one hour each. These two samples are independent.

Answer 3

False

Question 4

In testing the difference between the means of two normally distributed populations using large independent random samples, the sample sizes from the two populations must be equal.

Question 5

In testing the difference between two means from two independent populations, the sample sizes do not have to be equal.

Question 6

If the limits of the confidence interval of the difference between the means of two normally distributed populations were from −2.6 to 1.4 at the 95 percent confidence level, then we can conclude that we are 95 percent certain that there is a significant difference between the two population means.

Question 7

The controller of a chain of toy stores is interested in determining whether there is any difference in the weekly sales of store 1 and store 2. The weekly sales are normally distributed. This problem should be analyzed using an independent means method.

Question 8

In forming a confidence interval for μ₁ − μ₂, only two assumptions are required: independent samples and sample sizes of at least 30.

Question 9

In testing the equality of population variances, two assumptions are required: independent samples and normally distributed populations.

Question 10

When testing the difference between two proportions selected from populations with large independent samples, the z test statistic is used.

Question 11

When comparing two population means based on independent random samples, the pooled estimate of the variance is used when there is an assumption of equal population variances.

Question 12

Assume that we are constructing a confidence interval for the difference in the means of two populations based on independent random samples. If both sample sizes n₁ and n₂ =10, and the distributions of both populations are highly skewed, then a confidence interval for the difference in the means can be constructed using the t test statistic.

Question 13

If the limits of the confidence interval of the difference between the means of two normally distributed populations were 8.5 and 11.5 at the 95 percent confidence level, then we can conclude that we are 95 percent certain that there is a significant difference between the two population means.

Question 14

When comparing the variances of two normally distributed populations using independent random samples, if s₁² = s₁², the calculated value of F will always be equal to one.

Question 15

When we are testing a hypothesis about the difference in two population proportions based on large independent samples, we compute a combined (pooled) proportion from the two samples if we assume that there is no difference between the two proportions in our null hypothesis.

Question 16

In testing the difference between the means of two normally distributed populations using independent random samples, the alternative hypothesis always indicates no difference between the two specified means.

Question 17

When comparing two independent population means, if n₁ = 13 and n₂ = 10, degrees of freedom for the t statistic is 22.

Question 18

In testing the difference between the means of two normally distributed populations using independent random samples, we can only use a two-sided test.

Question 19

In testing the difference between the means of two independent populations, if neither population is normally distributed, then the sampling distribution of the difference in means will be approximately normal, provided that the sum of the sample sizes obtained from the two populations is at least 30.

Question 20

An independent samples experiment is an experiment in which there is no relationship between the measurements in the different samples.

Question 21

In testing the difference between the means of two normally distributed populations using independent random samples with equal variances, the correct test statistic to use is the

A) z statistic.
B) t statistic.
C) F statistic.
D) chi-square statistic.
E) None of the other choices is correct.

Question 22

In order to test the effectiveness of a drug called XZR designed to reduce cholesterol levels, the cholesterol levels of 9 heart patients are measured before they are given the drug. The same 9 patients use XZR for two continuous months. After two months of continuous use, the cholesterol levels are measured again. The comparison of cholesterol levels before versus after administering the drug is an example of testing the difference between

A) two means from independent populations.
B) two population variances from independent populations.
C) two population proportions.
D) matched pairs from two dependent populations.

Question 23

The F statistic can assume either a positive or a negative value.

Question 24

A new company is in the process of evaluating its customer service. The company offers two types of sales: (1) Internet sales and (2) store sales. The marketing research manager believes that the Internet sales are more than 10 percent higher than store sales. The null hypothesis would be

A) P_Internet − P_store > .10.
B) P_Internet − P_store < .10.
C) P_Internet − P_store ≥ .10.
D) P_Internet − P_store ≤ .10.
E) P_Internet − P_store = .10.

Question 25

A new company is in the process of evaluating its customer service. The company offers two types of sales: (1) Internet sales and (2) store sales. The marketing research manager believes that the Internet sales are more than 10 percent higher than store sales. The alternative hypothesis for this problem would be stated as

A) P_Internet − P_store > 0.
B) P_Internet − P_store < 0.
C) P_Internet − P_store ≥ .10.
D) P_Internet − P_store ≤ .10.
E) P_Internet − P_store > .10.

Question 26

When comparing two independent population means by using samples selected from two independent, normally distributed populations with equal variances, the correct test statistic to use is ________.

A) z
B) t
C) F
D) t²

Question 27

A financial analyst working for a financial consulting company wishes to find evidence that the average price-to-earnings ratio in the consumer industry is higher than the average price-to-earnings ratio in the banking industry. The alternative hypothesis is

A) μ_consumer = μ_banking.
B) μ_consumer ≤ μ_banking.
C) μ_consumer > μ_banking.
D) μ_consumer < μ_banking.
E) μ_consumer ≠ μ_banking.

Question 28

When testing a hypothesis about the mean of a population of paired differences in which two different observations are taken on the same units, the correct test statistic to use is ________.

A) z
B) t
C) F
D) chi-square
E) None of the other choices is correct.

Question 29

In testing the difference between two means from two normally distributed independent populations, the distribution of the difference in sample means will be

A) normally distributed only if sample sizes are equal.
B) normally distributed only if both population standard deviations are known.
C) normally distributed.
D) normally distributed if both sample sizes are very large.
E) normally distributed only if both population variances are equal.

Question 30

The value of F_α in a particular situation depends on the size of the right-hand tail area and the numerator degrees of freedom.

Question 31

The exact shape of the curve of the F distribution depends on two parameters, df₁ and df₂.

Question 32

If we are testing the difference between the means of two normally distributed independent populations with samples of n₁ = 10, n₂ = 10, the degrees of freedom for the t statistic is ________.

A) 19
B) 18
C) 9
D) 8
E) 20

Question 33

An experiment in which two different measurements are taken on the same units and inferences are made using the differences between the pairs of measurements is a(n) ________ experiment.

A) paired difference
B) equal variances
C) independent samples
D) dependent samples

Question 34

When testing the difference between two population proportions using large independent random samples, the ________ test statistic is used.

A) z
B) t
C) F
D) chi-square
E) None of the other choices is correct.

Question 35

Given the following information about a hypothesis test of the difference between two means based on independent random samples, which one of the following is the correct rejection region at a significance level of .05? H_A: μ_A > μ_B, ₁ = 12, ₂ = 9, s₁ = 4, s₂ = 2, n₁ = 13, n₂ = 10.

A) Reject H₀ if t > 1.96.
B) Reject H₀ if t > 1.645.
C) Reject H₀ if t > 1.721.
D) Reject H₀ if t > 2.08.
E) Reject H₀ if t > 1.782.

Question 36

When testing the difference between two population proportions, the ________ test statistic is used.

A) z
B) t
C) F
D) t²

Question 37

In testing for the equality of means from two independent populations, if the hypothesis of equal population means is rejected at α = .01, it will ________ be rejected at α = .05.

A) always
B) sometimes
C) never

Question 38

In which of the following tests is the variable of interest the difference between the values of the observations from the two samples, rather than the actual observations themselves?

A) a test of hypothesis about the mean of a population of paired differences selected from two related samples
B) a test of hypothesis about the difference between the means of two normally distributed populations using independent samples
C) a test of hypothesis about the difference between two population proportions, using large independent random samples
D) a test of hypothesis about the difference between the variances of two normally distributed populations using independent samples

Question 39

An experiment in which there is no relationship between the measurements on the different samples is a(n) ________ experiment.

A) paired difference
B) equal variances
C) independent samples
D) dependent samples

Question 40

If we are testing the hypothesis about the mean of a population of paired differences with samples of n₁ = 10, n₂ = 10, the degrees of freedom for the t statistic is ________.

A) 19
B) 18
C) 9
D) 8
E) 10

Question 41

Two independent samples selected from two normally distributed populations have variances of σ₁² and σ₂² with n₁ = 10 and n₂ = 15. The degrees of freedom for the F distribution when testing the equality of the two population variances are

A) 10 and 15.
B) 11 and 16.
C) 9 and 14.
D) 8 and 13.

Question 42

Given the following information about a hypothesis test of the difference between two means based on independent random samples, what is the standard deviation of the difference between the two means? Assume that the samples are obtained from normally distributed populations having equal variances. H_A: μ_A > μ_B, ₁ = 12, ₂ = 9, s₁ = 5, s₂ = 3, n₁ = 13, n₂ = 10.

A) 1.792
B) 1.679
C) 2.823
D) 3.210
E) 1.478

Question 43

In comparing the difference between two independent population means, the sampling distributions of the population means are at least approximately ________.

A) skewed right
B) skewed left
C) normal
D) binomial

Question 44

Construct a 95 percent confidence interval for μ₁ − μ₂, where ₁ = 34.36, ₂ = 26.45, s₁ = 9, s₂ = 6, n₁ = 10, n₂ = 16. (Assume equal population variances.)

Question 45

If we are testing the hypothesis about the mean of a population of paired differences with samples of n₁ = 8, n₂ = 8, the degrees of freedom for the t statistic is ________.

A) 16
B) 7
C) 14
D) 9

Question 46

When comparing the variances of two normally distributed populations using independent random samples, the correct test statistic to use is ________.

A) z
B) t
C) F
D) chi-square
E) None of the other choices is correct.

Question 47

In testing the difference between two independent population means, if the assumption is of unequal variances, the critical value of the t statistic is obtained by calculating the ________.

A) degrees of freedom
B) sum of the two sample sizes (n₁ + n₂)
C) p-value
D) pooled variance

Question 48

When comparing two independent population variances, the correct test statistic to use is ________.

A) z
B) t
C) F
D) t ²

Question 49

Given the following information about a hypothesis test of the difference between two means based on independent random samples, what is the calculated value of the test statistic? Assume that the samples are obtained from normally distributed populations having equal variances. H_A: μ_A > μ_B, = 12, = 9, s₁ = 5, s₂ = 3, n₁ = 13, n₂ = 10.

A) t = 1.96
B) t = 1.5
C) t = 2.823
D) t = 1.674
E) t = 1.063

Question 50

In testing the difference between the means of two independent populations, the variances of the two samples can be pooled if the population variances are assumed to ________.

A) be unequal
B) be greater than the mean
C) sum to 1
D) be equal

Question 51

The test of means for two related populations matches the observations (matched pairs) in order to reduce the ________ attributable to the difference between individual observations and other factors.

A) means
B) test statistic
C) degrees of freedom
D) variation

Question 52

When testing the difference for the population of paired differences in which two different observations are taken on the same units, the correct test statistic to use is ________.

A) z
B) t
C) F
D) t²

Question 53

Given the following information about a hypothesis, and the test of the difference between two variances based on independent random samples, what is the critical value of the test statistic at a significance level of 0.05? Assume that the samples are obtained from normally distributed populations. H_A: σ²_A > σ²_B, ₁ = 12, ₂ = 9, s₁ = 5, s₂ = 3, n₁ = 13, n₂ = 10.

A) 3.87
B) 2.67
C) 3.07
D) 2.80
E) 2.38

Question 54

In testing the equality of population variance, what assumption(s) should be considered?

A) independent samples
B) equal sample sizes
C) normal distribution of the populations
D) independent samples and equal sample sizes
E) independent samples and normal distribution of the populations

Question 55

In testing the difference between the means of two normally distributed populations, if μ₁ = μ₂ = 50, n₁ = 9, and n₂ = 13, the degrees of freedom for the t statistic equals ________.

A) 22
B) 21
C) 19
D) 20

Question 56

Parameters of the F distribution include

A) n₁.
B) degrees of freedom for the numerator and the denominator.
C) n₂.
D) n₁ and n₂.
E) None of the other choices is correct.

Question 57

In testing the difference between two independent population means, it is assumed that the level of measurement is ________.

A) a ratio variable
B) a qualitative variable
C) an interval variable
D) a categorical variable

Question 58

Given two independent normal distributions with s₁² − s₂²= 100, μ₁ = μ₂ = 50, and n₁ = n₂ = 50, the sampling distribution of the mean difference ₁ − ₂ will have a mean of ________.

A) 100
B) 1
C) 0
D) 50
E) 10

Question 59

In general, the shape of the F distribution is ________.

A) skewed right
B) skewed left
C) normal
D) binomial

Question 60

In order to test the effectiveness of a drug called XZR designed to reduce cholesterol levels, the cholesterol levels of 9 heart patients are measured before they are given the drug. The same 9 patients use XZR for two continuous months. After two months of continuous use, the cholesterol levels are measured again. The comparison of cholesterol levels before versus after the administration of the drug is an example of testing the difference between two ________.

A) samples of equal variances
B) independent samples
C) paired samples
D) samples of unequal variances

Question 61

Using a 90 percent confidence interval of [−.0076, .0276] for the difference between the proportions of failures in factory 1 and factory 2, where = 05, = .24, n₁ = 500, and
n₂ = 2000, can we reject the null hypothesis at α = .10?

Question 62

When we test H₀: μ₁ ≤ μ₂, H_A: μ₁ > μ₂ at α = .10, where ₁ = 77.4, ₂ = 72.2, s₁ = 3.3,
s₂ = 2.1, n₁ = 6, and n₂ = 6, can we reject the null hypothesis (using critical value rules)? (Assume equal variances.)

Question 63

When we test H₀: p₁ − p₂ ≤ .01, H_A: p₁ − p₂ > .01, at α = .05, where = .08, = .035,
n₁ = 200, and n₂= 400, what is the standard deviation used to calculate the test statistic?

Question 64

We are testing the hypothesis that the proportion of winter-quarter profit growth is more than 2 percent greater for consumer industry companies (CON) than for banking companies (BKG). At α = .10, given that = .20, = .14, n_CON = 300, and n_BKG = 400, can we reject the null hypothesis (using critical value rules)?

Question 65

Find a 90 percent confidence interval for the difference between the proportions of failures in factory 1 and factory 2, where = .05, = .04, n₁ = 500, and n₂ = 2000.

Question 66

When testing H₀: μ₁ − μ₂ = 2, H_A: μ₁ − μ₂ > 2, where ₁ = 522, ₂ = 516, σ₁² = 28,
σ₂² = 24, n₁ = 40, n₂ = 30, at α = .01, what is the test statistic? (Assume unequal variances.)

Question 67

Find a 95 percent confidence interval for the difference between means, where n₁ = 50,
n₂ = 36, ₁ = 80, ₂ = 75, s₁² = 5, and s₂² = 3. Assume unequal variances.

Question 68

Find a 95 percent confidence interval for the difference between the proportions of older and younger drivers who have tickets, where = .275, = .25, n₁ = 1000, and n₂ = 1000.

Question 69

When we test H₀: μ₁ − μ₂ ≤ 0, H_A: μ₁ − μ₂ > 0, ₁ = 15.4, ₂ = 14.5, σ₁ = 2, σ₂ = 2.28,
n₁ = 35, and n₂ = 18 at α = .01, what is the value of the test statistic?

Question 70

We are testing the hypothesis that the proportion of winter-quarter profit growth is more than 2 percent greater for consumer industry companies (CON) than for banking companies (BKG). At α = .10, given that = .20, = .14, n_CON = 300, and n_BKG = 400, calculate the estimated standard deviation for the model.

Question 71

When we test H₀: μ₁ − μ₂ ≤ 0, H_A: μ₁ − μ₂ > 0, ₁ = 15.4, ₂ = 14.5, s₁ = 2, s₂ = 2.28,
n₁ = 35, and n₂ = 18 at α = .01, can we reject the null hypothesis? (Assume unequal variances.)

Question 72

When we test H₀: p₁ − p₂ ≤ .01, H_A: p₁ − p₂ > .01 at α = .05 where = .08, = .035,
n₁ = 200, and n₂ = 400, can we reject the null hypothesis?

Question 73

When we test H₀: μ₁ ≤ μ₂, H_A: μ₁ > μ₂ at α = .10, where ₁ = 77.4, ₂ = 72.2, s₁ = 3.3,
s₂ = 2.1, n₁ = 6, and n₂ = 6, what is the estimated pooled variance?

Question 74

Find a 98 percent confidence interval for the paired difference d₁ − d₂ where = 1.6, S_d ²= 40.96, n = 30

Question 75

Find a 95 percent confidence interval for μ₁ − μ₂, where n₁ = 50, n₂ = 75, ₁ = 82, ₂ = 76, s₁² = 8, and s₂² = 6. Assume unequal variances.

Question 76

Find a 90 percent confidence interval for the difference between the proportions of group 1 and group 2. Let p₁ represent the population proportion of the people in group 1 who like a new mobile app, and let p₂ represent the population proportion of the people in group 2 who like a new mobile app. = .21, = .13, n₁ = 300, and n₂ = 400.

Question 77

Sample 1 has data: 16, 14, 19, 18, 19, 20, 15, 18, 17, 18; and sample 2 has data: 13, 19, 14, 17, 21, 14, 15, 10, 13, 15. Testing the equality of means at α = .05, can we reject the null hypothesis (using critical value rules)? (Assume equal population variances.)

Question 78

When testing H₀: μ₁ − μ₂ = 2, H_A: μ₁ − μ₂ > 2, where ₁ = 522, ₂ = 516, s₁² = 28, s₂² = 24, n₁ = 40, n₂ = 30, at α = .01, what can we conclude using critical value rules? (Assume unequal variances.)

Question 79

Find a 95 percent confidence interval for μ₁ − μ₂, where n₁ = 15, n₂ = 10, ₁ = 1.94,
₂ = 1.04, s₁² = .2025, and s₂² = .0676. (Assume equal population variances.)

Question 80

Find a 95 percent confidence interval for μ₁ − μ₂, where n₁ = 9, n₂ = 6, ₁ = 64, ₂ = 59,
s₁² = 6, and s₂² = 3. (Assume equal population variances.)