Deck 10: Comparing Two Means and Two Proportions

Question 1

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.

Answer 1

True

Question 2

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 $n _ { 1 } \text { and } n _ { 2 } = 10$
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.

Answer 2

False

Question 3

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

Answer 3

True

Question 4

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

Question 5

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

Question 6

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 7

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.

Question 8

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 9

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 10

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 11

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

Question 12

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 13

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 14

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

Question 15

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.

Question 16

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 17

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 18

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 19

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

Question 20

In order to test the effectiveness of a drug called XZR designed to reduce cholesterol levels,9 heart patients' cholesterol levels 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 9 patients' 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 21

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: ?_A > ?_B, $\bar { X } _ { 1 } = 12$
$\bar { X } _ { 2 } = 9$
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 22

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

A)z
B)t
C)F
D)t²

Question 23

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 24

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 these

Question 25

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 26

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

A)A ratio variable
B)A qualitative variable
C)An interval variable
D)A categorical variable

Question 27

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 these

Question 28

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 29

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 30

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 31

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 32

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 33

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

A)z statistic.
B)t statistic.
C)F statistiC.
D)Chi-square statistic.
E)None of thesE.

Question 34

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 35

In order to test the effectiveness of a drug called XZR designed to reduce cholesterol levels,9 heart patients' cholesterol levels 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 9 patients' 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 36

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 ≥ 0
D)P_Internet - P_store ≤ .10
E)P_Internet - P_store > .10

Question 37

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 38

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 39

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 40

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 41

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: ?_A > ?_B, $\bar { X } _ { 1 } = 12$
$\bar { X } _ { 2 } = 9$
S₁ = 5,s₂ = 3,n₁ = 13,n₂ = 10.

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

Question 42

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

Question 43

Using a 90 percent confidence interval for the difference between the proportions of failures in factory 1 and factory 2,where
= .05,
= .04,n₁ = 500,n₂ = 2000 of [-.0076,.0276],can we reject the null hypothesis at α = .10?

Question 44

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

Question 45

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 46

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

Question 47

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

Question 48

When we test H₀: μ₁ ≤ μ₂,H_A: μ₁ > μ₂ at α = .10,where
= 77.4,
= 72.2,s₁ = 3.3,s₂ = 2.1,n₁ = 6,n₂ = 6,can we reject the null hypothesis?

Question 49

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

Question 50

Given two independent normal distributions with s₁² - s₂² = 100,?₁ = ?₂ = 50,n₁ = n₂ = 50,the sampling distribution of the mean difference $\bar { X } _ { 1 } - \bar { X } _ { 2 }$
Will have a mean of _________.

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

Question 51

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

Question 52

When testing H₀: μ₁ - μ₂ = 2,H_A: μ₁ - μ₂ > 2,where
= 522,
= 516,s₁² = 28,s₂² = 24,n₁ = 40,n₂ = 30,at α = .01,what can we conclude?

Question 53

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: ?_A > ?_B, $\bar { Y } _ { 1 } = 12$
$\bar { X } _ { 2 } = 9$
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 54

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,n₂ = 2000.

Question 55

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,n₂ = 1000.

Question 56

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 57

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

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

Question 58

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 in the calculation of the test statistic?

Question 59

Find a 98 percent confidence interval for the paired difference.

Question 60

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

Question 61

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

Question 62

A fast food company uses two management-training methods.Method 1 is a traditional method of training and Method 2 is a new and innovative method.The company has just hired 31 new management trainees.15 of the trainees are randomly selected and assigned to the first method,and the remaining 16 trainees are assigned to the second training method.After three months of training,the management trainees took a standardized test.The test was designed to evaluate their performance and learning from training.The sample mean score and sample standard deviation of the two methods are given below.The management wants to determine if the company should implement the new training method.

Is there evidence at α = .05 to conclude that the new training method is more effective than the traditional training method?

Question 63

Let p₁ represent the population proportion of U.S.senatorial and congressional (House of Representatives)Democrats who are in favor of a new modest tax on "junk food".Let p₂ represent the population proportion of U.S.senatorial and congressional Republicans who are in favor of a new modest tax on "junk food." Out of the 265 Democratic senators and members of Congress,106 of them are in favor of a "junk food" tax.Out of the 285 Republican senators and members of Congress,only 57 are in favor a "junk food" tax.At α = .01,can we conclude that the proportion of Democrats who favor a "junk food" tax is more than 5 percent higher than the proportion of Republicans who favor the new tax?

Question 64

Find a 99 percent confidence interval for the difference between means,given that n₁ = 49,n₂ = 49,

Question 65

Calculate the t statistic for testing equality of means where

Question 66

At α = .10,testing the hypothesis that the proportion of consumer industry companies' (CON)winter-quarter profit growth is more than 2 percent greater than the proportion of banking companies' (BKG)winter-quarter profit growth,given that

n_CON = 300,and n_BKG = 400,can we reject the null hypothesis?

Question 67

A fast-food company uses two management-training methods.Method 1 is a traditional method of training,and Method 2 is a new and innovative method.The company has just hired 31 new management trainees.15 of the trainees are randomly selected and assigned to Method 1,and the remaining 16 trainees are assigned to Method 1 .After three months of training,the management trainees take a standardized test.The test is designed to evaluate their performance and learning from the training.The sample mean score and sample standard deviation of the two methods are given below.Company management wants to determine whether the company should implement the new training method.

What is the absolute value of the rejection point (critical value of the test statistic)at α = .05?

Question 68

At α = .10,testing the hypothesis that the proportion of consumer industry companies' (CON)winter-quarter profit growth is more than 2 percent greater than the proportion of banking companies' (BKG)winter-quarter profit growth,given that

n_CON = 300,and n_BKG = 400,calculate the estimated standard deviation for the model.

Question 69

A fast-food company uses two management-training methods.Method 1 is a traditional method of training,and Method 2 is a new and innovative method.The company has just hired 31 new management trainees.15 of the trainees are randomly selected and assigned to Method 1,and the remaining 16 trainees are assigned to Method 2.After three months of training,the management trainees take a standardized test.The test is designed to evaluate their performance and learning from the training.The sample mean score and sample standard deviation of the two methods are given below.Company management wants to determine whether the company should implement the new training method.

Write the null and alternative hypotheses.

Question 70

When we test H₀: μ₁ - μ₂ ≤ 0,H_A: μ₁ - μ₂ > 0,
= 15.4,
= 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 71

Find a 95 percent confidence interval for μ₁ - μ₂,where n₁ = 9,n₂ = 6,

s₁² = 6,and s₂² = 3.(Assume equal population variances. )

Question 72

In an opinion survey,a random sample of 1000 adults from the United States and 1000 adults from Germany were asked whether they supported the death penalty.590 American adults and 560 German adults indicated that they supported the death penalty.The researcher wants to know whether there is sufficient evidence to conclude that the proportion of adults who support the death penalty is higher in the United States than in Germany.What is the rejection point (critical value of the test statistic)at α = .10?

Question 73

Let p₁ represent the population proportion of U.S.senatorial and congressional (House of Representatives)Democrats who are in favor of a new modest tax on "junk food." Let p₂ represent the population proportion of U.S.senatorial and congressional Republicans who are in favor of a new modest tax on "junk food." Out of the 265 Democratic senators and members of Congress,106 of them are in favor of a "junk food" tax.Out of the 285 Republican senators and members of Congress,only 57 are in favor of a "junk food" tax.Find a 95 percent confidence interval for the difference between proportions l and 2.

Question 74

Calculate the pooled variance where 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.

Question 75

A fast food company uses two management-training methods.Method 1 is a traditional method of training and Method 2 is a new and innovative method.The company has just hired 31 new management trainees.15 of the trainees are randomly selected and assigned to the first method,and the remaining 16 trainees are assigned to the second training method.After three months of training,the management trainees took a standardized test.The test was designed to evaluate their performance and learning from training.The sample mean score and sample standard deviation of the two methods are given below.The management wants to determine if the company should implement the new training method.

What is the sample value of the test statistic?

Question 76

A fast food company uses two management-training methods.Method 1 is a traditional method of training and Method 2 is a new and innovative method.The company has just hired 31 new management trainees.15 of the trainees are randomly selected and assigned to the first method,and the remaining 16 trainees are assigned to the second training method.After three months of training,the management trainees took a standardized test.The test was designed to evaluate their performance and learning from training.The sample mean score and sample standard deviation of the two methods are given below.The management wants to determine if the company should implement the new training method.

What is the absolute value of the rejection point (critical value of the test statistic)at α = .01?

Question 77

A fast food company uses two management-training methods.Method 1 is a traditional method of training and Method 2 is a new and innovative method.The company has just hired 31 new management trainees.15 of the trainees are randomly selected and assigned to the first method,and the remaining 16 trainees are assigned to the second training method.After three months of training,the management trainees took a standardized test.The test was designed to evaluate their performance and learning from training.The sample mean score and sample standard deviation of the two methods are given below.The management wants to determine if the company should implement the new training method.

Is there evidence at α = .01 to conclude that the new training method is more effective than the traditional training method?

Question 78

Testing the equality of means at α = .05,where 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,can we reject the null hypothesis? (Assume equal population variances. )

Question 79

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

Question 80

Determine the 95 percent confidence interval for the difference between two population means,where 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.(Assume equal population variances. )