Deck 10: Hypothesis Testing: Additional Topics

Question 1

If testing for the difference between the means of two related populations,the appropriate null hypothesis is:

A)H₀ : μ_D ≠ 0
B)H₀ : μ_D = 0
C)H₀ : μ_D < 0
D)H₀ : μ_D > 0

Answer 1

H₀ : μ_D = 0

Question 2

Based on the given information,the p-value for the F test of equal variances can be calculated and shown to be 0.289.Based on this p-value,what is your conclusion?

A)The assumption of equal variances is correct.
B)The assumption of equal variances is incorrect.
C)The answer cannot be determined based on this p-value.
D)The variance of the first sample is greater than the other.

Answer 2

The assumption of equal variances is correct.

Question 3

The test of equality of the variances between two normally distributed populations relies on the random variable which has a(n):

A)chi-square distribution.
B)distribution.
C)Student's t distribution.
D)normal distribution.

Answer 3

distribution.

Question 4

What is the value of the test statistic?

A)2.92
B)0.79
C)3.62
D)1.74

Question 5

Given the following information, = 4,
= 6,n₁ = 16,n₂ = 25,calculate the pooled sample variance that should be used in the pooled-variance t test.

A)6.45
B)5.33
C)4.19
D)5.23

Question 6

If testing the difference between the means of two related populations with samples of sizes n₁ = 16 and n₂ = 16,what is the number of degrees of freedom?

A)30
B)28
C)15
D)14

Question 7

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
Two samples each of size 20 are taken from independent populations assumed to be normally distributed with equal variances.The first sample has a mean of 43.5 and standard deviation of 4.1 while the second sample has a mean of 40.1 and standard deviation of 3.2.A researcher would like to test if there is a difference between the population means at the 0.05 significance level.
State the null hypothesis.

A)H₀ : P_x = P_y
B)H₀ :
=

C)H₀ : μ_x - μ_y = 3.4
D)H₀ : μ_x - μ_y = 0

Question 8

In hypothesis testing,such as testing the difference between two population proportions,which of the following statements is correct when the significance level is 0.05?

A)only statistics that occur with a probability of less than .95 when H₀ is true are sufficient evidence to reject H₀.
B)only statistics that occur with a probability of less than .05 when H₀ is true are sufficient evidence to reject H₀.
C)only statistics that occur with a probability of .95 or more when H₀ is true are sufficient evidence to reject H₀.
D)only statistics that occur with a probability of .05 or more when H₀ is true are sufficient evidence to reject H₀.

Question 9

The F distribution is constructed as the ratio of two random variables following a ________ distribution.

A)normal
B)chi-square
C)Student's t
D)Poisson

Question 10

The t test for the mean difference between two related populations assumes that the:

A)population distribution of differences are approximately normal.
B)population sizes are equal.
C)sample variances are equal.
D)sample sizes are small.

Question 11

Which of the following is an appropriate null hypothesis?

A)H₀ : μ_x - μ_y ≠ 0
B)H₀ : P_x > P_y
C)H₀ : μ_x - μ_y < 0
D)H₀ : σ²_x =σ²_y

Question 12

When testing for the difference between the means of two normal populations using two independent samples with equal variances and with sizes of n_x = 20 and n_y = 15,what is the number of degrees of freedom for the t test statistic?

A)35
B)33
C)18
D)13

Question 13

What is the critical value(s)for this test? (Hint: Use Excel to find the critical value(s))

A)± 2.093
B)± 2.024
C)-1.729
D)-1.304

Question 14

If testing for the difference between the means of two related populations,with samples of sizes n₁ = n₂ = n,where the variance of the differences is unknown,what is the number of degrees of freedom?

A)n - 1
B)2n - 1
C)2(n - 1)
D)n - 2

Question 15

What is the pooled variance?

A)3.650
B)5.201
C)13.525
D)12.849

Question 16

What can the researcher conclude?

A)There is not sufficient evidence to reject the null hypothesis;the population means and standard deviations are equal.
B)The two population means are equal but the standard deviations are different.
C)There is sufficient evidence to reject the null hypothesis and conclude that the two population means are different.
D)The answer cannot be determined without calculating the p-value and comparing it to the significance level.

Question 17

In what type of test is the variable of interest the difference between the values of the observations rather than the observations themselves?

A)a test of the equality of variances from 2 independent populations
B)a test of the difference between the means of 2 related populations
C)a test of the difference between the means of 2 independent populations
D)a test for of equality of variances from two related populations.

Question 18

When testing for the difference between the means of two independent populations,with samples of sizes n₁ and n₂,where the population variances are unknown but assumed to be equal,what is the number of degrees of freedom?

A)n - 1
B)n₁ + n₂ - 1
C)n₁ + n₂ - 2
D)n - 2

Question 19

When testing for the difference between the means of two independent populations with equal variances and with samples of sizes n₁ = 20 and n₂ = 20,what is the number of degrees of freedom?

A)40
B)38
C)20
D)19

Question 20

The t test for the difference between the means of two independent populations assumes that the respective:

A)sample sizes are equal.
B)sample variances are equal.
C)populations are approximately normal.
D)population variances are known.

Question 21

What is the value of α,if the cutoff point for an F distribution F_9,23 = 2.32?

A)0.025
B)0.1
C)0.01
D)0.05

Question 22

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
Suppose two food preservatives are extensively tested and determined safe for use in meats.A processor wants to compare the preservatives for their effects on retarding spoilage.Suppose 16 cuts of fresh meat are treated with preservative A and another 12 cuts of meat are treated with preservative B.The number of hours until spoilage begins is recorded for each of the 28 cuts of meat.The results are summarized in the table below.

The most accurate statement that can be made about the p-value for testing whether there is a difference in the population variances for preservatives A and B is:

A)p-value > 0.05
B)p-value = 0.00
C)p-value < 0.01
D)0.01 < p-value < 0.05

Question 23

What conclusion can be made about the effectiveness of the diet,based on the test,at the 0.05 level of significance?

A)The diet is not effective.
B)The diet is effective.
C)The sample size is relatively small to make a conclusion.
D)A conclusion cannot be drawn without calculating the p-value.

Question 24

Consider a standard model with a random sample of n_x observations from a population with a proportion P_x of successes and a second independent random sample of n_y observations from a population with a proportion P_y of successes.Which of the following formulas illustrates normally distributed random variables when testing the difference between two population proportions?

A)Z =
B)Z =
C)Z =
D)Z =

Question 25

What is the value of α,if the cutoff point for an F distribution F_4,8 = 7.01?

A)0.025
B)0.1
C)0.05
D)0.01

Question 26

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
Suppose two food preservatives are extensively tested and determined safe for use in meats.A processor wants to compare the preservatives for their effects on retarding spoilage.Suppose 16 cuts of fresh meat are treated with preservative A and another 12 cuts of meat are treated with preservative B.The number of hours until spoilage begins is recorded for each of the 28 cuts of meat.The results are summarized in the table below.

What assumptions are necessary for a comparison of the population variances to be valid?

A)At least one of the population variances is known.
B)The samples are correlated.
C)One of the samples follows normal distribution and the other follows F distribution.
D)Both sampled populations are normally distributed.

Question 27

In testing for differences between the means of two independent populations,the null hypothesis is:

A)H₀ : μ₁ - μ₂ ≠ 1.
B)H₀ : μ₁ - μ₂ = 0.
C)H₀ : μ₁ - μ₂ > 0.
D)H₀ : μ₁ - μ₂ < 1.

Question 28

The statistical distribution that is used for testing the difference between two population variances is the:

A)Student's t distribution.
B)standard normal distribution.
C)binomial distribution.
D)distribution.

Question 29

Consider independent random samples from two normally distributed populations.The first population has a mean of μ_x and a variance of and we obtain a random sample of size n_x.The second population has a mean of μ_y and a variance of
And we obtain a random sample of size n_y.The sample means are denoted by
And
)Which of the following formulae calculates the random variable,Z?

A)Z =
B)Z =
C)Z =
D)Z =

Question 30

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
Suppose two food preservatives are extensively tested and determined safe for use in meats.A processor wants to compare the preservatives for their effects on retarding spoilage.Suppose 16 cuts of fresh meat are treated with preservative A and another 12 cuts of meat are treated with preservative B.The number of hours until spoilage begins is recorded for each of the 28 cuts of meat.The results are summarized in the table below.

The numerator and denominator degrees of freedom associated with the test statistic for determining if there is a difference in the population variances for preservatives A and B are,respectively:

A)16 and 12
B)12 and 16
C)15 and 11
D)11 and 15

Question 31

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
To investigate the effectiveness of a diet,a random sample of 16 female patients is drawn from a population of adult females using the diet.The weight of each individual in the sample is taken at the start and at the end of the diet.Assume that the population of differences in weight before and after the diet follows a normal distribution.Suppose the mean decrease in weights over all 16 subjects in the study is 4.0 pounds with the standard deviation of differences computed as 6.4 pounds.Let µ_x - µ_y = mean weight before the diet - mean weight after the diet.
In order to test if the diet is effective,what is the value of the test statistic?

A)2.5
B)2.0
C)1.8
D)1.3

Question 32

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
To investigate the effectiveness of a diet,a random sample of 16 female patients is drawn from a population of adult females using the diet.The weight of each individual in the sample is taken at the start and at the end of the diet.Assume that the population of differences in weight before and after the diet follows a normal distribution.Suppose the mean decrease in weights over all 16 subjects in the study is 4.0 pounds with the standard deviation of differences computed as 6.4 pounds.Let µ_x - µ_y = mean weight before the diet - mean weight after the diet.
In order to test if the diet is effective,what is the appropriate statistical test?

A)test of the difference between two normal population means using independent samples with known population variances
B)test of the difference between two normal population means using independent samples with unknown,but assumed to be equal,population variances
C)test of the difference between two normal population means using independent samples with unknown,but not equal,population variances
D)test of the difference between two normal population means using dependent (matched pairs)samples

Question 33

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
To investigate the effectiveness of a diet,a random sample of 16 female patients is drawn from a population of adult females using the diet.The weight of each individual in the sample is taken at the start and at the end of the diet.Assume that the population of differences in weight before and after the diet follows a normal distribution.Suppose the mean decrease in weights over all 16 subjects in the study is 4.0 pounds with the standard deviation of differences computed as 6.4 pounds.Let µ_x - µ_y = mean weight before the diet - mean weight after the diet.
In order to test if the diet is effective,what is the appropriate alternative hypothesis?

A)H₁ : μ_d < 0
B)H₁ : μ_d > 0
C)H₁ : μ_d = 0
D)H₁ : μ_d ≠ 0

Question 34

Given the following information, = 4,
= 6,n₁ = 15,n₂ = 20,calculate the degrees of freedom that should be used in the pooled-variance t test.

A)35
B)33
C)31
D)29

Question 35

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
Suppose two food preservatives are extensively tested and determined safe for use in meats.A processor wants to compare the preservatives for their effects on retarding spoilage.Suppose 16 cuts of fresh meat are treated with preservative A and another 12 cuts of meat are treated with preservative B.The number of hours until spoilage begins is recorded for each of the 28 cuts of meat.The results are summarized in the table below.

The value of the test statistic for determining if there is a difference in the population variances for preservatives A and B is equal to:

A)0.784
B)1.625
C)1.275
D)1.129

Question 36

What is the cutoff point for an F distribution with 10 numerator degrees of freedom and 13 denominator degrees of freedom at α = 0.01?

A)4.93
B)4.71
C)4.10
D)2.67

Question 37

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
To investigate the effectiveness of a diet,a random sample of 16 female patients is drawn from a population of adult females using the diet.The weight of each individual in the sample is taken at the start and at the end of the diet.Assume that the population of differences in weight before and after the diet follows a normal distribution.Suppose the mean decrease in weights over all 16 subjects in the study is 4.0 pounds with the standard deviation of differences computed as 6.4 pounds.Let µ_x - µ_y = mean weight before the diet - mean weight after the diet.
In order to test if the diet is effective,what is the appropriate critical value(s)at the 0.05 level of significance?

A)2.131
B)2.120
C)1.753
D)1.746

Question 38

The test statistic for the equality of two population variances is based on the:

A)ratio of the two sample variances.
B)difference between the two sample variances.
C)difference between the two population variances.
D)difference between the sample variances divided by the difference between the sample means.

Question 39

When testing for the difference between two population variances with sample sizes of n₁ = 10 and n₂ = 12,what is (are)the number (s)of degrees of freedom?

A)10 and 12
B)9 and 11
C)22
D)20

Question 40

What is the cutoff point for an F distribution with 20 numerator degrees of freedom and 30 denominator degrees of freedom at α = 0.05? (Hint: Round your answer to 2 decimal places. )

A)1.93
B)2.04
C)2.55
D)2.07

Question 41

Assume that we have independent random samples of size n_x and n_yobservations from normal populations with means μ_x and μ_y and unequal variances.The sample variances and
Are used and the means are
And
)Which of the following formulae calculates the degree of freedom,v,for the Student's t statistic where variance is unknown and considered unequal?

A)ν =
B)ν =
C)ν =
D)ν =

Question 42

When testing for differences between the means of two independent populations,only a t-test can be used.

Question 43

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
A manager is considering the purchase of new production equipment to improve the output of the plant.Suppose the equipment from two suppliers is extensively tested in multiple production runs.A total of 20 runs are recorded on Supplier A's equipment and another 15 runs are reported from Supplier B's equipment.The volume produced is recorded for each of the 35 runs.The results are summarized in the table below.

The value of the test statistic for determining if there is a difference in the population variances for Supplier A and B is equal to:

A)0.44
B)1.5
C)2.25
D)0.67

Question 44

A null hypothesis of the difference between two population proportions is rejected at the 5% level,but not at the 2.5% level.This means that the p-value of the test is between 0.025 and 0.05.

Question 45

Suppose that a t test of H₀ : μ₁ - μ₂ = 0 against H₁ : μ₁ - μ₂ ≠ 0 is based on 15 degrees of freedom.If the calculated value of the test statistic is t = -1.75,then the p-value for this test is approximately 0.10.

Question 46

In a hypothesis test of the difference between two population means,if the probability of Type II error is 0.25,then:

A)the probability of rejecting H₀,given that H₀ is false is 0.75.
B)the probability of Type I error is 0.75.
C)the probability of rejecting H₀,given that H₀ is true,is 0.75.
D)the probability of accepting H₁,given that H₀ is true is 0.75.

Question 47

If two population variances are known,the test for differences in the two population means does not require the populations to be normally distributed if the sample sizes are large.

Question 48

If the null hypothesis H₀ : μ_x - μ_y ≤ 0 is rejected against the alternative H₁ : μ_x - μ_y > 0 at the 5% level of significance,then using the same data,it must be rejected against that alternative at the 1% level.

Question 49

For two independent random samples that each follows a normal distribution,if the population variances are equal,then the random variable F is calculated using the formula:

A)F =
*

B)F =
C)Z =
D)F =

Question 50

The p-value of testing H₀ : P_x - P_y = 0 against H₁ : P_x - P_y ≠ 0 is the probability that the null hypothesis is true.

Question 51

For two independent random samples with n_x and n_y observations,and each following a normal distribution,which of the following would be the F distribution degrees of freedom in the numerator?

A)n_x - 1
B)n_x + 1
C)1 -
D)n_y - 1

Question 52

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
Are medical students more motivated than law students? A randomly selected group of each were administered a survey of attitudes toward Life,which measures motivation for upward mobility.The scores are summarized below.

Judging from the way the data were collected,which test would likely be most appropriate to employ?

A)independent samples Z-test for the difference in means
B)matched pairs t-test for mean difference
C)pooled-variance t-test for the difference in means
D)matched pairs Z-test for mean difference

Question 53

State the null and alternative hypotheses to determine if the mean score of medical students differs from the mean score of law students.

A)H₀ : μ_M - μ_L ≥ 0 versus H₁ : μ_M - μ_L < 0
B)H₀ : μ_M - μ_L = 0 versus H₁ : μ_M - μ_L ≠ 0
C)H₀ : μ_M - μ_L ≤ 0 versus H₁ : μ_M - μ_L > 0
D)H₀ :
_M -

_L = 0 versus H₁ :

_M -

_L ≠ 0

Question 54

Assuming the independent samples procedure was used,which of the following is the correct test statistic.

A)Z =
B)Z =
C)Z =
D)Z =

Question 55

The ________ is constructed as the ratio of two chi-square random variables,each divided by its degrees of freedom.

A)chi-square distribution
B)normal distribution
C)distribution
D)Student's t distribution

Question 56

Hypothesis tests that use the F distribution do not depend on the assumption that the populations are normally distributed as long as the sample sizes are large.

Question 57

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
A manager is considering the purchase of new production equipment to improve the output of the plant.Suppose the equipment from two suppliers is extensively tested in multiple production runs.A total of 20 runs are recorded on Supplier A's equipment and another 15 runs are reported from Supplier B's equipment.The volume produced is recorded for each of the 35 runs.The results are summarized in the table below.

The numerator and denominator degrees of freedom associated with the test statistic for determining if there is a difference in the population variances for Supplier A and B are,respectively:

A)20 and 15.
B)15 and 10.
C)14 and 9.
D)19 and 14.

Question 58

If the null hypothesis H₀ : μ_x - μ_y ≥ 0 is rejected against the alternative H₁ : μ_x - μ_y < 0 at the 2% level of significance,then using the same data,it must be rejected against that alternative at the 3% level.

Question 59

Given H₁ : μ_x - μ_y < 0,the hypothesis test is a(n):

A)two-sided alternative.
B)null hypothesis.
C)one-sided test.
D)upper-tailed test.

Question 60

Suppose that the test statistic is Z = 1.95.Find the p-value if we assume that the alternative hypothesis is a two-tailed test (H₁ : μ_M - μ_L ≠ 0).

A)0)0256
B)0)9488
C)0)0512
D)0)9744

Question 61

The sample size in each independent sample must be the same if we are to test for differences between the means of two independent populations.

Question 62

The test for the equality of two population variances assumes that each of the two populations is normally distributed.

Question 63

When testing for the differences between the means of two independent populations,only a two-tailed test can be used.

Question 64

In testing for the differences between the means of two independent populations,we assume that the 2 populations each follow a normal distribution.

Question 65

When the sample sizes are equal,the pooled variance of the two groups is the average of the two sample variances.

Question 66

When testing H₀ : μ₁ - μ₂ = 0 against H₁ : μ₁ - μ₂ ≠ 0,the observed value of the Z-score was found to be -2.13.The p-value for this test would be 0.4668.

Question 67

In testing H₀ : P_x - P_y = 0,an estimate of the common population proportion is a weighted average of the two sample proportions,the weights being equal to the sample sizes.

Question 68

Repeated measurements collected from the same individuals is an example of data collected from two related populations.

Question 69

The F distribution is constructed as the ratio of two Student's t random variables;each divided by its degrees of freedom.

Question 70

When testing for differences between the means of two related populations,either a one-tailed or two-tailed test can be used.

Question 71

The F test used for testing the difference in two population variances is always a one-tailed test.

Question 72

At 5% significance level,the upper-tailed critical value of an F test with 3 degrees of freedom in the numerator and 8 degrees of freedom in the denominator is equal to 4.07.

Question 73

An F distribution is an example of a symmetrical distribution.

Question 74

To use the Student's t distribution to compare population means when the population variances are unknown and sample sizes are small,the populations are assumed to be normally distributed.

Question 75

In testing for differences between the means of two independent populations if the test statistic is very large,then the p-value of the test should be very small.

Question 76

In a random sample of 150 music industry management majors,75 rated a sense of humor as very important trait to their career performance.The same view was held by 81 of an independent random sample of 180 professional golf management majors.Test at the 5% level against a two-sided alternative the null hypothesis that the population proportions of music industry management and professional golf management majors who rate a sense of humor as very important are the same.

Question 77

In testing for the differences between the means of two related populations,we assume that the differences follow a t distribution.

Question 78

In practical applications,the F ratio is arranged in a way such that the larger sample variance is in the denominator and the smaller is in the numerator.

Question 79

The procedure that utilizes the normal distribution to test the difference between two population proportions is valid for all sample sizes.

Question 80

In testing H₀ : P_x - P_y = 0,the p-value equals the probability,1 - β,of avoiding the Type II error of erroneously failing to reject a null hypothesis that is in fact false.