Question 1

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?

Accepted Answer

When we have unequal variances, we use the formula:

ν = [(s1^2/n1 + s2^2/n2)^2]/[1/(n1-1)(s1^2/n1)^2 + 1/(n2-1)(s2^2/n2)^2]

which simplifies to:

ν = [(s1^2/n1 + s2^2/n2)^2]/[s1^4/(n1^2(n1-1)) + s2^4/(n2^2(n2-1))]

where s1^2 and s2^2 are the sample variances, n1 and n2 are the sample sizes, and ν is the degrees of freedom for the t statistic.

Substituting the given values, we get:

ν = [(23.8/14 + 31.5/10)^2]/[23.8^4/(196(13.8)^2) + 31.5^4/(100(16.2)^2)]
ν = [(1.7006)^2]/[0.0429 + 0.0828]
ν = 15.546

Rounding down to the nearest integer, we get ν = 15, which corresponds to choice B.

Question 2

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

Accepted Answer

While a t-test is commonly used for testing differences between the means of two independent populations, other tests such as ANOVA or non-parametric tests like the Mann-Whitney U test can also be used depending on the nature of the data and assumptions of the test.

Question 3

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:

Accepted Answer

The test statistic for determining if there is a difference in the population variances for Supplier A and B is equal to the ratio of the sample variances. In this case, the sample variance for Supplier A is 0.25 and the sample variance for Supplier B is 0.16. Therefore, the test statistic is 0.25 / 0.16 = 1.5625, which is closest to choice A, 0.44.

Question 4

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.

Accepted Answer

Since the null hypothesis is rejected at the 5% level but not at the 2.5% level, this means that the p-value is greater than 0.025 but less than or equal to 0.05. Therefore, the statement in the question is true.

Question 5

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.

Accepted Answer

If the calculated value of the test statistic t = -1.75 and the test has 15 degrees of freedom, the p-value can be found using a t-distribution table or calculator. With a two-tailed test and a calculated t-value of -1.75, the approximate p-value is 0.0975 or 0.10 (rounded to two decimal places). Therefore, the statement is true.

Question 6

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

Accepted Answer

The probability of a Type II error is the probability of failing to reject the null hypothesis when it is false. Therefore, if the probability of a Type II error is 0.25, the probability of correctly rejecting the null hypothesis when it is false (the power of the test) is 1 - 0.25 = 0.75.

Question 7

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.

Accepted Answer

This is the assumption of the independent samples t-test when the sample sizes are large (n≥30) and the variances are known. However, if the sample sizes are small, then the populations need to be normally distributed, and if the variances are unknown and assumed to be equal, then the populations can have different shapes but have to be also normally distributed.

Question 8

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.

Accepted Answer

Rejecting a null hypothesis at the 5% level of significance does not automatically mean it will be rejected at the 1% level, as the 1% level requires stronger evidence against the null hypothesis.

Question 9

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:

Accepted Answer

The formula for calculating the F statistic for two independent random samples that each follow a normal distribution and have equal variances is F = (sample variance of the first sample) / (sample variance of the second sample). Therefore, option B is the correct formula for calculating the F statistic. Options A and D have incorrect characters and do not represent a valid formula. Option C shows the formula for calculating the Z statistic, which is used when the population standard deviation is known, but this is not the case in this scenario.

Question 10

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.

Accepted Answer

The p-value is not the probability that the null hypothesis is true, but rather the probability of obtaining a test statistic as extreme or more extreme than the observed test statistic, assuming the null hypothesis is true. Therefore, the statement in the question is false.

Question 11

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?

Accepted Answer

The F distribution degrees of freedom in the numerator are based on the sample sizes of the two independent groups being compared. In this case, the sample size of the first group is n_x and the sample size of the second group is n_y. Therefore, the F distribution degrees of freedom in the numerator would be n_x - 1. Option A is the only one that fits this criteria.

Question 12

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?

Accepted Answer

The answer of THE NEXT QUESTIONS ARE BASED ON THE...

Question 13

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

Accepted Answer

The null hypothesis is that the mean score of medical students is equal to the mean score of law students, which is represented by μ_M - μ_L = 0. The alternative hypothesis is that the mean score of medical students differs from the mean score of law students, which is represented by μ_M - μ_L ≠ 0. This is the best choice as it covers both the possibilities that the mean score of medical students could be greater than or less than the mean score of law students.

Question 14

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

Accepted Answer

The answer of Assuming the independent samples procedure was used,which...

Question 15

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

Accepted Answer

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

Question 16

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.

Accepted Answer

Hypothesis tests that use the F distribution, such as ANOVA, generally assume that the populations from which samples are drawn are normally distributed. This assumption is less critical when sample sizes are large due to the Central Limit Theorem, but it is not completely disregarded.

Question 17

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:

Accepted Answer

The answer of THE NEXT QUESTIONS ARE BASED ON THE...

Question 18

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.

Accepted Answer

If the null hypothesis is rejected at a lower level of significance (2%), it will definitely be rejected at a higher level of significance (3%). Therefore, the statement is true.

Question 19

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

Accepted Answer

The inequality suggests that the alternative hypothesis is one-sided (either the mean of x is less than the mean of y or vice versa). Hence, the correct choice is C, which represents a one-sided test. A two-sided alternative would be appropriate if there was no clear difference between the means of x and y, whereas a null hypothesis would be a statement that assumes no difference between the means of x and y. An upper-tailed test would be appropriate if the alternative hypothesis stated that the mean of x is greater than the mean of y, which is not the case in this scenario.

Question 20

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).

Accepted Answer

For a two-tailed test with a Z score of 1.95, the p-value is found by looking up the corresponding area in the standard normal distribution table or using a calculator. The area to the right of Z = 1.95 is approximately 0.0256. Since it's a two-tailed test, we double this value to get the total p-value, which is 0.0512.

Quiz 10: Hypothesis Testing: Additional Topics

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

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.

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.

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

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.

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.

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:

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.

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?

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

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

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

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.

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.

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

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) .

Quiz 10: Hypothesis Testing: Additional Topics

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

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.

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.

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

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.

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

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:

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

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

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

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

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

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.

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

Given H1 : μx - μy < 0,the hypothesis test is a(n) :

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 : μM - μL ≠ 0) .

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.

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.

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.

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?

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.

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

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) .