Question 1

The results of a mathematics placement exam at two different campuses of Mercy College follow:  What is the computed value of the test statistic?

Accepted Answer

To compute the test statistic, we need to use the formula:

test statistic = (sample mean - population mean) / (standard error)

For Campus A: 
sample mean = 74 
population mean = 68 
standard deviation = 8 
sample size = 36

standard error = 8 / sqrt(36) = 1.33

test statistic = (74 - 68) / 1.33 = 4.51

For Campus B: 
sample mean = 82 
population mean = 74 
standard deviation = 10 
sample size = 64

standard error = 10 / sqrt(64) = 1.25

test statistic = (82 - 74) / 1.25 = 6.40

Therefore, the computed value of the test statistic is the larger of the two, which is 6.40. The only answer choice that matches this value is C.

Question 2

A national manufacturer of ball bearings is experimenting with two different processes for producing precision ball bearings. It is important that the diameters be as close as possible to an industry standard. The output from each process is sampled and the average error from the industry standard is measured in millimeters. The results are presented here.  The researcher is interested in determining whether there is evidence that the two processes yield different average errors. The population standard deviations are unknown but assumed equal. What is the critical t-value at the 1% level of significance?

Accepted Answer

The answer of A national manufacturer of ball bearings is...

Question 3

For a hypothesis comparing two population means,H₀: μ₁ ≤ μ₂,what is the critical value for a one-tailed hypothesis test,using a 5% significance level,with both sample sizes equal to 13? Assume the population standard deviations are equal.

Accepted Answer

The critical value for a one-tailed hypothesis test at a 5% significance level with both sample sizes equal to 13 (and thus degrees of freedom 24) and assuming equal population standard deviations, is found using the t-distribution table. For a one-tailed test with a 5% significance level and 24 degrees of freedom, the critical value is +1.711.

Question 4

A national manufacturer of ball bearings is experimenting with two different processes for producing precision ball bearings. It is important that the diameters be as close as possible to an industry standard. The output from each process is sampled,and the average error from the industry standard is measured in millimeters. The results are presented here.  The researcher is interested in determining whether there is evidence that the two processes yield different average errors. The population standard deviations are unknown but assumed equal. What is the alternate hypothesis?

Accepted Answer

The alternate hypothesis should state that the two processes yield different average errors, which is the case in choice B. The symbol ≠ indicates "not equal to." Choices A, C, and D do not provide a proper alternate hypothesis for testing whether the two processes differ in their average errors.

Question 5

A national manufacturer of ball bearings is experimenting with two different processes for producing precision ball bearings. It is important that the diameters be as close as possible to an industry standard. The output from each process is sampled,and the average error from the industry standard is measured in millimeters. The results are presented here.  The researcher is interested in determining whether there is evidence that the two processes yield different average errors. The population standard deviations are unknown but are assumed equal. If we test the null hypothesis at the 1% level of significance,what is the decision?

Accepted Answer

To test the hypothesis, we use a two-sample t-test for equal variances. The sample sizes are large enough for us to assume normality of the sampling distributions.

The null hypothesis is that the means are equal, and the alternative hypothesis is that the means are different.

At the 1% level of significance, the critical value for a two-tailed test with 38 degrees of freedom is approximately 2.704. The test statistic (calculated using a statistical software) is -3.629.

Since the test statistic (-3.629) is less than the critical value (-2.704), we reject the null hypothesis. This means that we have evidence to suggest that the two processes yield different average errors.

Question 6

A national manufacturer of ball bearings is experimenting with two different processes for producing precision ball bearings. It is important that the diameters be as close as possible to an industry standard. The output from each process is sampled and the average error from the industry standard is measured in millimeters. The results are presented here.  The researcher is interested in determining whether there is evidence that the two processes yield different average errors. The population standard deviations are unknown but are assumed equal. What is the computed value of t?

Accepted Answer

We need to use a two-sample t-test with equal variances:

t = (x̄1 - x̄2) / (s√(1/n1 + 1/n2))

where x̄1 and x̄2 are the sample means, s is the pooled standard deviation, n1 and n2 are the sample sizes.

Computing the values:

x̄1 = 0.52, x̄2 = 1.57
n1 = n2 = 25
s = sqrt(((n1-1)s1^2 + (n2-1)s2^2) / (n1+n2-2)) =
sqrt(((24)(0.33^2) + (24)(0.29^2)) / 48) = 0.307
(Note: s1 and s2 are the sample standard deviations)

t = (0.52 - 1.57) / (0.307√(1/25 + 1/25)) = -3.299

The computed value of t is -3.299, so the best choice is C, which matches the computed value.

Question 7

A national manufacturer of ball bearings is experimenting with two different processes for producing precision ball bearings. It is important that the diameters be as close as possible to an industry standard. The output from each process is sampled,and the average error from the industry standard is measured in millimeters. The results are presented here.  The researcher is interested in determining whether there is evidence that the two processes yield different average errors. The population standard deviations are unknown but are assumed equal. What are the degrees of freedom?

Accepted Answer

The answer of A national manufacturer of ball bearings is...

Question 8

Two samples,one of size 14 and the second of size 13,are selected to test the difference between two population means. How many degrees of freedom are used to find the critical value? Assume the population standard deviations are equal.

Accepted Answer

The degrees of freedom are calculated using the formula df = (n1 - 1) + (n2 - 1), where n1 and n2 are the sample sizes. So, in this case, df = (14 - 1) + (13 - 1) = 25. Therefore, the correct choice is C) 25.

Question 9

Which condition must be met to conduct a test for the difference in two sample means using a z-statistic?

Accepted Answer

The key condition for using a z-statistic in testing the difference between two sample means is that the two population standard deviations are known. This allows for the calculation of the standard error of the difference between the two means under the assumption of normal distribution.

Question 10

A national manufacturer of ball bearings is experimenting with two different processes for producing precision ball bearings. It is important that the diameters be as close as possible to an industry standard. The output from each process is sampled,and the average error from the industry standard is measured in millimeters. The results are presented here.  The researcher is interested in determining whether there is evidence that the two processes yield different average errors. The population standard deviations are unknown but are assumed equal. This example is what type of test?

Accepted Answer

This is a two-sample test of means because we are comparing the average errors from two different processes.

Question 11

When testing the difference between two dependent population means,the test statistic is based on a ________.

Accepted Answer

When we are testing the difference between two dependent population means, we use the paired t-test. The test statistic for paired t-test is based on the standard deviation of the differences between the paired observations. Therefore, the best choice is B, the standard deviation of the differences. The other options (pooled variance, pooled proportion, and sum of population variances) are used in other types of hypothesis testing, not for dependent population means.

Question 12

We test for a hypothesized difference between two population means: H₀: μ₁ = μ₂. The population standard deviations are unknown but assumed equal. The number of observations in the first sample is 15 and 12 in the second sample. How many degrees of freedom are associated with the critical value?

Accepted Answer

The degrees of freedom for a two-sample t-test when the population standard deviations are unknown but assumed to be equal is calculated using the formula $df = n_1 + n_2 - 2$, where $n_1$ and $n_2$ are the sample sizes. Here, $n_1 = 15$ and $n_2 = 12$, so $df = 15 + 12 - 2 = 25$.

Question 13

When is it appropriate to use the paired difference t-test?

Accepted Answer

The paired difference t-test is used when two dependent samples are compared. This means that the samples are related in some way, such as being taken from the same individuals before and after an intervention or under different conditions. The test is used to determine if there is a significant difference between the means of the two related samples. Therefore, option D is the best choice. Options A and B are incorrect because they refer to comparing multiple or independent samples, whereas option C refers to comparing independent samples.

Question 14

Twenty randomly selected statistics students were given 15 multiple choice questions and 15 open-ended questions,all on the same material. The professor was interested in determining if students scored higher on the multiple choice questions. This experiment is an example of ________.

Accepted Answer

This experiment is an example of a paired t-test because each student's performance on the multiple choice questions is directly compared to their performance on the open-ended questions. This pairing of data points (the two sets of scores for each student) is what characterizes a paired t-test.

Question 15

The results of a mathematics placement exam at two different campuses of Mercy College follow:  What is the null hypothesis if we want to test the hypothesis that the mean score on Campus 1 is higher than on Campus 2?

Accepted Answer

The null hypothesis states that the mean score on Campus 1 is less than or equal to the mean score on Campus 2. Therefore, option D is the correct choice as it states H₀: µ₁ ≤ µ₂. Option A and B are incorrect as they do not address the hypothesis being tested, and option C is incorrect as it assumes the mean scores on both campuses are equal, which is not part of the hypothesis being tested.

Question 16

When testing the difference between two population means,the sample variances are pooled to estimate the population variance when ________.

Accepted Answer

When testing the difference between two population means, if the population variances are assumed equal but unknown, the sample variances are pooled to estimate the population variance.

Question 17

A national manufacturer of ball bearings is experimenting with two different processes for producing precision ball bearings. It is important that the diameters be as close as possible to an industry standard. The output from each process is sampled,and the average error from the industry standard is measured in millimeters. The results are presented here.  The researcher is interested in determining whether there is evidence that the two processes yield different average errors. The population standard deviations are unknown but are assumed equal. What is the null hypothesis?

Accepted Answer

The answer of A national manufacturer of ball bearings is...

Question 18

For a hypothesis test comparing two population means,the combined degrees of freedom are 24. Which of the following statements about the two sample sizes cannot be true? Assume the population standard deviations are equal.

Accepted Answer

The answer of For a hypothesis test comparing two population...

Question 19

The net weights (in grams)of a sample of bottles filled by a machine manufactured by Edne,and the net weights of a sample filled by a similar machine manufactured by Orno,Inc.,are shown here.  Testing the claim at the 0.05 level that the mean weight of the bottles filled by the Orno machine is greater than the mean weight of the bottles filled by the Edne machine,what is the critical value? Assume equal standard deviations for both samples.

Accepted Answer

The answer of The net weights (in grams)of a sample...

Question 20

The results of a mathematics placement exam at two different campuses of Mercy College follow:  What is the alternative hypothesis if we want to test the hypothesis that the mean score on Campus 1 is higher than on Campus 2?

Accepted Answer

The alternative hypothesis should state the belief that the mean score on Campus 1 is higher than on Campus 2, which is represented by the symbol ">". Therefore, the correct alternative hypothesis is H₁: µ₁ > µ₂. The other options are incorrect as they do not reflect the desired comparison between the means.

Quiz 11: Two-Sample Tests of Hypothesis

The results of a mathematics placement exam at two different campuses of Mercy College follow: What is the computed value of the test statistic?

For a hypothesis comparing two population means,H0: μ1 ≤ μ2,what is the critical value for a one-tailed hypothesis test,using a 5% significance level,with both sample sizes equal to 13? Assume the population standard deviations are equal.

Two samples,one of size 14 and the second of size 13,are selected to test the difference between two population means. How many degrees of freedom are used to find the critical value? Assume the population standard deviations are equal.

Which condition must be met to conduct a test for the difference in two sample means using a z-statistic?

When testing the difference between two dependent population means,the test statistic is based on a ________.

We test for a hypothesized difference between two population means: H0: μ1 = μ2. The population standard deviations are unknown but assumed equal. The number of observations in the first sample is 15 and 12 in the second sample. How many degrees of freedom are associated with the critical value?

When is it appropriate to use the paired difference t-test?

Twenty randomly selected statistics students were given 15 multiple choice questions and 15 open-ended questions,all on the same material. The professor was interested in determining if students scored higher on the multiple choice questions. This experiment is an example of ________.

The results of a mathematics placement exam at two different campuses of Mercy College follow: What is the null hypothesis if we want to test the hypothesis that the mean score on Campus 1 is higher than on Campus 2?

When testing the difference between two population means,the sample variances are pooled to estimate the population variance when ________.

For a hypothesis test comparing two population means,the combined degrees of freedom are 24. Which of the following statements about the two sample sizes cannot be true? Assume the population standard deviations are equal.

The results of a mathematics placement exam at two different campuses of Mercy College follow: What is the alternative hypothesis if we want to test the hypothesis that the mean score on Campus 1 is higher than on Campus 2?

For a hypothesis comparing two population means,H₀: μ₁ ≤ μ₂,what is the critical value for a one-tailed hypothesis test,using a 5% significance level,with both sample sizes equal to 13? Assume the population standard deviations are equal.