Question 1

Which of the following best describes the symbol $\bar { X } _ { D }$ ?
A)The difference in the means of two dependent populations.
B)The difference in the means of two independent populations.
C)The mean of the differences in the pairs of observations taken from two dependent samples.
D)The matched pairs differences.

Accepted Answer

$\bar { X } _ { D }$ represents the mean of the differences in the pairs of observations taken from two dependent samples, typically used in paired sample t-tests or related analyses.

Question 2

If some natural relationship exists between each pair of observations that provides a logical reason to compare the first observation of sample 1 with the first observation of sample 2, the second observation of sample 1 with the second observation of sample 2, and so on, the samples are referred to as:
A)matched samples.
B)independent samples.
C)weighted samples.
D)random samples.

Accepted Answer

Matched samples refer to samples where there is a natural relationship between each pair of observations. This allows for a logical reason to compare the observations within each sample. In contrast, independent samples refer to samples where there is no natural relationship between the observations in each sample, making direct comparison of observations between samples less straightforward. Weighted samples refer to samples where some observations carry more weight (or importance) than others, while random samples refer to samples that are selected randomly from a population.

Question 3

Two independent samples of sizes 40 and 50 are randomly selected from two normally distributed populations. Assume that the population variances are known. In order to test the difference between the population means, µ 1 - µ 2, which of the following test statistics should be used?
A)t-test
B)Chi-square test
C)z-test
D)None of these choices are correct.

Accepted Answer

Since population variances are known and the sample sizes are reasonably large (more than 30), a z-test can be used to test the difference between the population means. A t-test would have been used if the population variances were unknown or if the sample sizes were smaller. A chi-square test would have been used for testing the goodness of fit or independence of variables. Therefore, the best choice for testing the difference between the population means is a z-test.

Question 4

A political analyst in Perth surveys a random sample of Labor Party members and compares the results with those obtained from a random sample of Liberal Party members. This would be an example of:
A)independent samples.
B)dependent samples.
C)independent samples only if the sample sizes are equal.
D)dependent samples only if the sample sizes are equal.

Accepted Answer

The samples are independent because they come from two different populations (Labor Party members and Liberal Party members), and therefore, the results of one sample do not affect the results of the other sample. The sample sizes do not affect whether the samples are independent or dependent.

Question 5

In testing the difference between two population means, using two independent samples, the sampling distribution of the sample mean difference $\bar { X } _ { 1 } - \bar { X } _ { 2 }$ is normal if the:
A)sample sizes are both greater than 30.
B)populations are normal.
C)populations are non-normal and the sample sizes are large.
D)All of these choices are correct.

Accepted Answer

The answer of In testing the difference between two population...

Question 6

In testing the null hypothesis H0: p1 - p2 = 0, if H0 is true, the test could lead to:
A)a Type I error.
B)a Type II error.
C)either a Type I or a Type II error.
D)None of these choices are correct.

Accepted Answer

A Type I error occurs when the null hypothesis is true, but we incorrectly reject it.

Question 7

In testing the difference between the means of two normal populations with known population standard deviations the test statistic calculated from two independent random samples equals 2.56. If the test is two-tailed and the 1% level of significance has been specified, the conclusion should be:
A)to reject the null hypothesis.
B)not to reject the null hypothesis.
C)the test is inconclusive.
D)None of these choices are correct.

Accepted Answer

The answer of In testing the difference between the means...

Question 8

For testing the difference between two population proportions, the pooled proportion estimate should be used to compute the value of the test statistic when the:
A)populations are normally distributed.
B)sample sizes are small.
C)samples are independently drawn from the populations.
D)null hypothesis states that the two population proportions are equal.

Accepted Answer

The pooled proportion estimate should be used when testing the difference between two population proportions only when the null hypothesis states that the two population proportions are equal. It is not dependent on the distribution of populations or the sample sizes.

Question 9

A test is being conducted to test the difference between two population means, using data that are gathered from a matched pairs experiment. If the paired differences are normal, then the distribution used for testing is the:
A)normal.
B)binomial.
C)Student t.
D) F.

Accepted Answer

When conducting a test of difference between two population means with matched pairs data, the appropriate distribution is the Student t distribution, provided that the paired differences are normal.

Question 10

The number of degrees of freedom associated with the t-test, when the data are gathered from a matched pairs experiment with 40 pairs, is:
A)38.
B)39.
C)1.
D)2.

Accepted Answer

In a matched pairs t-test, the degrees of freedom are calculated as the number of pairs minus one. Therefore, with 40 pairs, the degrees of freedom are 40 - 1 = 39.

Question 11

A sample of size 150 from population 1 has 40 successes. A sample of size 250 from population 2 has 30 successes. The value of the test statistic for testing the null hypothesis that the proportion of successes in population 1 exceeds the proportion of successes in population 2 by 0.05 is:
A)1.645.
B)2.327.
C)1.960.
D)1.977.

Accepted Answer

We are testing the difference in proportions between two independent samples, so we will use the formula for the two-sample z-test:
$$\large z=\frac{(\hat{p}_1-\hat{p}_2)-\Delta}{\sqrt{\frac{\hat{p}(1-\hat{p})}{n_1}+\frac{\hat{p}(1-\hat{p})}{n_2}}}$$

where $\hat{p}_1$ and $\hat{p}_2$ are the sample proportions, $\Delta$ is the hypothesized difference between the population proportions, $\hat{p}$ is the pooled proportion, and $n_1$ and $n_2$ are the sample sizes.

We are given $\hat{p}_1=40/150=0.267$, $\hat{p}_2=30/250=0.12$, and $\Delta=0.05$. We will assume the null hypothesis and use the pooled proportion under the null hypothesis, which is:
$$\large \hat{p}=\frac{x_1+x_2}{n_1+n_2}=\frac{40+30}{150+250}=0.146$$

Plugging in these values, we get:
$$\large z=\frac{(0.267-0.12-0.05)}{\sqrt{\frac{(0.146)(1-0.146)}{150}+\frac{(0.146)(1-0.146)}{250}}}$$

Simplify this to get:
$$\large z=2.327$$

Using a significance level of 0.05 and a two-sided test, the critical z-value is 1.96. Since our calculated z-value is greater than the critical value, we reject the null hypothesis and conclude that the proportion of successes in population 1 is significantly different from the proportion of successes in population 2.

Question 12

In testing the difference between two population means, using two independent samples, we use the pooled variance in estimating the standard error of the sampling distribution of the sample mean difference $\bar { X } _ { 1 } - \bar { X } _ { 2 }$ if the:
A)sample sizes are both large.
B)populations are normally distributed.
C)populations have equal variances.
D)populations are normal with equal variances.

Accepted Answer

The answer of In testing the difference between two population...

Question 13

When the necessary conditions are met, a two-tail test is being conducted to test the difference between two population proportions. The two sample proportions are p1 = 0.25 and p2 = 0.20 and the standard error of the sampling distribution of $\hat { p } _ { 1 } - \hat { p } _ { 2 }$ is 0.04. The calculated value of the test statistic will be:
A)Z = 0.25.
B)Z = 1.25.
C)t = 0.25.
D)t = 0.80.

Accepted Answer

The test statistic for the difference between two proportions is calculated using the formula $Z = \frac{(\hat{p}_1 - \hat{p}_2) - (p_1 - p_2)}{SE}$, where $SE$ is the standard error of the sampling distribution of $\hat{p}_1 - \hat{p}_2$. Assuming the null hypothesis implies $p_1 = p_2$, the formula simplifies to $Z = \frac{\hat{p}_1 - \hat{p}_2}{SE}$. Plugging in the given values, $Z = \frac{0.25 - 0.20}{0.04} = \frac{0.05}{0.04} = 1.25$.

Question 14

In testing whether the means of two normal populations are equal, summary statistics computed for two independent samples are as follows: n1 = 25, $\bar { X } _ { 1 }$ = 7.30, s1 = 1.05.
N2 = 30, $\bar { X } _ { 2 }$ = 6.80, s2 = 1.20.
Assume that the population variances are equal. Then the standard error of the sampling distribution of the sample mean difference $\bar { X } _ { 1 } - \bar { X } _ { 2 }$ is equal to:
A)0.3072.
B)0.0917.
C)0.3028.
D)0.0944.

Accepted Answer

The standard error of the sampling distribution of the sample mean difference is:$$\sigma_{\bar{X}_1 - \bar{X}_2} = \sqrt{\frac{\sigma^2}{n_1} + \frac{\sigma^2}{n_2}} = \sqrt{\frac{s^2}{n_1} + \frac{s^2}{n_2}}$$where s is the pooled standard deviation, which is:$$s = \sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}}$$Plugging in the given values, we get:$$s = \sqrt{\frac{(25 - 1)1.05^2 + (30 - 1)1.20^2}{25 + 30 - 2}} = 1.125$$Therefore, the standard error of the sampling distribution of the sample mean difference is:$$\sigma_{\bar{X}_1 - \bar{X}_2} = \sqrt{\frac{1.125^2}{25} + \frac{1.125^2}{30}} = 0.3072$$So the correct answer is A.

Question 15

In testing the difference between two population means, for which the population variances are unknown and are not assumed to be equal, two independent samples of large sizes are drawn from the populations. Which of the following tests is appropriate?
A)z-test.
B)Pooled-variances t-test.
C)Unequal variances t-test.
D)Matched pairs t-test.

Accepted Answer

The appropriate test in this scenario is the Unequal variances t-test, also known as Welch's t-test. This test does not assume equal variances and is more reliable when the population variances are not known. It is also appropriate for large sample sizes, which are given in the question. The z-test is only appropriate when the population variances are known or when the sample sizes are very large. The Pooled-variances t-test assumes equal variances and is not appropriate when the variances are unknown or unequal. The Matched pairs t-test is used when comparing two samples that are paired, such as before and after measurements.

Question 16

In testing the difference between the means of two normal populations, using two independent samples, when the population variances are unknown and unequal, the sampling distribution of the resulting statistic is:
A)normal.
B)Student t.
C)approximately normal.
D)approximately Student t.

Accepted Answer

When the population variances are unknown and unequal, and two independent samples are used to test the difference between the means of two normal populations, the sampling distribution of the resulting statistic follows a Student t-distribution.

Question 17

A sample of size 100 selected from one population has 53 successes, and a sample of size 150 selected from a second population has 95 successes. The test statistic for testing the equality of the population proportions is equal to:
A)-11.1051.
B)-17.1107.
C)-0.0944.
D)-1.6286.

Accepted Answer

We can use the test statistic formula for testing the equality of population proportions, which is:

z = (p1 - p2) / sqrt [ p * (1 - p) * (1/n1 + 1/n2) ]

Where:
p1 = proportion of successes in population 1
p2 = proportion of successes in population 2
p = pooled proportion = (x1 + x2) / (n1 + n2)
x1 = number of successes in sample 1
x2 = number of successes in sample 2
n1 = sample size of sample 1
n2 = sample size of sample 2

Plugging in the values given in the problem, we get:

p1 = 53/100 = 0.53
p2 = 95/150 = 0.6333
p = (53 + 95) / (100 + 150) = 0.56
n1 = 100, n2 = 150

z = (0.53 - 0.6333) / sqrt [0.56 * (1 - 0.56) * (1/100 + 1/150)]
z = -1.6286

Therefore, the test statistic for testing the equality of population proportions is -1.6286, which is choice (D).

Question 18

When the necessary conditions are met, a two-tail test is being conducted to test the difference between two population means, but your statistical software provides only a one-tail area of 0.028 as part of its output. The p-value for this test will be:
A)0.972.
B)0.05.
C)0.056.
D)0.014.

Accepted Answer

The p-value for a two-tailed test is twice the one-tail area provided by the software because you need to account for both tails of the distribution. Therefore, 0.028 * 2 = 0.056.

Question 19

Which of the following is a required condition for using the normal approximation to the binomial distribution in testing the difference between two population proportions?
A)n1p1 > 5 and n2p2 > 5.
B)n2(1 - p2) > 5.
C)n1p1 > 5, n1(1 - p1) > 5, n2p2 > 5 and n2(1 - p2) > 5.
D)n1(1 - p1) > 5 and n2(1 - p2) > 5.

Accepted Answer

The normal approximation to the binomial distribution requires that both the success and failure conditions are met for each sample, which means n1p1 > 5, n1(1 - p1) > 5, n2p2 > 5, and n2(1 - p2) > 5.

Question 20

In testing the difference between the means of two normally distributed populations, the number of degrees of freedom associated with the unequal-variances t-test statistic usually results in a non-integer number. It is recommended that you: A)round up to the nearest integer. B)round down to the nearest integer. C)change the sample sizes until the number of degrees of freedom becomes an integer. D)assume that the population variances are equal, and then use df = n₁ + n₂ - 2.

Accepted Answer

The answer of In testing the difference between the means...

Quiz 13: Hypothesis Testing: Comparing Two Populations

Which of the following best describes the symbol XˉD\bar { X } _ { D }XˉD​ ?

Two independent samples of sizes 40 and 50 are randomly selected from two normally distributed populations. Assume that the population variances are known. In order to test the difference between the population means, µ 1 - µ 2, which of the following test statistics should be used?

A political analyst in Perth surveys a random sample of Labor Party members and compares the results with those obtained from a random sample of Liberal Party members. This would be an example of:

In testing the difference between two population means, using two independent samples, the sampling distribution of the sample mean difference Xˉ1−Xˉ2\bar { X } _ { 1 } - \bar { X } _ { 2 }Xˉ1​−Xˉ2​ is normal if the:

In testing the null hypothesis H0: p1 - p2 = 0, if H0 is true, the test could lead to:

In testing the difference between the means of two normal populations with known population standard deviations the test statistic calculated from two independent random samples equals 2.56. If the test is two-tailed and the 1% level of significance has been specified, the conclusion should be:

For testing the difference between two population proportions, the pooled proportion estimate should be used to compute the value of the test statistic when the:

A test is being conducted to test the difference between two population means, using data that are gathered from a matched pairs experiment. If the paired differences are normal, then the distribution used for testing is the:

The number of degrees of freedom associated with the t-test, when the data are gathered from a matched pairs experiment with 40 pairs, is:

A sample of size 150 from population 1 has 40 successes. A sample of size 250 from population 2 has 30 successes. The value of the test statistic for testing the null hypothesis that the proportion of successes in population 1 exceeds the proportion of successes in population 2 by 0.05 is:

In testing the difference between two population means, using two independent samples, we use the pooled variance in estimating the standard error of the sampling distribution of the sample mean difference Xˉ1−Xˉ2\bar { X } _ { 1 } - \bar { X } _ { 2 }Xˉ1​−Xˉ2​ if the:

In testing the difference between two population means, for which the population variances are unknown and are not assumed to be equal, two independent samples of large sizes are drawn from the populations. Which of the following tests is appropriate?

In testing the difference between the means of two normal populations, using two independent samples, when the population variances are unknown and unequal, the sampling distribution of the resulting statistic is:

A sample of size 100 selected from one population has 53 successes, and a sample of size 150 selected from a second population has 95 successes. The test statistic for testing the equality of the population proportions is equal to:

When the necessary conditions are met, a two-tail test is being conducted to test the difference between two population means, but your statistical software provides only a one-tail area of 0.028 as part of its output. The p-value for this test will be:

Which of the following is a required condition for using the normal approximation to the binomial distribution in testing the difference between two population proportions?

In testing the difference between the means of two normally distributed populations, the number of degrees of freedom associated with the unequal-variances t-test statistic usually results in a non-integer number. It is recommended that you:

Which of the following best describes the symbol $\bar { X } _ { D }$ ?

In testing the difference between two population means, using two independent samples, the sampling distribution of the sample mean difference $\bar { X } _ { 1 } - \bar { X } _ { 2 }$ is normal if the:

In testing the difference between two population means, using two independent samples, we use the pooled variance in estimating the standard error of the sampling distribution of the sample mean difference $\bar { X } _ { 1 } - \bar { X } _ { 2 }$ if the: