Question 1

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 2

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 3

In testing the null hypothesis H₀ = $p _ { 1 } - p _ { 2 }$ = 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 4

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 $\bar { x } _ { 1 } - \bar { x } _ { 2 }$ is equal to:$$ \sqrt { \frac { \sigma ^ { 2 } } { n _ { 1 } } + \frac { \sigma ^ { 2 } } { n _ { 2 } } } = \sqrt { \frac { s ^ { 2 } } { n _ { 1 } } + \frac { s ^ { 2 } } { n _ { 2 } } } $$where s^2 is the pooled variance, which is estimated by:$$ s ^ { 2 } = \frac { ( n _ { 1 } - 1 ) s _ { 1 } ^ { 2 } + ( n _ { 2 } - 1 ) s _ { 2 } ^ { 2 } } { n _ { 1 } + n _ { 2 } - 2 } $$Plugging in the given values, we get:$$ s ^ { 2 } = \frac { ( 25 - 1 ) ( 1.05 ) ^ { 2 } + ( 30 - 1 ) ( 1.20 ) ^ { 2 } } { 25 + 30 - 2 } = 1.125 $$Therefore, the standard error of the sampling distribution of the sample mean difference is:$$ \sqrt { \frac { 1.125 } { 25 } + \frac { 1.125 } { 30 } } = 0.3072 $$

Question 5

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) -1.0944.

Accepted Answer

The formula for calculating the test statistic for testing the equality of population proportions is:

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

where p = (x1 + x2) / (n1 + n2)

Here, n1 = 100, x1 = 53, n2 = 150, x2 = 95.

Thus,

p1 = x1 / n1 = 0.53 
p2 = x2 / n2 = 0.6333 
p = (x1 + x2) / (n1 + n2) = (53 + 95) / (100 + 150) = 0.5867

So,

z = (0.53 - 0.6333) / sqrt{0.5867(1 - 0.5867)[(1/100) + (1/150)]}

= -1.09 (rounded to two decimal places)

Therefore, the test statistic for testing the equality of the population proportions is -1.0944, which is closest to choice D.

Question 6

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

The symbol $\bar { x } _ { D }$ represents the mean of the differences between the pairs of observations taken from two dependent samples. This is also known as the matched pairs differences or paired differences. It is used in inferential statistics to test whether there is a significant difference between the means of two dependent populations.

Question 7

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

When population variances are unknown and assumed to not be equal, the appropriate test is the unequal variances t-test. This test allows for the use of two independent samples of large sizes without assuming that the variances are equal. Option A and B assume equal population variances, and option D assumes paired samples.

Question 8

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 9

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 analyst is comparing two groups (Labor Party members and Liberal Party members) that are independent of each other, meaning that the characteristics of one group do not affect the characteristics of the other group. Therefore, this is an example of independent samples. The sample sizes being equal or unequal does not affect whether the samples are dependent or independent.

Question 10

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

It is recommended to round the number of degrees of freedom down to the nearest integer in order to select the appropriate critical values for the t-test. This conservatively estimates the variability in the data and reduces the chances of making a Type I error (rejecting the null hypothesis when it is actually true).

Question 11

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 12

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 can be used if both n*p and n*(1-p) are greater than or equal to 5 for each group being compared. Therefore, both conditions must be met for both groups (total of 4 conditions). This is expressed in choice C.

Question 13

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 14

Two independent samples of sizes 30 and 40 are randomly selected from two populations to test the difference between the population means, µ$\mu$₂ - µ$\mu$₁_,where the population variances are unknown. Which of the following best describes the sampling distribution of the sample mean difference $\bar { x } _ { 1 } - \bar { x } _ { 2 }$ ? A) Normal. B) t-distributed with 68 degrees of freedom. C) Approximately normal. D) None of these choices are correct.

Accepted Answer

The answer of Two independent samples of sizes 30 and...

Question 15

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 16

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 $p _ { 1 } - p _ { 2 }$ 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

To calculate the test statistic for a two-tail test of difference between two population proportions, we use the formula:

z = (p1 - p2) / SE(p1 - p2)

where SE(p1 - p2) is the standard error of the sampling distribution of p1 - p2.

Here, p1 = 0.25, p2 = 0.20 and SE(p1 - p2) = 0.04.

Substituting these values in the formula, we get:

z = (0.25 - 0.20) / 0.04 = 1.25

Therefore, the calculated value of the test statistic is z = 1.25.

The correct answer is B.

Question 17

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 involve pairing observations in a meaningful way, making it appropriate to compare each pair of observations directly. This is in contrast to independent samples, where there is no natural pairing or relationship between observations. Weighted and random samples do not describe the relationship or pairing between observations.

Question 18

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 use of pooled proportion estimate is appropriate when the null hypothesis assumes that the two population proportions are equal. The other options are not relevant in determining whether or not to use the pooled proportion estimate.

Question 19

We can design a matched pairs experiment when the data collected are:
A) observational.
B) experimental.
C) controlled.
D) All these choices are correct.

Accepted Answer

Matched pairs experiments are a type of controlled experiment in which the subjects are matched on one or more characteristics before being assigned to different treatment groups. This helps to ensure that the groups are comparable and that any differences between them are due to the treatment, not to other factors. Matched pairs experiments can be used to collect either observational or experimental data.

Question 20

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$\mu$₁ - µF$\mu$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

The Z-test should be used when the sample size is large (>30) and the population variance is known. Since both conditions are met in this scenario, the Z-test is the appropriate choice. The t-test should be used when the population variance is unknown or the sample size is small (<30). The Chi-square test is used for testing categorical data, not continuous data like population means. Therefore, choices A and B are incorrect.

Quiz 14: Hypothesis Testing: Comparing Two Populations

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:

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:

In testing the null hypothesis H0 = p1−p2p _ { 1 } - p _ { 2 }p1​−p2​ = 0, if H0 is true, the test could lead to:

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:

Which of the following best describes the symbol xˉD\bar { x } _ { D }xˉD​ ?

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 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 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 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:

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:

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 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 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:

We can design a matched pairs experiment when the data collected are:

In testing the null hypothesis H₀ = $p _ { 1 } - p _ { 2 }$ = 0, if H₀ is true, the test could lead to:

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

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:

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: