Question 1

Although it is often associated only with small-sample tests,the t distribution is appropriate when the population standard deviations are unknown,regardless of how large or small the samples happen to be.

Accepted Answer

The t distribution is appropriate when the population standard deviations are unknown, regardless of sample size.

Question 2

An example of dependent samples occurs when subjects are randomly assigned to the experimental and control groups of an experiment.

Accepted Answer

Dependent samples typically involve matched pairs or the same subjects being tested under different conditions, not random assignment to groups.

Question 3

The number of degrees of freedom associated with the unequal-variances t-test for comparing the means of two independent samples is n₁ + n₂ - 2.

Accepted Answer

The correct formula for degrees of freedom associated with the unequal-variances t-test is (n1 - 1) + (n2 - 1), where n1 and n2 are the sample sizes of the two groups being compared.

Question 4

The comparison of sample proportions from two independent samples relies on the t-distribution and assumes that the two populations are normally distributed.

Accepted Answer

The comparison of sample proportions typically relies on the normal distribution (or a z-test) under the Central Limit Theorem, assuming that the sample sizes are large enough, not the t-distribution, and does not require the populations to be normally distributed.

Question 5

The z-test approximation to the unequal variance t-test assumes that the population standard deviations, $\text { ? }$ ₁ and $\text { ? }$ ₂,are known.

Accepted Answer

The z-test approximation to the unequal variance t-test assumes that the population standard deviations are unknown and estimated using the sample standard deviations.

Question 6

Independent samples are those for which the selection process for one is not related to the selection process for the other.

Accepted Answer

Independent samples are selected without regard to each other, meaning the selection process for one sample does not affect the selection process for the other sample. Therefore, the statement is true.

Question 7

When either test can be applied to the same data,the z-test for comparing the means of two independent samples is preferable to the unequal-variances t-tests,especially when doing the test with computer assistance.

Accepted Answer

The choice between a z-test and an unequal-variances t-test (also known as Welch's t-test) depends on the known or assumed characteristics of the populations from which the samples are drawn. The z-test is appropriate when the population variances are known and the sample size is large. In contrast, the unequal-variances t-test is designed for situations where the population variances are unknown and assumed to be unequal, making it more appropriate in many real-world scenarios, especially with smaller sample sizes or unequal variances. Computer assistance does not inherently make the z-test preferable; it simply facilitates the calculation for either test.

Question 8

Comparing the means or proportions from two independent samples requires comparing the calculated value of a test statistic with the computed p-value,then deciding whether the null hypothesis should be rejected.

Accepted Answer

Comparing means or proportions from two independent samples involves comparing the test statistic to a critical value or comparing the p-value to a significance level (alpha) to decide whether to reject the null hypothesis.

Question 9

An important factor in choosing between the pooled-variances t-test and the unequal-variances t-test is whether we can assume the population means might be equal.

Accepted Answer

The choice between the pooled-variances t-test and the unequal-variances t-test depends on whether we can assume the population variances are equal, not the means.

Question 10

The z-test requires no assumptions about the shape of the population distributions as long as both n₁ and n₂ are ? 30.

Accepted Answer

When both n₁ and n₂ are ≥ 30, the sample size is considered large enough for the central limit theorem to apply, making the z-test valid regardless of the shape of the population distributions.

Question 11

When the population standard deviations are unknown and are not assumed to be equal,s₁ and s₂ must be used to estimate their respective population standard deviations, $\text { ? }$ ₁ and $\text { ? }$ _2.

Accepted Answer

When the population standard deviations are unknown and are not assumed to be equal, the sample standard deviations s₁ and s₂ must be used to estimate their respective population standard deviations, σ₁ and σ₂. This is because we do not have any other information about the population standard deviations, so we have to rely on the sample standard deviations as estimates.

Question 12

The F distribution is the sampling distribution of $s _ { 1 } ^ { 2 } / s _ { 2 } ^ { 2 }$ that would result if two samples were repeatedly drawn from the same,normally distributed population.

Accepted Answer

This statement accurately describes the F distribution.

Question 13

A typical example of independent samples occurs with before-and-after measurements for the same individuals or test units.

Accepted Answer

Independent samples refer to samples that are not related to each other in any way. Before-and-after measurements for the same individuals or test units are dependent samples, as the second measurement is related to the first one.

Question 14

When comparing the mean of two dependent samples,we are interested in only one variable: the difference between measurements for each person or object.

Accepted Answer

When comparing the means of two dependent samples, we are interested in the difference between measurements for each person or object in order to calculate the mean difference, which is used in the test of significance.

Question 15

Unlike the t distribution,the exact shape of the F distribution is determined by two different degrees of freedom instead of just a single value.

Accepted Answer

The F distribution is determined by two different degrees of freedom, one for the numerator and one for the denominator.

Question 16

When the population standard deviations are unknown and not assumed to be equal,pooling the sample standard deviations into a single estimate of their common population value is no longer applicable.

Accepted Answer

When population standard deviations are unknown and not assumed to be equal, it is not appropriate to pool the sample standard deviations into a single estimate of their common population value. Instead, separate estimates of the population standard deviations must be calculated for each group or sample being compared.

Question 17

If we wish to examine the before-and-after productivity of individual employees after a change in their workstation layout,a matched-pairs t-test is applicable.

Accepted Answer

A matched-pairs t-test is appropriate when we want to compare the means of two related or matched samples, such as the same group of employees before and after a change in workstation layout. Therefore, this type of test is suitable for examining the before-and-after productivity of individual employees.

Question 18

If we wish to compare the before-and-after reading speeds of individual participants in a speed-reading course,the z-test for comparing the means of two independent samples is appropriate.

Accepted Answer

The correct test for comparing the before-and-after reading speeds of individual participants in a speed-reading course is a paired t-test, not a z-test for independent samples, because the same participants are measured twice, making the samples dependent, not independent.

Question 19

A typical example of dependent samples occurs when we have before-and-after measures of the same individuals.

Accepted Answer

This is because dependent samples refer to samples that are related to one another, such as in the example given of before-and-after measures of the same individuals.

Question 20

One of the most useful applications of business statistics involves comparing two samples to examine whether a difference between them is significant or more likely due to chance variation from one sample to the next.

Accepted Answer

Comparing two samples to determine whether a difference between them is statistically significant is a common use of business statistics. This involves conducting hypothesis tests and calculating p-values to determine the likelihood that the observed difference is due to chance variation.

Quiz 11: Hypothesis Tests Involving Two Sample Means