Chapter 8, Problem 8.89LM

List the assumptions necessary for each of the following inferential techniques:

1. a. Large-sample inferences about the difference (μ₁ – μ₂) between population means using a two-sample z-statistic
2. b. Small-sample inferences about (μ₁ – μ₂) using an independent samples design and a two-sample t-statistic
3. c. Small-sample inferences abojt (μ₁ – μ₂) using a paired diference design and a single sample t-statistic to analyze the differences
4. d. Large-sample inferences about the differences (μ₁ – μ₂) between binomial proportions using a two-sample z-statistic
5. e. Inferences about the ratio $\frac{{\sigma }_1^2}{{\sigma }_2^2}$ of two population variances using an F-test.

To determine

To find: The assumptions required for the analysis of the difference of two population means $\left({\mu }_1-{\mu }_2\right)$ using the two sample z-statistic.

Explanation of Solution

For the z-statistic of the difference of two means to be valid, the following assumptions are required.

Requirements for a large sample test for difference of two population means:

• The samples must be independent and drawn randomly.
• The sample sizes must be greater than 30 that is $n_1>30$ and $n_2>30$.

Thus, the requirements must be satisfied for the z-statistic of the difference of two means to be valid.

To determine

To find: The assumptions required for the analysis of the difference of two population means $\left({\mu }_1-{\mu }_2\right)$ using the two sample t-statistic.

Explanation of Solution

For the t-statistic of the difference of two means to be valid, the following assumptions are required.

Assumptions about the two populations:

• The variances of both the populations must be equal.
• The relative frequency distribution of the two populations must be approximately normal.

Assumptions about the two samples:

• The two samples must be drawn from the desired population independently in a random process.

Thus, the requirements must be satisfied for the t-statistic of the difference of two means to be valid.

To determine

To find: The assumptions required for the analysis of the paired difference of two population means $\left({\mu }_1-{\mu }_2\right)$.

Explanation of Solution

For the paired t test of the difference of two means to be valid, the following assumptions are required.

Requirements for a small sample paired difference test:

• The difference sample must be drawn randomly from the target population of difference.
• The population of difference must be approximately normal.

Thus, the requirements must be satisfied for the z-statistic of the difference of two means to be valid.

To determine

To find: The assumptions required for the analysis of the difference of two population proportions $\left({\mu }_1-{\mu }_2\right)$ using the two sample z-statistic.

Explanation of Solution

For the z-statistic of the difference of two population proportions to be valid, the following assumptions are required.

Requirements for a large sample test for difference of two population means:

• The samples must be independent and drawn randomly from the binomial population.
• The sample sizes must be large for the normal approximation using the central limit theorem.

Conditions to check large sample for difference of population proportions $\left({\widehat{p}}_1-{\widehat{p}}_2\right)$:

• $n_1{\widehat{p}}_1\ge 15$ and $n_1{\widehat{q}}_1\ge 15$.
• $n_2{\widehat{p}}_2\ge 15$ and $n_2{\widehat{q}}_2\ge 15$.

Thus, the requirements must be satisfied for the z-statistic of the difference of two population proportions to be valid.

To determine

To find: The assumptions required for the analysis of the difference of two population variances using the two sample F-statistic.

Explanation of Solution

For the F-statistic of the difference of two population variances to be valid, the following assumptions are required.

Requirements for a large sample test for difference of two population means:

• The samples must be independent and drawn randomly.
• The relative frequency distribution of the two populations must be approximately normal.

Thus, the requirements must be satisfied for the F-statistic of the difference of two population variances to be valid.