Chapter 9, Problem 118UP

a.

To determine

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

b.

To determine

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

c.

To determine

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

d.

To determine

Provide 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.

e.

To determine

Give 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.