Question

In each of Exercises 10.35-10.38, we have provided summary statistics for independent simple random samples from two populations. Preliminary data analyses indicate that the variable under consideration is normally distributed on each population. Decide, in each case, whether use of the pooled $t$-lest and pooled $t$-interval procedure is reasonable. Explain your answer.

10.36 ${\overline{x}}_1=115.1,s_1=79.4,n_1=51$
${\overline{x}}_2=24.3,s_2=10.5,n_2=19$

Short answer

Not reasonable. pooled $t$- test and pooled $t$- interval test are not appropriate for comparing two population means.

Step-by-step solution

Given information

To explain the use of pooled $t$- test and pooled $t$-interval test is reasonable with the given sample data.

${\overline{x}}_1=115.1,s_1=79.4,n_1=51$

${\overline{x}}_2=24.3,s_2=10.5,n_2=19$

Explanation

For comparing two population means, the following assumptions are made when using the pooled $t$- test and pooled $t$- interval test.
(1) Simple random variables
(2) Independent samples
(3) Normal population or large samples.

In the case of large samples, this pooled $t$- test is resilient to moderate violation.
(4) Equal population standard deviation.

In the case of nearly equal sample sizes, this pooled $t$- test is robust for modest violation.
Assumptions $1$and $2$ are correct based on the information provided.
Because both populations are regularly distributed and the sample sizes are high $(n_1=57,n_2=19)$, assumption $3$ is also confirmed.
However, the standard deviation of two populations $s_1=79.4$and $s_2=10.5$ is vastly different.
This violates assumption 4 .
As a result, pooled $t$- test and pooled $t$- interval test are not reasonable for comparing two population means.