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.
10.35 ${\overline{x}}_1=468.3,s_1=38.2,n_1=6$

$\overline{x_2}=394.6,s_2=84.7,n_2=14$

Short answer

No, that isn't reasonable. Because the sample standard deviations indicate that the two population standard deviations are not identical and the sample sizes are not roughly equal.

Step-by-step solution

Given information

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

${\overline{x}}_1=468.3,s_1=38.2,n_1=6$
${\overline{x}}_2=394.6,s_2=84.7,n_2=14$

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.

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

Note that: 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.
Assumption $3$ is also confirmed because two populations are regularly distributed.
Yet, there is a significant difference in standard deviation between the two populations.
$s_1=38.2$and $s_2=84.7$.
Assumption $4$ is violates.
As a result, Not reasonable.
Because the sample standard deviations indicate that the two population standard deviations are not identical and the sample sizes are not roughly equal.