Step 1

It is given that for a sample of n = 16 sheets, mean value and standard deviation for aluminum alloy sheets are 70 GPa and 1.6 GPa respectively.

 
Step 2

a.

If $\bar X$ is the sample mean Young’s modulus for a random sample of n = 16 sheets, required to find where is the sampling distribution of $\bar X$ centered, and what is the standard deviation of the $\bar X$ distribution.

The sample mean Young’s modulus is $\bar X$.

The population parameters, mean and standard deviation of random variable X are $\mu$ and $\sigma$ respectively.

The sample mean $E\left( {\bar X} \right)$ and sample standard deviation ${\sigma _{\bar X}}$ are given as:

$E\left( {\bar X} \right) = \mu $ and ${\sigma _{\bar X}} = \frac{\sigma }{{\sqrt n }}$

Substituting the given values, the sample mean $E\left( {\bar X} \right)$ is obtained as follows:

\[\begin{array}{c} E\left( {\bar X} \right) = \mu \\ = 70 \end{array}\]

And, the sample standard deviation ${\sigma _{\bar X}}$ is obtained as follows:

\[\begin{array}{c} {\sigma _{\bar X}} = \frac{\sigma }{{\sqrt n }}\\ = \frac{{1.6}}{{\sqrt {16} }}\\ = 0.4 \end{array}\]
 
Step 3

b.

to answer the questions posed in part (a) for a sample size of n = 64 sheets.

Given that, for a sample size of n = 64 sheets, $\mu = 70\,\,and\,\sigma = 1.6$.

Now, the sample mean $E\left( {\bar X} \right)$remains the same irrespective of change in sample size.

Therefore, $E\left( {\bar X} \right) = 70$.

And, the sample standard deviation ${\sigma _{\bar X}}$ for n= 64, is given as:

\[\begin{array}{c} {\sigma _{\bar X}} = \frac{\sigma }{{\sqrt n }}\\ = \frac{{1.6}}{{\sqrt {64} }}\\ = 0.2 \end{array}\]
 
Step 4

c.

Required to find for which of the two random samples, the one of part (a) or the one of part (b), is $\bar X$ more likely to be within 1 GPa of 70 GPa.

In part (a) the sample size is 16, and the obtained sample standard deviation is 0.4 GPa.

In part (b) the sample size is 64, and the obtained sample standard deviation is 0.2 GPa.

The sample standard deviation is less in part (b) compared to part (a), but the sample size is large for part (b) compared to part (a).

Data is less variable when standard deviation is less, which implies more observations falling within 1 standard deviation. As a result of this, the sample mean $\bar X$ will fall within 1 standard deviation of population mean.

Thus, $\bar X$ is more likely to be within 1 GPa of 70 GPa in the random sample in part (b).