Part A:
The z score of the hypothesis testing of n samples of a normally distributed data set is given by:

Given that the population mean is 75 and the population standard deviation is 7, then the number of standard deviation of the mean for x = 72.3 is given by the z score:

Therefore, 72.3 is 1.93 standard deviations below the mean.
Part B:
The test statistics of the hypothesis testing of n samples of a normally distributed data set is given by:

Thus given that x = 72.3, μ = 75, <span>σ = 7 and n = 25,

The p-value is given by:
</span><span>

Since </span>α = 0.01 and p-value = 0.03, this means that the p-value is greater than the <span>α, ant thus, we will faill to reject the null hypothesis.
Therefore, the conclussion is </span>
do not reject the null hypothesis. there is not sufficient evidence to conclude that the mean drying time is less than 75.
<span>Part C
</span>
In order to reject the null hypothesis, the p-value must be less than or equal to <span>α.
The p-value for z ≤ -2.8 is 0.00256.
Therefore, the </span>α <span>for the test procedure that rejects the null hypothesis when z ≤ −2.8</span> is 0.0026
Part D:
In general for the alternative hypothesis ,

<span>

</span>So for the test procedure with α = <span>0.0026
</span>

Part E:
For α = 0.0026, and a general sample size n <span>we have that
</span>

Since, we want n so that β(70) = 0.01, thus<span>
</span><span>

</span>so we need n = 52.
Part F:
p-value =

<span>
</span>Since p-value is larger that .01 we fail to reject µ = 76.