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 73 and the population standard
deviation is 9, then the number of standard deviation below the null value of x =
72.3 is given by the z score:
Therefore, 72.3 is 0.39 standard deviations below the null value.
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 = 73, <span>σ = 9 and n = 25,
</span><span>
</span><span>The p-value is given by:
P(-0.39) = 0.3483
</span><span>Since α = 0.005 and p-value = 0.3483, 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 73.
</span>Part C:
In general for the alternative hypothesis,
So for the test procedure with α = <span>0.005
</span>Part D:
For α = 0.005, and a general sample size n we have that
Since, we want n so that β(70) = 0.01, thus
so we need n = 217.
Part E