Suppose a sample of O-rings was obtained and the wall thickness (in inches) of each was recorded. Use a normal probability plot to assess whether the sample data could have come from a population that is normally distributed.
\begin{tabular}{llll}
0.155 & 0.181 & 0.206 & 0.216 \\
\hline 0.226 & 0.232 & 0.240 & 0.244 \\
\hline 0.249 & 0.258 & 0.277 & 0.272 \\
\hline 0.292 & 0.293 & 0.321 & 0.337
\end{tabular}
Using the correlation coefficient of the normal probability plot, is it reasonable to conclude that the population is normally distributed? Select the correct choice below and fill in the answer boxes within your choice. (Round to three decimal places as needed:)
A. No. The correlation between the expected $z$-scores and the observed data, $\square$, does not exceed the critical value, $\square$. Therefore, it is not reasonable to conclude that the data come from a normal population.
B. Yes. The correlation between the expected z-scores and the observed data, _. , exceeds the critical value, _. Therefore, it is reasonable to conclude that the data come from a normal population.
C. No. The correlation between the expected z-scores and the observed data, _ . does.not exceed the critical value, ( . . Therefore, it is reasonable to conclude that the data come from a normal population.
D. Yes. The correlation between the expected z-scores and the observed data, $\square$, exceeds the critical value, —. Therefore, it is not reasonable to conclude that the data come from a normal population.
Answer
Final Answer: \(\boxed{\text{Yes. The correlation between the expected z-scores and the observed data, 0.996, exceeds the critical value, 1.96. Therefore, it is reasonable to conclude that the data come from a normal population.}}\)
Step 1 ：First, we need to generate a normal probability plot for the given data and calculate the correlation coefficient of the plot. The correlation coefficient will tell us how closely the data follows a normal distribution. If the correlation coefficient is close to 1, it means the data is closely following a normal distribution. If it's far from 1, it means the data is not following a normal distribution.
Step 2 ：We also need to find the critical value for the correlation coefficient. If the calculated correlation coefficient is greater than the critical value, we can conclude that the data comes from a normal population. Otherwise, we can't make that conclusion.
Step 3 ：The given data is [0.155, 0.181, 0.206, 0.216, 0.226, 0.232, 0.24, 0.244, 0.249, 0.258, 0.277, 0.272, 0.292, 0.293, 0.321, 0.337].
Step 4 ：The correlation coefficient is calculated to be approximately 0.996, which is very close to 1. This indicates that the data closely follows a normal distribution.
Step 5 ：The critical value is approximately 1.96. Since the correlation coefficient is greater than the critical value, we can conclude that the data comes from a normal population.
Step 6 ：Final Answer: \(\boxed{\text{Yes. The correlation between the expected z-scores and the observed data, 0.996, exceeds the critical value, 1.96. Therefore, it is reasonable to conclude that the data come from a normal population.}}\)
