Chapter 23, Problem 22E

To determine

To explain how these two results are consistent and your explanation should discuss the confidence level, the P-value and the decision.

Explanation of Solution

In this question, a nutrition lab tested $40$ hot dogs to see if their mean sodium content was less than the $325$ mg upper limit set by regulations for “reduced sodium” franks. The confidence interval has confidence level $95\%$ which corresponds with a two-sided significance test with significance level of:

  $\begin{array}{l}\alpha =1-90\%\\ =1-0.90\\ =0.10\end{array}$

The confidence interval $(317.20,326.80)$ contains $325$ mg, which indicates that the mean of $325$ mg is likely to occur and thus we would fail to reject the null hypothesis that states that the population mean is $325$ mg.

The P-value of the significance test is $0.142$ . Since the P-value is larger than the significance level $\alpha =0.10$ , we would in this case fail to reject the null hypothesis that states that the population mean is $325$ mg.

Thus, we note that the results of the significance test and the confidence interval are consistent.