Problem
Height and age: Are older men shorter than younger men? According to a national report, the mean height for U.S. men is 69.4 inches. In a sample of 307 men between the ages of 60 and 69 , the mean height was $\bar{x}=69.1$ inches. Public health officials want to determine whether the mean height $\mu$ for older men is less than the mean height of all adult men. Assume the population standard deviation to be $\sigma=3.01$. Use the $\alpha=0.10$ level of significance and the $P$-value method with the TI-84 calculator. Part 1 of 4 State the appropriate null and alternate hypotheses. \[ \begin{array}{l} H_{0}: \mu=69.4 \\ H_{1}: \mu<69.4 \end{array} \] This hypothesis test is a left-tailed $\quad$ test. Part: $1 / 4$ Part 2 of 4 Compute the $P$-value. Round the answer to at least four decimal places. \[ P \text {-value }=\square \]
Solution
Step 1 ：First, we need to calculate the test statistic (z-score) using the formula: \(Z = \frac{{\bar{X} - \mu}}{{\sigma/\sqrt{n}}}\)
Step 2 ：Substitute the given values into the formula: \(Z = \frac{{69.1 - 69.4}}{{3.01/\sqrt{307}}}\)
Step 3 ：After calculating, we get: \(Z = -1.41\)
Step 4 ：Now, we can use the TI-84 calculator to find the P-value. The P-value is the probability that a z-score is less than -1.41 under the standard normal distribution.
Step 5 ：Using the TI-84 calculator, we find that the P-value is approximately 0.0793
Step 6 ：So, the final answer is \(\boxed{0.0793}\)
