Questions: 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 117 men between the ages of 60 and 69, the mean height was (barx=68.9) 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=2.76). Use the (alpha=0.01) level of significance and the (P)-value method with Excel. (a) State the appropriate null and alternate hypotheses. [ H0: mu=69.4 H1: mu<69.4 ] This hypothesis test is a left-tailed test. (b) Compute the value of the test statistic. Round the answer to at least two decimal places.

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 117 men between the ages of 60 and 69, the mean height was (barx=68.9) 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=2.76). Use the (alpha=0.01) level of significance and the (P)-value method with Excel. (a) State the appropriate null and alternate hypotheses. [ H0: mu=69.4 H1: mu<69.4 ] This hypothesis test is a left-tailed test. (b) Compute the value of the test statistic. Round the answer to at least two decimal places.
Transcript text: 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 117 men between the ages of 60 and 69 , the mean height was $\bar{x}=68.9$ 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 $\sigma=2.76$. Use the $\alpha=0.01$ level of significance and the $P$-value method with Excel. (a) 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 test. (b) Compute the value of the test statistic. Round the answer to at least two decimal places.
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Solution

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Solution Steps

Step 1: State the Hypotheses

The null and alternative hypotheses for this test are defined as follows: \[ H_{0}: \mu = 69.4 \] \[ H_{1}: \mu < 69.4 \] This is a left-tailed test.

Step 2: Calculate the Standard Error

The standard error \(SE\) is calculated using the formula: \[ SE = \frac{\sigma}{\sqrt{n}} = \frac{2.76}{\sqrt{117}} \approx 0.26 \]

Step 3: Compute the Test Statistic

The test statistic \(Z\) is computed using the formula: \[ Z = \frac{\bar{x} - \mu_0}{SE} = \frac{68.9 - 69.4}{0.26} \approx -1.96 \]

Step 4: Determine the P-value

For a left-tailed test, the P-value corresponding to the test statistic \(Z = -1.96\) is: \[ P = T(z) \approx 0.03 \]

Final Answer

The test statistic is \(Z \approx -1.96\) and the P-value is approximately \(0.03\).

Thus, the final boxed answer is: \[ \boxed{Z = -1.96} \]

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