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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 304 men between the ages of 60 and 69, the mean height was [tex]\bar{x} = 68.8[/tex] inches. Public health officials want to determine whether the mean height [tex]\mu[/tex] for older men is less than the mean height of all adult men. Assume the population standard deviation to be [tex]\sigma = 3.13[/tex]. Use the [tex]\alpha = 0.05[/tex] level of significance and the critical value method.

Part 0 / 5

Part 1 of 5:

(a) State the appropriate null and alternate hypotheses.

[tex]\[
\begin{array}{l}
H_0: \mu = 69.4 \\
H_1: \mu \ \textless \ 69.4
\end{array}
\][/tex]

This hypothesis test is a left-tailed test. [tex]$\square$[/tex]

Part 1 / 5

Part 2 of 5:

Find the critical value(s). Round the answer(s) to three decimal places, if necessary. If there is more than one critical value, separate them with commas.

Critical value(s): [tex]$\square$[/tex]



Answer :

GinnyAnswer
To determine the critical value for a left-tailed test at the [tex]\(\alpha = 0.05\)[/tex] level of significance, we use the standard normal distribution (Z-distribution).

For a left-tailed test:
The critical value corresponding to [tex]\(\alpha = 0.05\)[/tex] is the value of Z such that the area to the left of Z is 0.05.

From standard normal distribution tables or using statistical tools, the critical value for [tex]\(\alpha = 0.05\)[/tex] is:
[tex]\[ \text{Critical value} = -1.645 \][/tex]

Therefore, the critical value at the [tex]\(\alpha = 0.05\)[/tex] level of significance is:
Critical value(s): [tex]\(-1.645\)[/tex]

Thus, for a left-tailed test, our critical value is -1.645.

[tex]$\square$[/tex]

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