According to a survey, the mean height for men is $69.7$ inches. In a sample of 280 men between the ages of 60 and 69 , the mean height was $\bar { x } = 69.4$ 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. Assuming the population standard deviation to be $\sigma = 2.86$ inches, do you reject $H _ { 0 }$ at the $\alpha = 0.05$ level?
A) Yes
B) There is not enough information to draw a conclusion.
C) No

Question

Quizplus · Accepted Answer

We want to test the hypothesis:
$$H _ { 0 }: \mu \geq 69.7$$
$$H _ { 1 }: \mu < 69.7$$
where $\mu$ is the mean height of all adult men.

Since the population standard deviation $\sigma$ is known and the sample size $n = 280$ is large enough, we can use a z-test.

The test statistic is given by:
$$z = \frac{(\bar{x} - \mu)}{(\sigma/\sqrt{n})} = \frac{(69.4 - 69.7)}{(2.86/\sqrt{280})} = -1.311$$

Using a standard normal distribution table, the p-value for this test is approximately 0.095 (or 9.5%). Since this p-value is greater than the significance level $\alpha = 0.05$, we fail to reject the null hypothesis.

Therefore, we do not have enough evidence to suggest that the mean height of older men is significantly less than the mean height of all adult men at the 5% significance level.

According to a Survey, the Mean Height for Men Is 69.769.769.7

According to a Survey, the Mean Height for Men Is $69.7$