Answer to Question #345093 in Statistics and Probability for Derek

Question

2. A simple random sample of 15 people from a certain population has a

mean age of 35 with a standard deviation of 20. Can we conclude that

the mean age of the population is younger than 35? Let alpha = .05.

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu\ge35$

$H_1:\mu<35$

This corresponds to a left-tailed test, for which a t-test for one mean, with unknown population standard deviation, using the sample standard deviation, will be used.

Based on the information provided, the significance level is $\alpha = 0.05,$ $df=n-1=14$ and the critical value for a left-tailed test is $t_c =-1.76131.$

The rejection region for this left-tailed test is $R = \{t:t<-1.76131\}.$

The t-statistic is computed as follows:

$$t=\dfrac{\bar{x}-\mu}{s/\sqrt{n}}=\dfrac{35-35}{20/\sqrt{15}}=0$$

Since it is observed that $t=0>-1.76131=t_c,$ it is then concluded that the null hypothesis is not rejected.

Using the P-value approach:

The p-value for left-tailed, $df=14$ degrees of freedom, $t=0$ is $p=0.5,$ and since $p=0.5>0.05=\alpha,$ it is concluded that the null hypothesis is notrejected.

Therefore, there is not enough evidence to claim that the population mean $\mu$

is less than 35, at the $\alpha = 0.05$ significance level.