Answer to Question #347490 in Statistics and Probability for jouuuuuu

Question

A sports trainer wants to know whether the true average time of his athletes who do 100-meter sprint is 98 seconds. He recorder 18 trials of his team and found that the average time is 98.2 seconds with a standard deviation of 0.4 second

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu=98$

$H_1:\mu\not=98$

This corresponds to a two-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 $\alpha = 0.05,$ $df=n-1=17$ and the critical value for a two-tailed test is $t_c =2.109816.$

The rejection region for this two-tailed test is $R = \{t:|t|>2.109816\}.$

The t-statistic is computed as follows:

$$t=\dfrac{\bar{x}-\mu}{s/\sqrt{n}}=\dfrac{98.2-98}{0.4/\sqrt{18}}=2.1213$$

Since it is observed that $|t|=2.1213>2.109816=t_c,$ it is then concluded that the null hypothesis is rejected.

Using the P-value approach:

The p-value for two-tailed, $df=17$ degrees of freedom, $t=2.1213$ is $p=0.048898,$ and since $p= 0.048898<0.05=\alpha,$ it is concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population mean $\mu$

is different than 98, at the $\alpha = 0.05$ significance level.