Answer to Question #349452 in Statistics and Probability for jjj

Question

A researcher knows that the average height of Filipino women is 1.53 m. A random sample of

26 women was taken and found to have a mean height if 1.56 m with a standard deviation of

0.1 m. Is there a reason to believe that the 26 women in the sample are significantly taller than

the others?

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu\le1.53$

$H_1:\mu>1.53$

This corresponds to a right-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 $\alpha = 0.05,$ $df=n-1=25$ and the critical value for aright-tailed test is $t_c =1.708141.$

The rejection region for thisright-tailed test is $R = \{t:t>1.708141\}.$

The t-statistic is computed as follows:

$$t=\dfrac{\bar{x}-\mu}{s/\sqrt{n}}=\dfrac{1.56-1.53}{0.1/\sqrt{26}}=1.5297$$

Since it is observed that $t=1.5297<1.708141=t_c,$ it is then concluded that the null hypothesis is not rejected.

Using the P-value approach:

The p-value for right-tailed, $df=25$ degrees of freedom, $t=1.5297$ is $p=0.069324,$ and since $p=0.069324>0.05=\alpha,$ it is concluded that the null hypothesis is not rejected.

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

is greater than 1.53, at the $\alpha = 0.05$ significance level.