Answer to Question #345166 in Statistics and Probability for asas

Question

The head of the PE Department of a certain high school claims that the mean height ofGrade 7 students is 163 cm. The mean height of 45 randomly selected Grade 7 studentsis 161 cm. Using 0.01 significance level, can it be concluded that the mean height ofGrade 7 students is different from 163 cm as claimed by the Head of the PE Department?

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu=163$

$H_a:\mu\not=163$

This corresponds to a two-tailed test, for which a t-test for two population means, with two independent samples, with unknown population standard deviations will be used.

Based on the information provided, the significance level is $\alpha = 0.01,$ and the degrees of freedom are $df = 45-1=44$

Hence, it is found that , using t-table, the critical value for this two-tailed test is $\:\:t_c\:=2.69$ for $\alpha =0.01$ and $\:\:df=44$ .

The rejection region for this two-tailed test is $R=\left\{t:∣t∣>2.69\right\}.$

The z-statistic is computed as follows:

$$t=\dfrac{\bar{x}-\mu}{s/\sqrt{n}}=\dfrac{161-163}{\sqrt{45}}\approx-0.3$$

Since it is observed that $z = -0.066> -2.3263=z_c,$ it is then concluded that the null hypothesis is not rejected.

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

is less than 89, at the $\alpha = 0.01$ significance level.

Since it is observed that $∣t∣=0.3<t c ​ =2.69$ it is then concluded that the null hypothesis is not rejected.