Question

Conduct a traditional hypothesis test for the null hypothesis H0: µ = 100 against the alternative

hypothesis H1: µ > 100 based on the 35 random observations. The sample mean is 105 and the

sample standard deviation is 15. Use α = 0.01.

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu=100$

$H_1:\mu>100$

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 $\alpha = 0.01,$ $df=n-1=34$ and the critical value for a right-tailed test is $t_c =2.44115.$

The rejection region for this right-tailed test is $R = \{t:t>2.44115\}.$

The t-statistic is computed as follows:

$$t=\dfrac{\bar{x}-\mu}{s/\sqrt{n}}=\dfrac{105-100}{15/\sqrt{35}}=1.9720$$

Since it is observed that $t=1.9720<2.44115=t_c,$ it is then concluded that the null hypothesis is not rejected.

Using the P-value approach:

The p-value for right-tailed, $df=34$ degrees of freedom, $t=1.9720$ is $p= 0.028395,$ and since $p= 0.028395>0.01=\alpha,$ 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 100, at the $\alpha = 0.01$ significance level.