Answer to Question #343680 in Statistics and Probability for Nipuli

Question

A school has 1000 students. The principal of the school thinks that the average IQ of its students is at least 110. To prove her point, she administers an IQ test to 200 randomly selected students. Among the sampled students, the average IQ is 108 with a standard deviation of 10. Based on these results, should the principal accept or reject her original hypothesis? Assume a significance level of 1%

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu\ge110$

$H_a:\mu<110$

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 $\alpha = 0.01,$ $df=n-1=199$ degrees of freedom, and the critical value for a left-tailed test is $t_c =-2.345232.$

The rejection region for this right-tailed test is $R = \{t:t<-2.345232\}.$

The t-statistic is computed as follows:

$$t=\dfrac{108-110}{10/\sqrt{200}}=-2.8284$$

Since it is observed that $t =-2.8284<-2.345232=t_c,$ it is then concluded that the null hypothesis is rejected.

Using the P-value approach:

The p-value for left-tailed $df=199$ degrees of freedom, $t=-2.8284$ is $p=0.002578,$ and since $p=0.002578<0.01=\alpha,$ it is concluded that the null hypothesis is rejected.

Therefore, there is not enough evidence to claim that the population mean $\mu$ is at least 110, at the $\alpha = 0.01$ significance level.