Question

Supposed that the IQ scores of students in certain college has a mean of 500. A random sample of 35 students were administered the IQ Test and it was found out that the group mean 555. If the population standard deviation was 100, is the mean of this group differs from the population?

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu=500$

$H_1:\mu\not=500$

This corresponds to a two-tailed test, for which a z-test for one mean, with known population standard deviation, using the sample standard deviation, will be used.

Based on the information provided, the significance level is $\alpha=0.05,$ and the critical value for a two-tailed test is $z_c=1.96.$

The rejection region for this two-tailed test is $R=\{z: |z|>1.96\}$

The z-statistic is computed as follows:

$$z=\dfrac{\bar{x}-\mu}{\sigma/\sqrt{n}}=\dfrac{555-500}{100/\sqrt{35}}=3.2538$$

Since it is observed that $|z|=3.2538>1.96=z_c,$ it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is $p=2P(Z>3.2538)=0.001138,$ and since $p=0.001138<0.05=\alpha,$ it is concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population mean $\mu$ is different than $500,$ at the $\alpha=0.05$ significance level.

Therefore, there is enough evidence to claim that the the mean of this group differs from the population, at the $\alpha=0.05$ significance level.