Question

the mathematics teacher claims that the mean iq of statistics students is 110 with standard deviation of 12. The mean IQ of 28 randomly selected statistics students is 112 . Test the difference of the population and sample means at 5%level of significance.

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu=110$

$H_1:\mu\not=110$

This corresponds to a two-tailed test, for which a z-test for one mean, with known population 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:

$$z=\dfrac{\bar{x}-\mu}{\sigma/\sqrt{n}}=\dfrac{112-110}{12/\sqrt{28}}=0.8919$$

Since it is observed that $|z|=0.8919<1.96=z_c,$ it is then concluded that the null hypothesis is not rejected.

Using the P-value approach:

The p-value is $p=2P(z>0.8919)=0.372447,$ and since $p= 0.372447>0.05=\alpha,$ it is concluded that the null hypothesis is not rejected.

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

is different than 110, at the $\alpha = 0.05$ significance level.