Answer to Question #345094 in Statistics and Probability for Main

Question

The Head of the Mathematics Department announced that the mean score of Grade 11

students in the first validating test in Mathematics was 65 and the standard deviation was 12.

One student who believed that the mean score was less than this, interviewed 50 randomly

selected students and obtained a mean score of 60. At 0.01 level of significance, test the

student's belief.

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu\ge65$

$H_a:\mu<65$

This corresponds to a left-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 $\alpha = 0.01,$ and the critical value for a left-tailed test is (using t-table) $z_c = -2.3263.$

The rejection region for this left-tailed test is $R = \{z: z < -2.3263\}.$

The z-statistic is computed as follows:

$$z=\dfrac{\bar{x}-\mu}{\sigma/\sqrt{n}}=\dfrac{60-65}{12/\sqrt{50}}\approx-0.059$$

Since it is observed that $z = -0.059> -2.3263=z_c,$ 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 65, at the $\alpha = 0.01$ significance level.