Answer to Question #338375 in Statistics and Probability for sheshe

Question

A TV manufacturer claims that the life span of its regular TV sets is longer than 10 years with a standard deviation of 1.5 years. Using a random sample of their 16 TV sets, the average life span is found to be 11.5 years. Test the hypothesis that the TV sets' life span is longer than 10 years at a = 0.01.

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu\le 10$

$H_a:\mu>10$

This corresponds to a right-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 right-tailed test is $z_c = 2.3263.$

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

The z-statistic is computed as follows:

$$z=\dfrac{\bar{x}-\mu}{\sigma/\sqrt{n}}=\dfrac{11.5-10}{1.5/\sqrt{16}}=4$$

Since it is observed that $z = 4 >2.3263=z_c,$ it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is $p=P(Z>4)=0.000032,$ and since $p=0.000032<0.01=\alpha,$ it is concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population mean $\mu$ is greater than 10, at the $\alpha = 0.01$ significance level.