Answer to Question #337679 in Statistics and Probability for April Joy

Question

A TV manufacturer claims that the life span of its regular TV sets is 12 years with a standard

deviation of 1.2 years. Using a random sample of their 20 TV sets, the average life span is found

to be 11.2 years. Test the hypothesis that the T sets’ life span of 12 years as claimed by the

manufacturer is true at 0.05 level.

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu=12$

$H_a:\mu\not=12$

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:

$$z=\dfrac{\bar{x}-\mu}{\sigma/\sqrt{n}}=\dfrac{11.2-12}{1.2/\sqrt{20}}=-2.9814$$

Since it is observed that $|z| = 2.981 >1.96= z_c,$ it is then concluded that the null hypothesis is rejected.

Using the P-value approach:

The p-value is $p=2P(Z<-2.9814)=0.002869,$and since $p=0.002869<0.05=\alpha,$ it is concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population mean $\mu$

is different than 12, at the $\alpha = 0.05$ significance level.