Answer to Question #350891 in Statistics and Probability for vrix

Question

A TV manufacturer claims that the life span of its regular TV sets is longer than 10 years. Using a random sample of their 25 TV sets, the average life span is found to be 11.9 years with a standard deviation of 1.8 years. Test the hypothesis that the TV sets' life span is longer than 10 years at cx = 0.10

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu=10$

$H_1:\mu>10$

This corresponds to a right-tailed test, for which a t-test for one mean, with unknown population standard deviation, using the sample standard deviation, will be used.

Based on the information provided, the significance level is $\alpha = 0.10,$ $df=n-1=24$ and the critical value for a right-tailed test is $t_c =1.317836.$

The rejection region for this right-tailed test is $R = \{t:t>1.317836\}.$

The t-statistic is computed as follows:

$$t=\dfrac{11.9-10}{1.8/\sqrt{25}}=\dfrac{110-100}{10/\sqrt{16}}=5.2778$$

Since it is observed that $t=5.2778>1.317836=t_c,$ it is then concluded that the null hypothesis is rejected.

Using the P-value approach:

The p-value for right-tailed, $df=24$ degrees of freedom, $t=5.2778$ is $p=0.00001,$ and since $p=0.00001<0.10=\alpha,$ 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 $\alpha = 0.10$ significance level.