Question #337679

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.

1
Expert's answer
2022-05-06T13:00:09-0400

The following null and alternative hypotheses need to be tested:

H0:μ=12H_0:\mu=12

Ha:μ12H_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 α=0.05,\alpha = 0.05,

and the critical value for a two-tailed test is zc=1.96.z_c = 1.96.

The rejection region for this two-tailed test is R={z:z>1.96}.R = \{z: |z| > 1.96\}.

The z-statistic is computed as follows:


z=xˉμσ/n=11.2121.2/20=2.9814z=\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=zc,|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,p=2P(Z<-2.9814)=0.002869,and since p=0.002869<0.05=α,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 μ\mu

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


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