Question

A company that produces computers claims that that the average life of one of their computers is 8 years. A sample of 12 computers shows a sample mean life of 7.8 years, with a population standard deviation of 0.2 years. Does the data suggest that the average life of one of the computers is not 8 years at 0.05 level significance? Assume computer lifetimes are normally distributed.

1. Give null and alternate hypothesis

2. Give test statistic and P value, state conclusion

3. State what type 1 and type 2 errors would be

Answer 1

n = 12

1,) The test hypothesis is

2.) This is a two-sided test because the alternative hypothesis is formulated to detect differences from the hypothesized mean value of 30 on either side.
Now, the value of test static can be found out by following formula:

Since P-value of a two tailed test is equal to 2(1 - \phi(|Z_0|)

P = 2(1 - 0.0002660043497499743)
P = 1.999468
Since P = 1.999468 > 0.05, we fail to reject the null hypothesis
For , . Since , we reject the null hypothesis in favor of the alternative hypothesis .

3.) Since we rejected the null hypothesis, type 1 error would come into play. Type 1 errors are equivalent to false positives. Since the value of alpha is 0.05. This means that there is 5% probability that we will reject a true null hypothesis.