Answer:
There is a probability of P=0.02 of making a Type II error if the true mean is μ=1130.
Step-by-step explanation:
This is an hypothesis test for the lifetime of a certain ype of light bulb.
The population distribution is normal, with mean of 1,000 hours and STD of 110 hours.
The sample size for this test is n=10.
The significance level is assumed to be 0.05.
In this case, when the claim is that the new light bulb model has a longer average lifetime, so this is a right-tailed test.
For a significance level, the critical value (zc) that is bound of the rejection region is:
This value of zc is zc=1.645.
This value, for a sample with size n=10 is:
That means that if the sample mean (of a sample of size n=10) is bigger than 1057.22, the null hypothesis will be rejected.
The Type II error happens when a false null hypothesis failed to be rejected.
We now know that the true mean of the lifetime is 1130, the probability of not rejecting the null hypothesis (H0: μ=1100) is the probability of getting a sample mean smaller than 1057.22.
The probability of getting a sample smaller than 1057.22 when the true mean is 1130 is:
Then, there is a probability of P=0.02 of making a Type II error if the true mean is μ=1130.