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:
![P(z>z_c)=0.05](https://tex.z-dn.net/?f=P%28z%3Ez_c%29%3D0.05)
This value of zc is zc=1.645.
This value, for a sample with size n=10 is:
![z_c=\dfrac{X_c-\mu}{\sigma/\sqrt{n}}\\\\\\X_c=\mu+\dfrac{z_c\cdort\sigma}{\sqrt{n}}=1000+\dfrac{1.645*110}{\sqrt{10}}=1000+57.22=1057.22](https://tex.z-dn.net/?f=z_c%3D%5Cdfrac%7BX_c-%5Cmu%7D%7B%5Csigma%2F%5Csqrt%7Bn%7D%7D%5C%5C%5C%5C%5C%5CX_c%3D%5Cmu%2B%5Cdfrac%7Bz_c%5Ccdort%5Csigma%7D%7B%5Csqrt%7Bn%7D%7D%3D1000%2B%5Cdfrac%7B1.645%2A110%7D%7B%5Csqrt%7B10%7D%7D%3D1000%2B57.22%3D1057.22)
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:
![z=\dfrac{X-\mu}{\sigma/\sqrt{n}}=\dfrac{1057.22-1130}{110/\sqrt{10}}=\dfrac{-72.78}{34.7851}=-2.0923 \\\\\\P(M](https://tex.z-dn.net/?f=z%3D%5Cdfrac%7BX-%5Cmu%7D%7B%5Csigma%2F%5Csqrt%7Bn%7D%7D%3D%5Cdfrac%7B1057.22-1130%7D%7B110%2F%5Csqrt%7B10%7D%7D%3D%5Cdfrac%7B-72.78%7D%7B34.7851%7D%3D-2.0923%09%09%09%09%09%09%09%5C%5C%5C%5C%5C%5CP%28M%3C1057.22%29%3DP%28z%3C-2.0923%29%3D0.01821)
Then, there is a probability of P=0.02 of making a Type II error if the true mean is μ=1130.