Please help me solve this problem: 34. Suppose that a lightbulb manufacturing pl
ID: 3361897 • Letter: P
Question
Please help me solve this problem:
34. Suppose that a lightbulb manufacturing plant produces bulbs with a mean life of 2000 hours and a standard deviation of 200 hours. An inventor claims to have developed an improved process that produces bulbs with a longer mean life and the same standard deviation. The plant manager randomly selects 100 bulbs produced by the process She says that she will believe the inventor's claim if the sample mean life of the bulbs is greater than 2100 hours; otherwise, she will conclude that the new process is no better than the old process. Let denote the mean of the new process. Consider the null and alternative hypothesis Ho : -2000 vs H, 2000 (a) What is the probability of making a Type 1 error if the new procedure is no (b) Suppose the new process is in fact better and has a mean bulb life of 2150 hours. (c) What testing procedure should the plant manager use if she wants the size of better than the old one? What is the power of the plant manager's testing procedure? her test to be 5%?Explanation / Answer
Question 34 (a)
Population mean life = 2000 hours
standard deviation of mean life = 200 hours
we will reject the null hypothesis if the sample mean is greater than 2100 hours
Here type I error is the probability of rejecting null hypothesis when it is true.
Here critical test statistic Z = (2100 - 2000)/ (200/ sqrt(100) = 100/ 20 = 5
Here Probability of type I error = Pr(Z > 5) = 2.8 x 10-8 or 0.
(b) Here true mean of the new process = 2150 hours
population standard deviation = 200 hours
standard error of mean = 200/ sqrt(100) = 20 hours
Here probability of type II error = Pr(x < 2100 ) = NORM (x < 2100 ; 2150 ; 20)
Z = (2100 - 2150)/20 = -2.5
Pr(Type II error) = Pr(Z < -2.5) = 0.0062
(c) the size of a test is the probability of falsely rejecting the null hypothesis. here the size of the test is 5%, So, we will use the simple random sampling of Z test for one sample mean.