Question

A hard Drive manufacturers is required to ensure that the mean time between failures for its new hard drive is 1 million hours. A stress test is designed that can simulate the workload at a much faster rate. The test is designed so that a test lasting 10 days is equivalent to the hard drive lasting 1 million hours. In stress tests of 15 hard drives, the average is 9.5 days with a standard deviation of 1 day. Can the manufacturer conclude that it meets the requirement?

a.) Identify and define the population parameter.

b.) Set up the hypotheses to be tested by the manufacturer.

c.) Calculate the test statistic for the hypothesis in part b.

d.) What distribution does the test statistic in part c follow?

e.) Calculate the p-value associated with the test statistic in part c.

f.) Based on these results explain if its reasonable for the manufacturer to claim that it meets the requirement.

Answer 1

Solution: a) Here, the population parameter is time between failures or the lasting time for all new hard drive.

b) The null and alternative hypotheses to be tested by the manufacturer is given by--

H0: mu = 10 vs Ha: mu < 10 (Since we are to test if the hard drives can meet the requirements. If the null is rejected, it means we can't say that the requirement is met)

c) The test statistic for the hypothesis in part b is T= (xbar-mu0)/(s/sqrt(n)) ; where xbar = sample mean, mu0 = the hypothesized value of the population mean, n = sample size, s = sample standard deviation, sqrt refers to the square root function.

Here, mu0 = 10, xbar = 9.5, s = 1, n = 15 and T(observed) =-1.936492

d) Under H0,the test statistic in part c follows a Student's t distribution, i.e., T ~ t(n-1)

e) P-value is the probability of obtaining a test statistic more extreme than the observed test statistic assuming the null hypothesis is true. Here p-value is obtained by looking at the student's t distribution.

The p-value associated with the test statistic in part c. comes out to be = 0.0366

f) We reject H0 iff p-value is less than the level of significance. It level of significance is considered to be 0.05, then the obtained p-value < level of significance.

Hence we reject H0 and conclude at a 5% level of significance on the basis of the given sample measures that there is enough evidence to say that the lasting time of the hard drive is significantly less than 10 hours.

Thus, it is not reasonable for the manufacturer to claim that it meets the requirement.