A company manufacturing pacemakers is testing a new electrode. The electrodes mu

Question

A company manufacturing pacemakers is testing a new electrode. The electrodes must adhere to a silicone substrate for at least 20 years. The company is going to test the hypothesis that the mean adherence time is 20 years vs. the alternative that it is less than 20 years at the significance level alpha =0.05 . The experiment will be conducted with a sample of 25 volunteers. Assume that the population distribution for the adherence time is approximately normally distributed. The average adherence time for the pacemakers in the 25 volunteers is found to be 18.8 years and the standard deviation of the sample is found to be 3 years.
(a) Is the null hypothesis rejected?
(b) If the company wants to decrease the probability of making a type I error without increasing the sample size, should the critical value be increased or decreased? Justify your answer.

Explanation / Answer

a)The hypothesis are

Ho:=20

H1:<20

Since the population variance is unknown, the sampling distribution for X is thet-distribution. The degrees of freedom is

v=n-1=24-1=23

Therefore the null hypothesis (Ho) is rejected when

X<o-to(s/n) =20-1.711*(3/25) =18.97

Since X = 18.8 is less than the critical value 18.97 Ho is rejected.

b)Decrease the critical value. The probability of a type I error is the probability of rejecting Ho when it is true. Therefore it can always be decreased by decreasing the size of the critical region (the region where Ho is rejected). Decreasing the critical region corresponds to decreasing the critical value in this case since we reject Ho when X is less than the critical value.

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