A consumer electronics firm has developed a new type of remote control button that is designed to operate longer before becoming intermittent. A random sample of 35 of the new buttons is selected and each is tested in continuous operation until becoming intermittent. The resulting lifetimes are found to have a sample mean of 1,241.2 hours and a sample standard deviation of 110.8.
a. Independent tests reveal that the mean lifetime (in continuous operation) of the best remote control button on the market is 1,200 hours. Letting μ be the mean lifetime of the population of all new remote control buttons that will or could potentially be produced, set up the null and alternative hypotheses needed to attempt to provide evidence that the new button’s mean lifetime exceeds the mean lifetime of the best remote button currently on the market.
b. Using the previously given sample results, use critical values to test the hypotheses you set up in part a by setting α equal to .10, .05, .01, and .001. What do you conclude for each value of α?
c. Suppose that a sample mean of 1,241.2 and a sample standard deviation of 110.8 had been obtained by testing a sample of 100 buttons. Use critical values to test the hypotheses you set up in part a by setting α equal to .10, .05, .01, and .001. Which sample (the sample of 35 or the sample of 100) gives a more statistically significant result? That is, which sample provides stronger evidence that Ha is true?
d. If we define practical importance to mean that μ exceeds 1,200 by an amount that would be clearly noticeable to most consumers, do you think that the results of parts b and c have practical importance? Explain why the samples of 35 and 100 both indicate the same degree of practical importance.
e. Suppose that further research and development effort improves the new remote control button and that a random sample of 35 buttons gives a sample mean of 1,524.6 hours and a sample standard deviation of 102.8 hours. Test your hypotheses of part a by setting a equal to .10, .05, .01, and .001.
(1) Do we have a highly statistically significant result? Explain.
(2) Do you think we have a practically important result? Explain.