In recent years, several companies have been formed to compete with AT&T in long-distance calls. All advertise that their rates are lower than AT&T's. AT&T has responded by arguing that there will be no dierence in billing for the average consumer. Suppose that the average bill for current AT&T customers is $21. A statistician takes a random sample of 100 customers and recalculates their last month's AT&T bill using the rates quoted by a competitor. Let X be the random variable for the amount of a recalculated bill, and µ be the unknown population mean of X. The sample mean and standard deviation for X are $20 and $6. The statistician wants to test if µ is signi cantly dierent from $21, which is the average bill for current AT&T customers. (a) The statistician wants to formulate hypotheses in favor of AT&T. State the hypotheses. (b) Calculate the t-statistic. Under what signi cance level, among 1%, 5% and 10%, is the null hypothesis rejected? (c) Calculate the p-value. Under what signi cance level, among 1%, 5% and 10%, is the null hypothesis rejected? (d) Using 5% signi cance level, calculate the critical values of the test. Is the null hypothesis rejected? (e) The competitor would react by using dierent set of hypotheses. For example, consider H0 : µ = 19 and H1 : µ 6= 19. What is the conclusion of this test for α = 5%?
In recent years, several companies have been formed to compete with AT&T in long-distance calls. All advertise
that their rates are lower than AT&T's. AT&T has responded by arguing that there will be no dierence in billing
for the average consumer.
Suppose that the average bill for current AT&T customers is $21. A statistician takes a random sample of 100
customers and recalculates their last month's AT&T bill using the rates quoted by a competitor. Let X be the
random variable for the amount of a recalculated bill, and µ be the unknown population mean of X. The sample
mean and standard deviation for X are $20 and $6. The statistician wants to test if µ is signi cantly dierent from
$21, which is the average bill for current AT&T customers.
(a) The statistician wants to formulate hypotheses in favor of AT&T. State the hypotheses.
(b) Calculate the t-statistic. Under what signi cance level, among 1%, 5% and 10%, is the null hypothesis rejected?
(c) Calculate the p-value. Under what signi cance level, among 1%, 5% and 10%, is the null hypothesis rejected?
(d) Using 5% signi cance level, calculate the critical values of the test. Is the null hypothesis rejected?
(e) The competitor would react by using dierent set of hypotheses. For example, consider H0 : µ = 19 and
H1 : µ 6= 19. What is the conclusion of this test for α = 5%?
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