Vermont-based Green Mountain Coffee Roasters dominates the market for single-serve coffee in the United States,with its subsidiary Keurig accounting for approximately 70% of sales ("Rivals Try to Loosen Keurig's Grip on Single-Serve Coffee Market," Chicago Tribune,February 26,2011).But Keurig's patent on K-cups,the plastic pods used to brew the coffee,is expected to expire in 2012,allowing other companies to better compete.Suppose a potential competitor has been conducting blind taste tests on its blend and finds that 47% of consumers strongly prefer its French Roast to that of Green Mountain Coffee Roasters.After tweaking its recipe,the competitor conducts a test with 144 tasters,of which 72 prefer its blend.The competitor claims that its new blend is preferred by more than 47% of consumers to Green Mountain Coffee Roasters' French Roast.At the 1% significance level,does the evidence support the claim?
A) No,because the value of the test statistic is less than the critical value.
B) Yes,because the value of the test statistic is less than the critical value.
C) No,because the value of the test statistic is greater than the critical value.
D) Yes,because the value of the test statistic is greater than the critical value.

Question

Quizplus · Accepted Answer

We are given that the competitor claims that its new blend is preferred by more than 47% of consumers to Green Mountain Coffee Roasters' French Roast. This is a one-tailed test: the competitor is claiming superiority of its blend, compared to Green Mountain's blend. The null hypothesis is that the competitor's blend is not preferred by more than 47% of consumers. The alternative hypothesis is that the competitor's blend is preferred by more than 47% of consumers.

We can use a z-test for proportions to test this claim. The test statistic is calculated as:

z = (p - P) / sqrt(P * (1 - P) / n)

where p is the proportion of tasters who preferred the competitor's blend (72/144 = 0.5), P is the hypothesized proportion under the null (0.47), and n is the sample size (144).

Plugging in the values, we get:

z = (0.5 - 0.47) / sqrt(0.47 * 0.53 / 144) = 1.16

At the 1% significance level, the critical value for a one-tailed z-test is 2.33. Since our calculated test statistic (1.16) is less than the critical value (2.33), we fail to reject the null hypothesis.

Therefore, we do not have sufficient evidence to support the claim that the competitor's blend is preferred by more than 47% of consumers compared to Green Mountain Coffee Roasters' French Roast.

Vermont-Based Green Mountain Coffee Roasters Dominates the Market for Single-Serve