In there a relationship between confidence intervals and two-tailed hypothesis tests? The answer is yes. Let c be the level of confidence used to construct a confidence interval from sample data. Let * be the level of significance for a two-tailed hypothesis test. The following statement applies to hypothesis tests of the mean: For a two-tailed hypothesis test with level of significance a and null hypothesis \(H_0 : mu = k\) we reject Ho whenever k falls outside the \(c = 1 — \alpha\) confidence interval for mu based on the sample data. When A falls within the \(c = 1 — \alpha\) confidence interval. we do reject \(H_0\). For a one-tailed hypothesis test with level of significance Ho : mu = k and null hypothesiswe reject Ho whenever A falls outsidethe \(c = 1 — 2\alpha\) confidence interval for p based on the sample data. When A falls within the \(c = 1 — 2\alpha\) confidence interval, we do not reject \(H_0\). A corresponding relationship between confidence intervals and two-tailed hypothesis tests is also valid for other parameters, such as \(p,\mu_1 — \mu_2,\) and \(p_1, - p_2\). (b) Consider the hypotheses \(H_0 : p_1 — p_2 = O\) and \(H_1 : p_1 — p_2 =\) Suppose a 98% confidence interval for \(p_1 — p_2\) contains only positive numbers. Should you reject the null hypothesis when alpha = 0.05? Why or why not?
Although tea is the world’s most widely consumed beverage after water, little is known about its nutritional value. Folacin is the only B vitamin present in any significant amount in tea, and recent advances in assay methods have made accurate determination of folacin content feasible. Consider the accompanying data on folacin content for randomly selected specimens of the four leading brands of green tea. Brand 1234amp; Observations amp;7.9amp;5.7amp;6.8amp;6.4amp;6.2amp;7.5amp;7.5amp;7.1amp;6.6amp;9.8amp;5.0amp;7.9amp;8.6amp;6.1amp;7.4amp;4.5amp;8.9amp;8.4amp;5.3amp;5.0amp;10.1amp;amp;6.1amp;4.0amp;9.6amp;amp;amp;
(Data is based on “Folacin Content of Tea,” J. Amer. Dietetic Assoc., 1983: 627–632.) Does this data suggest that true average folacin content is the same for all brands? a. Carry out a test using \(\alpha:=.05\) via the P-value method. b. Assess the plausibility of any assumptions required for your analysis in part (a). c. Perform a multiple comparisons analysis to identify significant differences among brands.