The data in the following table represent the number of hours of relief provided

Question

The data in the following table represent the number of hours of relief provided by five different brands of headache tablets administered to 25 subjects experiencing fevers of 38degree C or more. Carry out a 1-way ANOVA (alpha =.05) to test the hypothesis that the mean number of hours of relief provided by the tablets is the same for all five brands by: a) Specifying the null and alternative hypothesis. b) Specifying the decision rule. c) Calculating the test statistic by filling in the ANOVA table below. d) Do you reject or fail to reject the null? e) Do post-hoc tests need to be conducted? If so, use Tukey's test. If not, explain why not. f) Summarize your conclusion (including the results of the post-hoc analysis, if applicable) using words of the problem.

Explanation / Answer

a) Null Hypothesis: The number of hours of relief by the five different brands of headache tablets are same.

Alternative Hypothesis: At least, the number of hours of relief by the five different brands of headache tablets are different.

b) Ans:

The decision rule for the F test in ANOVA is set up in a similar way to decision rules we established for t tests. The decision rule again depends on the level of significance and the degrees of freedom. The F statistic has two degrees of freedom. These are denoted df1 and df2, and called the numerator and denominator degrees of freedom, respectively. The degrees of freedom are defined as follows:

df1 = k-1 and df2=N-k,

where k is the number of comparison groups and N is the total number of observations in the analysis. In this question k=5 and N=25.

c) Ans:

One-way ANOVA: Hour versus Tablet

Source DF SS MS F P
Tablet 4 78.42 19.61 6.59 0.001
Error 20 59.53 2.98
Total 24 137.95

S = 1.725 R-Sq = 56.85% R-Sq(adj) = 48.22%

d) Fail to reject the null hypothesis at 0.05 level of significance.

e) Ans: Yes, the post-hoc test is need to compare the different brands of tablets.

Tukey 95% Simultaneous Confidence Intervals
All Pairwise Comparisons among Levels of Tablet

Individual confidence level = 99.28%

Tablet = A subtracted from:

Tablet Lower Center Upper ------+---------+---------+---------+---
B -0.804 2.460 5.724 (------*-----)
C -4.404 -1.140 2.124 (------*-----)
D -5.724 -2.460 0.804 (-----*------)
E -1.744 1.520 4.784 (-----*------)
------+---------+---------+---------+---
-5.0 0.0 5.0 10.0

Comment: The CI include the zero. Hence, tablest A is not different with remaining tabletes.

Tablet = B subtracted from:

Tablet Lower Center Upper ------+---------+---------+---------+---
C -6.864 -3.600 -0.336 (------*-----)
D -8.184 -4.920 -1.656 (-----*------)
E -4.204 -0.940 2.324 (-----*------)
------+---------+---------+---------+---
-5.0 0.0 5.0 10.0

Comment: Tablet B is different from C and D.

Tablet = C subtracted from:

Tablet Lower Center Upper ------+---------+---------+---------+---
D -4.584 -1.320 1.944 (-----*------)
E -0.604 2.660 5.924 (-----*------)
------+---------+---------+---------+---
-5.0 0.0 5.0 10.0

Comment: C is not different from D and E

Tablet = D subtracted from:

Tablet Lower Center Upper ------+---------+---------+---------+---
E 0.716 3.980 7.244 (------*-----)
------+---------+---------+---------+---
-5.0 0.0 5.0 10.0

Comment: D is different from E.

f) Ans: The calculated p-value from ANOVA is 0.001 and less than 0.05 level of significance. Therefore, the alternative hypothesis is accepted and can say that the mean hours are different among the different brands of tablets.

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