A researcher wants to know if there is a difference in the average number of hours a person sleeps based on four different types of natural sleep supplements (melatonin, herbal tea, tart cherry juice, and lemon balm). A random sample of 104 participants was assigned to use one of these supplements for a 2-week period and asked to estimate the average number of hours they slept each night during this period. Table 10.10 gives the number of hours of sleep (rounded to the nearest hour) based on the type of natural sleep supplement used.

Is there a difference in the mean number of hours slept based on the different types of natural supplements? In order to answer this question,

a. State the appropriate null and alternative hypotheses.

b. How would you describe the factor? And how many factor levels are there?

c. Find the SSE and the MSE.

d. Find the SSTR and the MSTR.

e. Find the number of degrees of freedom for the numerator and the number of degrees of freedom for the denominator.

f. Find the \(p\)-value and interpret it within the context of the problem.

g. Check any relevant model assumptions.

h. What is the minimum difference in the number of hours slept between natural sleep aids that the given sample can find, provided such a difference does actually exist?

Table 10.10

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Number of Hours Slept (Rounded to the Nearest Hour) by the Type of Natural Sleep Supplement Melatonin (1) Herbal Tea (2) Tart Cherry Juice (3) Lemon Balm (4) 5 7 9 8 5 a a 6 9 8 9 9 8 8 8 5 8 9 9 7 7 7 8. 666900592652298556 160 00 00 00 00 o94010117652726 7 5 9 5 56695NN806957NON IN 101000101016 8 7 8 9 8 8 9 7 7 8 9 9 8 8 8 8 8 160 00 00 00 a 00 00 00 00 00 0 9 8 5 7 5 8186696 00 8