Can your diet make you happier? A researcher wants to know if the food you eat has an effect on your happiness. A random sample of 41 participants was asked to complete a survey on happiness and a questionnaire on their diet. The happiness survey is scored on a scale of 50-100, where higher scores are associated with someone who has a higher degree of happiness and satisfaction with their life. The data from the survey and the type of diet are provided in Table 10.8. Is there a difference in the responses to the happiness survey based on the type of diet? 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.

Table 10.8

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Responses to Happiness Survey by Type of Diet RE No Specific Diet 70 Vegetarian Vegan Low Carb Low Fat 92 78 74 58 55 56 77 74 84 94 RN 73 94 72 97 50 92 62 69 73 53 74 61 55 75 76 86 86 71 81 90 50 86 66 53 72 73 66 - 58 | | | | 1 67