Question

Suppose a growing health and wellness company is rolling out three new nutritional diet plans. The research department wants to test the effectiveness of these three new nutritional plans, a low-fat plan and a low-carb plan, as well as its standard low-calorie plan. Marcia, a nutrition researcher with the company, randomly assigns 18 new weight loss program participants to each of the three plans, assigning 6 to each plan. After administering the plans, she follows up with the participants a month later and records the amount of weight, in pounds, that each participant lost.

As Marcia examines the collected data, she observes that the distribution of weight loss results for each sample are approximately normal without any outliers. As population variances are unknown, Marcia assumes equal variances based on the sample data because side-by-side boxplots of the samples indicate relatively equal sizes of the variances.

Marcia intends to conduct a one-way ANOVA ?- test at a significance level of ?=0.05 to test if the mean weight losses for the three diet plans are all equal. The mean square between groups, MSG, is 13.1667 , and the mean square within groups, also referred to as the mean square error, MSE, is 8.4444 . Marcia computes the ?- statistic, which is ?=1.5592 .

QUESTION 1: Select the statement that accurately evaluates whether or not the requirements of a one-way ANOVA ?-F-test have been met in Marcia's experiment. If the requirements have not been met, do not proceed to answer subsequent questions.

a) The requirements have not been met because the sample sizes are not properly selected and are not large enough.

b)The requirements have not been met because the population variances are unknown.

c) The requirements have been met because the samples are properly selected and are approximately normally distributed with equal variances.

d) The requirements have been met because the samples are properly selected, are approximately normally distributed, and each contains at least 5 observations.

e) The requirements have been met because the sampling distributions are approximately normal and the dependent variable is categorical.

QUESTION 2: If the test requirements have been met, compute the degrees of freedom for the means square between groups, MSG, and the degrees of freedom for the mean square error, MSE.

df1 =

df2 =

QUESTION 3: Determine the critical value of ? for Marcia's hypothesis test at the significance level of ?=0.05 using software or a ?‑ distribution table . Provide the result with precision to two decimal places.

QUESTION 4: Select the accurate statement regarding Marcia's hypothesis test decision and conclusion.

a) Marcia should reject the null hypothesis. There is sufficient evidence to reject the null hypothesis because the ?-F-statistic is greater than the critical value. Therefore, Marcia should conclude that one or more of the mean weight losses are different.

b) Marcia should fail to reject the null hypothesis. There is insufficient evidence to reject the null hypothesis because the ?-F-statistic is greater than the critical value. Therefore, Marcia should conclude that there is no evidence that the mean weight loss from any nutritional program is different.

c) Marcia should fail to reject the null hypothesis. There is insufficient evidence to reject the null hypothesis because the ?-F-statistic is less than the critical value. Therefore, Marcia should conclude that each pair of mean weight losses from the nutritional programs differs.

d) Marcia should fail to reject the null hypothesis. There is insufficient evidence to reject the null hypothesis because the ?-F-statistic is less than the critical value. Therefore, Marcia should conclude that there is no evidence that the mean weight loss among any of the nutritional programs is different.

e) Marcia should reject the null hypothesis. There is sufficient evidence to reject the null hypothesis because the ?-F-statistic is less than the critical value. Therefore, Marcia should conclude that the mean weight loss for at least one of the programs is different from the others.

Answer 1

QUESTION 1: c) The requirements have been met because the samples are properly selected and are approximately normally distributed with equal variances.

QUESTION 2: df1 = 2

df2 = 15

QUESTION 3: The critical value for an F-distribution with 2 and 15 degrees of freedom at a significance level of 0.05 is F_0.05,2,15=3.68

QUESTION 4: d) Marcia should fail to reject the null hypothesis. There is insufficient evidence to reject the null hypothesis because the ?-F-statistic is less than the critical value. Therefore, Marcia should conclude that there is no evidence that the mean weight loss among any of the nutritional programs is different.