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.

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 24 new weight loss program participants to each of
the three plans, assigning 8 to each plan. After administering the
plans, she follows up with the participants a...

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