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Use the following partially complete ANOVA table to answer questions 5-12.

Fifteen people are randomly assigned to three groups, so that 5 people exercise in the (1) morning, (2) afternoon, or (3) night. After 8 weeks, their weight loss is measured in pounds. The different types of sums of squared errors are reported below.

\begin{tabular}{|lc|}
\hline Source & SS df MS F Decision \\
\hline Between & 2.009 \\
Within & 2.602 \\
Total & 4.611 \\
\hline
\end{tabular}

Fifteen people are randomly assigned, so that 5 people exercise in the morning, afternoon, or night. After 8 weeks, their weight loss is measured in pounds. State the null hypothesis for this study.

A. [tex]$H_0: \mu_1 \neq \mu_2=\mu_3$[/tex]
B. [tex]$H_0: \mu_1=\mu_2 \neq \mu_3$[/tex]
C. [tex]$H_0: \mu_1 \neq \mu_2 \neq \mu_3$[/tex]
D. [tex]$H_0: \mu_1=\mu_2=\mu_3$[/tex]



Answer :

GinnyAnswer
To determine the null hypothesis for this ANOVA study, we need to understand the question in the context of an ANOVA (Analysis of Variance) test. ANOVA is used to compare the means of three or more groups to see if at least one of them is significantly different from the others.

Here is a step-by-step explanation of how to determine the correct null hypothesis for the provided ANOVA table:

1. Identify the Groups:
There are three groups based on the time of day they exercise (morning, afternoon, night).

2. Formulate the Null Hypothesis (H0):
The null hypothesis in ANOVA is that all group means are equal. This means there is no significant difference in the means of the groups being compared. In other words, the weight loss in pounds for the three groups should be the same.

3. Compare the Given Choices:
- Option A: [tex]\( H_0: \mu_1 \neq \mu_2=\mu_3 \)[/tex]
- Option B: [tex]\( H_0: \mu_1=\mu_2 \neq \mu_3 \)[/tex]
- Option C: [tex]\( H_0: \mu_1 \neq \mu_2 \neq \mu_3 \)[/tex]
- Option D: [tex]\( H_0: \mu_1=\mu_2=\mu_3 \)[/tex]

We see that Options A, B, and C suggest that at least one pair of means is not equal, which contradicts the null hypothesis for ANOVA.

4. Choose the Correct Null Hypothesis:
The correct null hypothesis should state that the means of all three groups are equal.

Therefore, the correct null hypothesis for this study is:

[tex]\[ H_0: \mu_1 = \mu_2 = \mu_3 \][/tex]

Thus, the answer is D.