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In a test of weight loss programs, 128 subjects were divided such that 32 subjects followed each of 4 diets. Each was weighed a year after starting the diet, and the results are in the ANOVA table below. Use a 0.025 significance level to test the claim that the mean weight loss is the same for the different diets.

\begin{tabular}{lccccccc}
Source of Variation & SS & df & MS & F & P-value & F crit \\
\hline Between Groups & 216.291 & 3 & 72.09694 & 2.3007 & 0.080531 & 3.223243 \\
Within Groups & 3885.783 & 124 & 31.33696 & & & \\
Total & 4102.074 & 127 & & & & \\
\hline
\end{tabular}

Should the null hypothesis that all the diets have the same mean weight loss be rejected?

A. No, because the [tex]$P$[/tex]-value is less than the significance level.
B. Yes, because the [tex]$P$[/tex]-value is greater than the significance level.
C. Yes, because the [tex]$P$[/tex]-value is less than the significance level.
D. No, because the [tex]$P$[/tex]-value is greater than the significance level.



Answer :

GinnyAnswer
To determine whether we should reject the null hypothesis that all diets have the same mean weight loss, we will follow the steps of hypothesis testing using the information provided in the ANOVA table and the given significance level.

### Step-by-Step Solution:

1. State the Null and Alternative Hypotheses:
- Null Hypothesis ([tex]\( H_0 \)[/tex]): The mean weight loss is the same for the different diets.
- Alternative Hypothesis ([tex]\( H_1 \)[/tex]): At least one diet leads to a different mean weight loss compared to the others.

2. Identify the Given Data:
- [tex]\( P \)[/tex]-value = 0.080531
- Significance level ([tex]\( \alpha \)[/tex]) = 0.025

3. Comparison of the [tex]\( P \)[/tex]-value and the Significance Level:
- If the [tex]\( P \)[/tex]-value is less than the significance level ([tex]\( P < \alpha \)[/tex]), we reject the null hypothesis.
- If the [tex]\( P \)[/tex]-value is greater than the significance level ([tex]\( P > \alpha \)[/tex]), we fail to reject the null hypothesis.

4. Decision Rule:
- Compare the [tex]\( P \)[/tex]-value (0.080531) with the significance level (0.025).

Since [tex]\( P \)[/tex]-value (0.080531) is greater than the significance level (0.025), we do not have enough evidence to reject the null hypothesis.

5. Conclusion:
- Based on the comparison, we fail to reject the null hypothesis that the mean weight loss is the same for the different diets.

### Answer:
D. No, because the P-value is greater than the significance level.

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