In A Test Of Weight Loss Programs

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In A Test Of Weight Loss Programs, 228 Subjects Were Divided Such That 38 Subjects Followed Each Of 6 Diets. Each Was Weighed A Year After Starting The Diet And The Results Are In The ANOVA Table Below. Use A 0.05 Significance Level To Test The Claim That The Mean Weight Loss Is The Same For The Different Diets Source Of Variation SS Df MS F P-Value F Crit

The Correct Answer and Explanation is :

To test the claim that the mean weight loss is the same for the different diets, we need to analyze the ANOVA table. The purpose of ANOVA (Analysis of Variance) is to compare the means of different groups to determine if there is a statistically significant difference between them. Below is the breakdown of how to interpret the given ANOVA table.

Step 1: Set Up Hypotheses

  • Null Hypothesis ((H_0)): The mean weight loss is the same for all the different diets.
  • Alternative Hypothesis ((H_1)): At least one of the diets results in a different mean weight loss.

Step 2: Given Information

From the ANOVA table, we are typically provided with the following information:

  • SS (Sum of Squares): This measures the variability in the data, split into different components (between groups and within groups).
  • Df (Degrees of Freedom): This tells us how many independent pieces of information are available for calculating the statistics.
  • MS (Mean Square): This is calculated by dividing the sum of squares by the degrees of freedom ((MS = SS / Df)).
  • F-Statistic: This is the ratio of the between-group variance (mean square between) to the within-group variance (mean square within).
  • P-Value: This indicates the probability of obtaining an F-statistic as extreme as, or more extreme than, the one observed if the null hypothesis were true.
  • F Crit (Critical F-Value): This is the value from the F-distribution table that we compare the observed F-statistic against to decide whether to reject the null hypothesis.

Step 3: Test the Hypothesis

  1. Calculate the F-statistic:
    [
    F = \frac{\text{MS Between}}{\text{MS Within}}
    ]
    The MS values are found by dividing the SS values by their corresponding degrees of freedom.
  2. Compare the F-statistic to the critical value:
  • If (F \geq F_{crit}), reject the null hypothesis.
  • If (F < F_{crit}), fail to reject the null hypothesis.
  1. Interpret the P-value:
  • If the P-value is less than the significance level ((0.05)), reject the null hypothesis.
  • If the P-value is greater than (0.05), fail to reject the null hypothesis.

Conclusion:

After conducting these steps, if we reject the null hypothesis, it means that at least one diet results in a different mean weight loss compared to the others. If we fail to reject the null hypothesis, it suggests that there is no statistically significant difference in the mean weight loss between the diets.

Unfortunately, the actual values from the ANOVA table aren’t provided here, so I cannot calculate the exact result, but following these steps will allow you to conduct the test properly.

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