Popular Diets. In the article "Comparison of the Atkins, Ornish, Weight Watchers, and Zone Diets for Weight Loss and Heart Disease Risk Reduction" (Journal of the American Medical Association, Vol. $293$, No. 1, Pp, $43-53$), M. Dansinger et al. conducted a randomized trial to assess the effectiveness of four popular diets for weight loss. Overweight adults with an average body mass index of$35$and ages$22-72$years participated in the randomized trial for 1 year. The weight losses, in kilograms, based on the results of the experiment are given on the WeissStats site. Negative losses are gains. $WW=$Weight Watchers.

a. Obtain individual normal probability plots and the standard deviation of the samples.

b. Perform a residual analysis.

c. Use your results from parts (a) and (b) to decide whether conducting a one-way ANOVA test on the data is reasonable. If so. also do parts (d) and (e).

d. Use a one-way ANOVA test to decide, at the $5\%$significance level, Whether the data provide sufficient evidence to conclude that a difference exists among the means of the populations fewer than the samples were taken.

e. Interpret your results from part (d)

ANOVA, or non - parametric tests, is a potent quantitative approach for revealing the disparity amongst two or more phenomena or aspects using variable importance. It also illustrates how to determine statistical significance among several demographics.

The following MINITAB output shows the individual normal probability maps of the samples:

The mean and standard deviation of the samples are given in the following MINITAB output:

ANOVA, or non - parametric tests, is a potent quantitative approach for revealing the disparity amongst two or more phenomena or aspects using variable importance. It also illustrates how to determine statistical significance among several demographics.

The normal probability of the residual analysis is plotted as given below,

ANOVA, or non - parametric tests, is a potent quantitative approach for revealing the disparity amongst two or more phenomena or aspects using variable importance. It also illustrates how to determine statistical significance among several demographics.

It appears plausible to assume that the assumption of normal populations is met based on the normal probability plots.

We can see that the highest sample standard deviation $(7.28)$is less than twice the lowest sample standard deviation from the standard deviations of the four samples $(4.80)$.

As a result, the criteria of equal standard deviations in the populations is met.

It appears acceptable to assume that the assumption of normal populations is met based on the normal probability plot of residuals.

ANOVA, or non - parametric tests, is a potent quantitative approach for revealing the disparity amongst two or more phenomena or aspects using variable importance. It also illustrates how to determine statistical significance among several demographics.

The null and alternative hypotheses are

(Mean weight losses are equal)

$H_a:$Not all the means are equal

We are to perform the test at the $5\%$significance level;

so $\alpha =0.05$.

The MINITAB output is as follows:

ANOVA, or non - parametric tests, is a potent quantitative approach for revealing the disparity amongst two or more phenomena or aspects using variable importance. It also illustrates how to determine statistical significance among several demographics.

We do not reject $H_0$since the P-value $(0.903)$is greater than the significance level.

Thus, at the$5\%$ significance level, the data do not provide adequate evidence to infer that the four diets differ in mean weight loss.