Because about two-thirds of Americans are considered overweight, weight loss is big business. There are many different types of diets, but do some work better than others? Is low fat better than low carb or is some combination best? Researchers (Garnder et al. 2007) conducted a study involving four popular diets: Atkins (very low carb), Zone (40:30:30 ratio of carbs, protein, fat), LEARN (high carbohydrate, low fat), and Ornish (low fat). They randomly assigned women aged 25-50 with a body mass index (BMI) of 27-40 (overweight and obese) to one of the four diets. The 311 women who volunteered for the program were educated on their assigned diet and were observed periodically as they stayed on the diet for a year. At the end of the year, the researchers calculated the change in BMI (e.g., negative means reduction in BMI) for each woman and compared the results across the four diets.

1.) What is the overarching research question the researchers hoped to answer?

2.)

a.) What are the observational units in this study?

b.) Identify the explanatory and response variables. Classify them as categorical or quantitative. For categorical variables, indicate how many categories are used.

Explanatory: Type: Explanatory: Type:

3.)

a. Does this study make use of random sampling, random assignment, both, or neither? What are the implications of your answer with regard to scope of inference?

b. Did the researchers collect the data as paired data or as independent samples? In other words, according to the study design, are the responses from one treatment group paired with or independent of the responses from other treatment groups?

4.) State the null and alternative hypotheses, both in words and symbols, for testing the research conjecture. (Recall that the response variable is change in BMI; positive values indicate an increase in BMI and negative values indicate a decrease in BMI from the beginning to the end of the study.)

5.) The data for this investigation appear in a fi le called Diets, available on the textbook website. Copy and paste the data into the Multiple Means applet. (Recall that the response variable is change in BMI.) Press Use Data, and the applet will produce dotplots and summary statistics. You can also check the Boxplots box to overlay those as well.

a.) Report the means, standard deviations, and sample sizes for the change in BMI amounts for each diet.

b.) Describe what the graphs and statistics reveal about whether the diets appear to differ with regard to change in BMI amounts and, if so, which diet appears to be best and which worst.

6.) Conduct a simulation-based randomization test, as was done in the last section, to compare the four group means using the mean of the absolute values of the differences (MAD) as the statistic. Report the observed value of this statistic. Then determine and report a p-value based on at least 1,000 shuffles.

7.) Based on this p-value, evaluate the strength of evidence against the null hypothesis provided by the experimental data. In other words, do the data provide strong evidence that at least one of the four diets differs with regard to average change in BMI?

Although the MAD statistic is fairly easy to understand and calculate, it is not commonly used, in part because there is no theory-based model for the null distribution. Another downside is that the MAD statistic is not standardized so it is not comparable across studies (e.g., aMAD statistic of 1 in one study might be strong evidence but in another study might not show convincing evidence against the null hypothesis).

A much more commonly used statistic for comparing multiple groups on a quantitative response, which does have a theory-based distribution, is called an F-statistic. As with the MAD statistic, the F-statistic equals zero only when the group means are all identical. Otherwise the F-statistic is positive, with larger values indicating larger differences across the group means.

The F-statistic (see Formula section at the end of Example 9.2 for details) is a ratio of "between-group" and "within-group" variability. Thus, F = between-group variability within-group variability The numerator is a measure of how much the group means differ from each other, and the denominator is a measure of how much variation there is within the groups (related to the SD within the groups).13. Now compare your answer to #11 to your answer to #12 by finding the ratio of the squares of these values ("variance" = standard deviation squared), multiplying the numerator by 77, roughly the sample size of each group: 77 x variability between the group means variability within the groups Note: There is some more discussion about the F-statistic in Example 9.2 and FAQ 9.2.2. If you haven't already done so, we encourage you to read more about the F-statistic there. The formula for the F-statistic is given at the end of Example 9.2