In Exercise 3.60 on page 191, we describe a study in which participants ate significantly more and exercised significantly less for a month. Two and half years later, participants weighed an average of 6.8 pounds more than at the start of the experiment (while the weights of a control group had not changed). Is the amount of weight gained over the following 2.5 years directly related to how much weight was gained during the one-month period? For the 18 participants, the correlation between increase of body weight during the one-month intervention and increase of body weight after 30 months is r = 0.21. We want to estimate, for the population of all adults, the correlation between weight gain over one month of bingeing and the effect of that month on a person’s weight 2.5 years later.

(a) What is the population parameter of interest? What is the best point estimate for that parameter?

(b) To find the sample correlation r = 0.21, we used a dataset containing 18 ordered pairs (weight gain over the one month and weight gain 2.5 years later for each individual in the study). Describe how to use this data to obtain one bootstrap sample.

(c) What statistic is recorded for the bootstrap sample?

(d) Suppose that we use technology to calculate the relevant statistic for 1000 bootstrap samples. Describe how to find the standard error using those bootstrap statistics.

(e) The standard error for one set of bootstrap statistics is 0.14. Calculate a 95% confidence interval for the correlation.

(f) Use the confidence interval from part (e) to indicate whether you are confident that there is a positive correlation between amount of weight gain during the one-month intervention and amount of weight gained over the next 2.5 years, or whether there is a reasonable possibility that there is no correlation at all. Explain your reasoning.

(g) Will a 90% confidence interval most likely be wider or narrower than the 95% confidence interval found in part (e)?

Exercise 3.60

Overeating for just four weeks can increase fat mass and weight over two years later, a Swedish study shows. Researchers recruited 18 healthy and normal-weight people with an average age of 26. For a four-week period, participants increased calorie intake by 70% (mostly by eating fast food) and limited daily activity to a maximum of 5000 steps per day (considered sedentary). Not surprisingly, weight and body fat of the participants went up significantly during the study and then decreased after the study ended. Participants are believed to have returned to the diet and lifestyle they had before the experiment. However, two and a half years after the experiment, the mean weight gain for participants was 6.8 lbs with a standard error of 1.2 lbs. A control group that did not binge had no change in weight.