Question

Effect of Overeating for One Month: Average Long-Term Weight Gain Overeating for just four weeks can increase fat mass and weight over two years later, a Swedish study shows \(^{35}\) 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. (a) What is the relevant parameter? (b) How could we find the actual exact value of the parameter? (c) Give a \(95 \%\) confidence interval for the parameter and interpret it. (d) Give the margin of error and interpret it.

Short answer

a) The relevant parameter is the long-term average weight gain.\nb) The actual exact value of the parameter can theoretically be found by measuring the weight gain of all individuals who overeat for a month.\nc) The 95% confidence interval of the weight gain is \(6.8 \pm 1.96*1.2\) lbs.\nd) The margin of error is \(1.96 * 1.2\) lbs. It indicates the range within which the true mean weight gain would fall 95% of the time if the study were repeated many times.

Step-by-step solution

Identify the Relevant Parameter

The relevant parameter for this study is the long-term weight gain of the participants. Parameter is a quantity that is a property of the population that is being studied. In this case, they are interested in how much weight on average a person will gain if they overeat for a month. The mean weight gain reported for participants after two and a half years is 6.8 lbs.

Finding the Actual Exact Value of the Parameter

The actual exact value of this parameter would require data from all individuals who have overeaten for a month to see how that affects their weight gain over the long term. In reality, this is nearly impossible because it would require measuring the weight gain of every single individual who overeats for a month, which is unfeasible. Therefore, we use the sample data (e.g., the 18 healthy and normal-weight people in the study) as a way to estimate the parameter of interest. The mean weight gain reported in the study is an estimate of the parameter.

Calculate the 95% Confidence Interval for the Parameter

A 95% confidence interval can be calculated using the formula \(confidence \, interval = sample \, mean \pm Z * SE \), where \(Z\) is the Z-value from the standard normal distribution corresponding to the desired level of confidence (1.96 for 95% confidence) and \(SE\) is the standard error. The standard error is given as 1.2 lbs. Therefore, the confidence interval is \(6.8 \pm 1.96*1.2\).

Calculate the Margin of Error and Interpret It

The margin of error for the 95% confidence level can be calculated using the formula \(MOE = Z * SE \). Using the values from the previous step, \(MOE = 1.96 * 1.2\). The margin of error indicates how much uncertainty there is in our estimate of the parameter. It says that if we were to repeat the study many times, then 95% of the time the average weight gain would fall within this interval.