Question

Effect of Overeating for One Month: Correlation between Short-Term and Long- Term Weight Gain In Exercise 3.70 on page \(227,\) we describe a study in which participants ate significantly more and exercised significantly less for a month. Two and a 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 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 it is plausible 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)?

Short answer

The population parameter of interest is the correlation between short-term and long-term weight gain, with an estimate of \(r = 0.21\). Bootstrap samples are created by randomly drawing from the original dataset. The bootstrap statistic recorded is the correlation coefficient \(r\). The standard error is calculated as the standard deviation of the bootstrap statistics. The \(95\%\) confidence interval for the correlation calculated from the given standard error is \(-0.07 , 0.49\). The confidence interval includes 0, so we cannot confidently prove a positive correlation. The \(90\%\) confidence interval will be narrower than the \(95\%\) one.

Step-by-step solution

Identify the Population Parameter and Its Best Estimate

The population parameter of interest in this study is the correlation between short-term and long-term weight gain in adults. The best estimate for this parameter, given by the correlation coefficient in our sample, is \(r = 0.21\).

Describe Bootstrapping Approach

To obtain a bootstrap sample, one should use the original set of 18 ordered pairs, and randomly draw with replacement 18 ordered pairs. This is repeated for a certain number of bootstrap samples; each contains the same amount of data as the original, but certain data points can be repeated due to the nature of bootstrapping.

Identify Bootstrap Statistic

The statistic that is recorded for the bootstrap sample is the correlation coefficient \(r\). It is the measure of the linear correlation between short-term and long-term weight gain in the bootstrap samples.

Calculate Standard Error

To find the standard error, calculate the correlation coefficient for each of the 1000 bootstrap samples. The standard error is the standard deviation of these 1000 values.

Calculate Confidence Interval

Given a standard error of \(0.14\), to calculate a \(95\%\) confidence interval for the correlation, use the formula: estimate ± (2 * standard error). Here, fill in our given values, estimate is \(0.21\) and the standard error is \(0.14\). So the interval would be \(0.21 ± (2 * 0.14)\) resulting in an interval of \(-0.07 , 0.49\).

Analyze Confidence Interval and Correlation

Key note to consider is that the interval includes 0, and thus it is plausible that there could be no correlation at all. So one cannot confidently say there is a positive correlation between short-term and long-term weight gain given the current dataset.

Compare Confidence Intervals

A \(90\%\) confidence interval will be narrower than a \(95\%\) one because we are less confident, and hence we need a smaller range that likely contains the true parameter.

Key concepts

Correlation Coefficient

When we consider the relationship between two variables, we often use the term 'correlation coefficient' to quantify the strength and direction of this relationship. In simple terms, this statistical measure tells us whether, and how strongly, pairs of variables are associated.

For instance, in the exercise where participants' short-term and long-term weight gain are examined, the correlation coefficient—denoted as 'r'—is reported to be 0.21. This indicates a weak positive relationship; as the weight gain in the short-term goes up, so does the weight gain in the long-term, but not very strongly. Correlation coefficients range from -1 to 1, where 1 means a perfect positive correlation, -1 means a perfect negative correlation, and 0 indicates no correlation at all.

Confidence Interval

A confidence interval is a range of values, derived from the sample statistics, which is likely to contain the population parameter of interest. It’s a tool used to estimate the precision and uncertainty of a sample statistic. In the context of our exercise, we constructed a 95% confidence interval for the population correlation coefficient based on our sample.

The interval was calculated to be from -0.07 to 0.49. This range means that we are 95% confident that the true correlation coefficient for the entire population lies within this interval. However, since zero is included, we cannot rule out the possibility that there is no correlation at all in the population.

Statistical Significance

Statistical significance is a determination of whether the results of a study can be considered to reflect a true effect, or whether they might be due to random chance. In our exercise, the confidence interval can be used to assess statistical significance. If the interval does not include zero, we can say the results are statistically significant at the given confidence level.

The 95% confidence interval we calculated includes zero, which suggests that the correlation we found may not be statistically significant. This implies that it is possible that there is no real association between short-term and long-term weight gain.

Standard Error

Standard error measures the variability of a sample statistic, such as the mean or in this case, the correlation coefficient. The standard error informs us about how much we can expect the sample statistic to vary from the true population parameter. In the bootstrap method described in the exercise, the standard error of the correlation coefficients from 1000 bootstrap samples is 0.14.

This value is a key component in calculating the confidence interval and aids in understanding the reliability of our estimate; a smaller standard error usually leads to a narrower confidence interval, indicating a more precise estimate of the population parameter.