Question

The paper "Effects of Fast-Food Consumption on Energy Intake and Diet Quality Among Children in a National Household Survey" (Pediatrics [2004]: \(112-118\) ) investigated the effect of fast-food consumption on other dietary variables. For a representative sample of 663 teens who reported that they did not eat fast food during a typical day, the mean daily calorie intake was 2,258 and the sample standard deviation was \(1,519 .\) For a representative sample of 413 teens who reported that they did eat fast food on a typical day, the mean calorie intake was 2,637 and the standard deviation was \(1,138 .\) Use the given information and a \(95 \%\) confidence interval to estimate the difference in mean daily calorie intake for teens who do eat fast food on a typical day and those who do not.

Short answer

The estimated 95% confidence interval for the difference in daily calorie consumption between teens who eat fast food and those who don't is between -202.25 and -555.75 calories per day.

Step-by-step solution

1. Identify the given data

First, we need to identify and organize the given data. We have two groups: group 1 is teens who did not eat fast food (n1 = 663, X̄1 = 2258, s1 = 1519) and group 2 is teens who ate fast food (n2 = 413, X̄2 = 2637, s2 = 1138)

2. Calculate the sample difference

The sample mean difference is \(X̄1 - X̄2\), so this is \(2258 - 2637 = -379\)

3. Calculate the standard error of the difference

The formula for the standard error of the difference is \(\sqrt{{{s1}^2/n1 + {s2}^2/n2}}\), substituting we get \(\sqrt{{(1519)^2/663 + (1138)^2/413}} = 89.91 \)

4. Calculate the 95% confidence interval

For a 95% confidence interval, the critical value is approximately 1.96. The interval is then \((X̄1 - X̄2) ± 1.96(standard error of the difference)\), which gives \(-379 ± 1.96*89.91\), or \(-379 ± 176.25\)

Key concepts

Sample Mean Difference

Understanding the concept of the sample mean difference is fundamental when comparing two different groups in a study. It represents the difference between the average outcomes of these groups. In our dietary study example, we compare the mean daily calorie intake between teens who did eat fast food on a typical day and those who did not. This is a pivotal step as it sets the foundation for estimating how distinct the two groups are from each other in terms of calorie intake. The calculation is straightforward: subtract one group's mean from the other's, which in this case results in a difference of erall calculation.

Standard Error of the Difference

The standard error of the difference is a bit more complex, as it measures the variability in the estimation of the sample mean difference. Put simply, it provides insight into how much we can expect the sample mean difference to vary if we were to repeat the study multiple times. To calculate the standard error of the difference, we use the formula with the sample standard deviations and sizes from both groups. For our specific dietary study, the calculation yielded a standard error of erall calculation.

Statistical Significance

Determining statistical significance is essential in discerning whether the observed difference in our study is likely real or just a result of random chance. It's a measure of how confident we can be in the results. Generally, we look at p-values or confidence intervals to assess this. A 95% confidence interval means we are 95% sure that the real mean difference lies within this range. The calculated interval from the dietary study was quite wide, indicating considerable variability, but still allowed us to infer that there is a meaningful difference in calorie intake between the two groups.

Dietary Study Statistics

In dietary study statistics, it's typical to look at caloric and nutrient intake differences among various groups to understand dietary patterns and their health implications. By calculating measures such as the sample mean difference and standard error, we can make inferences about the population. This is exactly what was done in the example study, where statistical techniques were applied to estimate the difference in calorie intake between teens consuming fast food and those not. Such studies can provide valuable insights for nutrition policy and dietary guidelines.