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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 sample of 663 teens who reported that they did not eat fast food during a typical day, the mean daily calorie intake was 2258 and the sample standard deviation was \(1519 .\) For a sample of 413 teens who reported that they did eat fast food on a typical day, the mean calorie intake was 2637 and the standard deviation was 1138 . a. What assumptions about the two samples must be reasonable in order for the use of the two-sample \(t\) confidence interval to be appropriate? b. Use the given information 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.

Step by step solution

01

Understanding Assumptions for two-sample t-test

For a two-sample t-test to be appropriate, the following assumptions about the two samples must be reasonably met: 1. The two populations from which the samples are drawn are independent of each other. 2. The two populations are normally distributed. 3. The variances of the two populations are equal.

02

Estimating the Difference in Mean Daily Calorie Intake

To estimate the difference in mean daily calorie intake for teens who do eat fast food on a typical day and those who don’t, we subtract the mean calorie intake of the group who do not eat fast food from the group who do. Using the information given: mean calorie intake (fast food) = 2637 and mean intake (no fast food) = 2258. So, the estimated difference in mean daily calorie intake is \(2637 - 2258 = 379\) calories.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mean Daily Calorie Intake

Understanding mean daily calorie intake involves recognizing that it is an average number of calories consumed by individuals each day. In the context of the study you read about, the reported values were calculated separately for teens who ate fast food and those who didn't. Here's how it breaks down:

• For teens who did not eat fast food, the mean was 2258 calories.
• For those who ate fast food, it was 2637 calories.

These values are crucial because they helped frame the research question: Does fast food consumption increase daily calorie intake? By comparing these averages, researchers could later analyze whether the observed difference was meaningful or just due to random variation.

Assumptions of Statistical Tests

When performing a two-sample t-test, certain assumptions ensure the test's validity. Firstly, both samples must be independent of one another. This means that the calorie intake of teens in one group does not affect those in the other. Also, the populations from which the samples are drawn need to be normally distributed. In reality, particularly with larger sample sizes, this assumption is often relaxed due to the Central Limit Theorem. However, it is an important consideration. The final assumption is homogeneity of variance, meaning the variabilities in both groups should be similar, as indicated by their standard deviations (1519 and 1138, respectively). Failing to meet these assumptions can lead to incorrect conclusions.

Confidence Interval Estimation

A confidence interval gives a range of values within which we can say with a certain degree of confidence that the true difference in population means lies. To calculate the confidence interval for the difference in means, you first compute the actual difference in sample means, which in this case was 379 calories. However, we don't just stop there. To finalize a confidence interval:

• Identify the standard deviation of each group.
• Use these to calculate the standard error of the difference.
• Apply the standard error in a t-distribution setting to establish your interval.

This process will guide you to construct a range that estimates how different the average calorie intakes might truly be, beyond just the sample we examined. This way, researchers can discuss the findings with more confidence, knowing they have accounted for sample variability.