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Exercises 6.192 and 6.193 examine the results of a study \(^{45}\) investigating whether fast food consumption increases one's concentration of phthalates, an ingredient in plastics that has been linked to multiple health problems including hormone disruption. The study included 8,877 people who recorded all the food they ate over a 24 -hour period and then provided a urine sample. Two specific phthalate byproducts were measured (in \(\mathrm{ng} / \mathrm{mL}\) ) in the urine: DEHP and DiNP. Find and interpret a \(95 \%\) confidence interval for the difference, \(\mu_{F}-\mu_{N},\) in mean concentration between people who have eaten fast food in the last 24 hours and those who haven't. The mean concentration of DEHP in the 3095 participants who had eaten fast food was \(\bar{x}_{F}=83.6\) with \(s_{F}=194.7\) while the mean for the 5782 participants who had not eaten fast food was \(\bar{x}_{N}=59.1\) with \(s_{N}=152.1\)

Step by step solution

01

Identify Given Data

First, you need to identify the given data. For the people who have eaten fast food, you have \( n_{F} = 3095, \bar{x}_{F} = 83.6, s_{F} = 194.7 \). For those who have not eaten fast food, you have \( n_{N} = 5782, \bar{x}_{N} = 59.1, s_{N} = 152.1 \). These values will be used to calculate the difference between the two means, along with the standard error for that difference.

02

Calculate Difference Between Means

Next, calculate the difference between the two sample means, \( \bar{x}_{F} - \bar{x}_{N} = 83.6 - 59.1 = 24.5 \) ng/mL. This is the observed difference in mean concentration of DEHP between the two groups.

03

Calculate Standard Error for Difference

The standard error for the difference between two means is given by the formula \[ SE = \sqrt { \frac {s_{F}^2}{n_{F}} + \frac {s_{N}^2}{n_{N}} } \], substituting the given values gives \[ SE = \sqrt { \frac {194.7^2}{3095} + \frac {152.1^2}{5782} } = 5.54 \] ng/mL.

04

Construct Confidence Interval

Lastly, construct the 95% confidence interval for the difference between the means using the formula \[ CI = (\bar{x}_{F} - \bar{x}_{N}) \pm Z*SE \] where Z is the Z-score for desired confidence level (1.96 for 95% confidence level). So \[ CI = 24.5 \pm 1.96*5.54 = 24.5 \pm 10.86 \]. Thus, the 95% confidence interval for the difference between the mean concentrations of DEHP for people who have eaten fast food and those who haven't ranges from 13.64 ng/mL to 35.36 ng/mL.

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Key Concepts

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

Statistical Analysis

Statistical analysis involves the process of collecting, reviewing, and interpreting data to make informed conclusions. In this context, it becomes essential to make sense of the data gathered from a significant number of participants in the study. Here, we gathered important details such as the number of participants in each group, along with their mean DEHP concentrations and standard deviations.

• People who consumed fast food had a mean DEHP concentration of 83.6 ng/mL.
• People who did not consume fast food had a mean DEHP concentration of 59.1 ng/mL.

Using these values, statistical analysis helps in determining whether the observed differences in the DEHP levels are significant or could be due to random chance. Ultimately, the method helps researchers decide if there's a substantial association between fast food consumption and DEHP levels.
A key element of statistical analysis is determining confidence intervals, which represent the range where the true population parameter is expected to fall for a given confidence level.

Mean Differences

When comparing two groups in statistical studies, the difference in means between the groups is critical in identifying possible effects or trends. We often calculate these differences to see if one group has significantly higher or lower values than another.
For our exercise, the mean difference is calculated as:

• The mean for fast food consumers (\(ar{x}_F=83.6g/mL\)) minus the mean for non-consumers (\(ar{x}_N=59.1g/mL\)).
• This results in a mean difference of 24.5 ng/mL, indicating that on average, fast food consumers have higher DEHP levels.

This step is fundamental, as it provides a basic understanding of the extent of variation due to the variable under examination (in this case, fast food consumption).
However, the mean difference alone does not provide us with information about the reliability or significance of this difference, which is where further statistical analysis, such as calculating the confidence interval, comes into play.

Standard Error

Standard error (SE) is a crucial concept in statistics that helps measure the accuracy with which a sample represents a population. It estimates how much the sample mean is likely to fluctuate from the true population mean. In simpler terms, it tells us about the precision of the mean difference in our context.
Calculating the standard error involves the formula:\[ SE = \sqrt { \frac {s_F^2}{n_F} + \frac {s_N^2}{n_N} } \]For this exercise:

• \(s_F\) and \(n_F\) are the standard deviation and sample size for fast food consumers.
• \(s_N\) and \(n_N\) are for non-consumers.
• The calculated SE is 5.54 ng/mL.

The standard error tells us that the observed difference in mean concentrations could vary by this much due to random sampling variability.
In more formal statistical testing, the SE is used along with confidence intervals and hypothesis testing to understand more about the impact of study factors, such as fast food consumption on phthalate levels. It represents the potential error involved in estimating the true mean difference between larger populations.