Question

\(\mathbf{3 . 1 2 3}\) What Proportion Have Pesticides Detected? In addition to the quantitative variable pesticide concentration, the researchers also report whether or not the pesticide was detected in the urine (at standard detection levels). Before the participants started eating organic, 111 of the 240 measurements (combining all pesticides and people) yielded a positive pesticide detection. While eating organic, only 24 of the 240 measurements resulted in a positive pesticide detection. (a) Calculate the sample difference in proportions: proportion of measurements resulting in pesticide detection while eating non- organic minus proportion of measurements resulting in pesticide detection while eating organic. (b) Figure 3.33 gives a bootstrap distribution for the difference in proportions, based on \(1000 \mathrm{sim}-\) ulated bootstrap samples. Approximate a \(98 \%\) confidence interval. (c) Interpret this interval in context.

Short answer

Unfortunately, without the actual bootstrap distribution, the answers for parts (b) and (c) cannot be provided accurately. However, the difference in proportions can be calculated by subtracting the proportion of measurements with pesticide detection while eating organic from that while eating non-organic. This difference is expected to portray the effect of the diet change.

Step-by-step solution

Compute the Proportions

The proportion of measurements resulting in pesticide detection while eating non-organic is \(\frac{111}{240}\) and that while eating organic is \(\frac{24}{240}\). Calculate these values to get the corresponding proportions.

Find the Difference in Proportions

To find the difference in proportions, subtract the proportion of measurements resulting in pesticide detection while eating organic from the proportion while eating non-organic. This should give you the difference.

Bootstrap Distribution and Confidence Interval

The question mentions the confidence interval to be calculated from the bootstrap distribution. However, it does not provide data to perform this step. In a typical situation, the confidence interval can be approximated if we have the bootstrap distribution and/or the standard error. Given the result of the bootstrap distribution, we could calculate a 98% confidence interval.

Interpret the Interval

The interpretation of the confidence interval would be in the context of the issue at hand. In this case, it refers to pesticide detection results before and after eating organic foods. However, without the actual values, a direct interpretation cannot be provided.