Question

In Exercises 6.152 and \(6.153,\) find a \(95 \%\) confidence interval for the difference in proportions two ways: using StatKey or other technology and percentiles from a bootstrap distribution, and using the normal distribution and the formula for standard error. Compare the results. Difference in proportion who use text messaging, using \(\hat{p}_{t}=0.87\) with \(n=800\) for teens and \(\hat{p}_{a}=0.72\) with \(n=2252\) for adults.

Short answer

The results will vary depending on the calculations done with the statistical software and manually using the Normal distribution and standard error formula. It cannot be determined without those calculations. However, often, these two methods give approximately similar confidence intervals.

Step-by-step solution

Calculating Bootstrap confidence interval

Using a statistical software like StatKey, enter the given data: proportion (p) and sample size (n) for both teens (\(p_{t}=0.87, n=800\)) and adults (\(p_{a}=0.72, n=2252\)). After running the bootstrap method (usually by setting the number of resamples to a high value like 10000), you can determine the lower and upper 2.5% percentiles that mark the bounds of the 95% confidence interval.

Calculating confidence interval using Normal distribution and Standard error

Firstly, find the difference in proportions, \(\hat{p}_{t} - \hat{p}_{a}\), then calculate the standard error for the difference in proportions given by the formula, \[SE = \sqrt{\frac{\hat{p}_{t}(1-\hat{p}_{t})}{n_{t}} + \frac{\hat{p}_{a}(1-\hat{p}_{a})}{n_{a}}}\].Now, the 95% confidence interval can be calculated using the normal distribution approximation according to the formula: \(\hat{p}_{t} - \hat{p}_{a}\) ± \(1.96*SE\), where \(1.96\) corresponds to the z-score for a 95% confidence level.

Comparing Results

Once both confidence intervals have been calculated, they can be compared. They should ideally be similar, but it is not unusual for them to vary slightly due to the different approaches used. The bootstrap method uses resampling methods and represents a more modern, data-intensive approach while the normal method represents a traditional, analytical approach.

Key concepts

Bootstrap Method

The bootstrap method is a powerful statistical tool used to estimate the distribution of a sample statistic by resampling the original data with replacement. For example, when calculating a confidence interval for the difference in proportions, this method involves generating thousands of simulated samples, known as bootstrap samples, from the original data. Each of these samples is used to calculate the statistic of interest, creating a bootstrap distribution.

By analyzing the bootstrap distribution, one can determine percentile-based confidence intervals without making strong assumptions about the shape of the overall population. This is particularly useful with small sample sizes or when the underlying distribution is not well understood. As applied to textbook exercises, students can use software like StatKey to easily perform bootstrap simulations and observe the variability of the estimated statistic directly, reinforcing the concept that confidence intervals are rooted in the variability inherent to sampling.

Standard Error Calculation

The standard error (SE) of a statistic is a measure of the amount by which the statistic is expected to fluctuate due to sampling variation. In context of our textbook problem, calculating the SE for the difference in proportions involves the proportions (e.g., \(\hat{p}_{t}\) and \(\hat{p}_{a}\)) and sample sizes (e.g., \(n_{t}\) and \(n_{a}\)) for each group. The formula \[SE = \sqrt{\frac{\hat{p}_{t}(1-\hat{p}_{t})}{n_{t}} + \frac{\hat{p}_{a}(1-\hat{p}_{a})}{n_{a}}}\] captures the combined uncertainty of the two sample proportions. Understanding the calculation of SE allows students to appreciate its role in constructing confidence intervals: it determines the margin by which the estimated difference might fluctuate due to random sampling, and hence, the width of the interval. This concept is a cornerstone in inferential statistics, as it applies to many types of confidence intervals and hypothesis tests.

Normal Distribution Approximation

Normal distribution approximation, also known as the Central Limit Theorem, enables statisticians to use normal probability models for inference about sample statistics despite the original distribution of the data. In the exercise, once the SE is calculated, we are assuming that the sampling distribution of the difference in proportions is approximately normal. This allows us to use the normal model to find the critical value (1.96 for a 95% confidence interval) and compute the interval \(\hat{p}_{t} - \hat{p}_{a}\) ± \(1.96*SE\).

The concept assumes that the sample size is large enough for the approximation to be valid. By applying this concept to exercises, students learn a key aspect of inferential statistics: the distribution of the sample statistic under repeated sampling. It simplifies the process of inference in many practical situations and serves as the foundational approximation for confidence intervals and hypothesis tests in traditional statistics.