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 favor a gun control proposal, using \(\hat{p}_{f}=0.82\) for 379 out of 460 females and \(\hat{p}_{m}=0.61\) for 318 out of 520 for males. (We found a \(90 \%\) confidence interval for this difference in Exercise 6.144.)

Short answer

The 95% confidence intervals for the difference in proportions are calculated using both the bootstrap method and the normal distribution method. The results from both methods should be similar, demonstrating that different statistical approaches can provide similar results.

Step-by-step solution

Calculate the proportions

First, the proportions need to be calculated. The proportion for females, \(\hat{p}_{f}\), is already given as 0.82 and the proportion for males, \(\hat{p}_{m}\), as 0.61.

Bootstrap method

To calculate the 95% confidence interval using the bootstrap method, one would typically use a software like StatKey. However, since we cannot use software here, we can say that the method involves creating a bootstrap distribution by resampling the data multiple times, then finding the 2.5th and 97.5th percentiles of this distribution.

Normal distribution method

For the normal distribution method, start by calculating the standard error, which is given by the formula: \[ SE = \sqrt{ \frac{{\hat{p}_{f}(1-\hat{p}_{f})}}{{n_f}} + \frac{{\hat{p}_{m}(1-\hat{p}_{m})}}{{n_m}}}\], where \(n_f\) and \(n_m\) are the number of females and males respectively. Here, \(n_f = 460\) and \(n_m = 520\). Calculate the difference in proportions: \(\hat{p}_{f} - \hat{p}_{m}\). Then, the 95% confidence interval is given by \((\hat{p}_{f} - \hat{p}_{m}) \pm 1.96(\text{ SE })\).

Compare the results

The confidence intervals calculated using the bootstrap method and normal distribution should be similar. Any differences could be due to variations in the bootstrap sampling process. The comparison of these results provides a practical understanding of how different statistical methods can lead to similar conclusions.

Key concepts

Bootstrap Method

When students encounter the bootstrap method for the first time, it might seem quite complicated, but it's actually a powerful non-parametric approach used to estimate confidence intervals and standard errors. Let's break this down:

The essence of the bootstrap method lies in creating many samples from the data you already have. This process, known as resampling, involves repeatedly drawing samples (with replacement) from the original data, each time calculating the statistic of interest – in this case, the difference in proportions. Imagine pulling names from a hat, recording the name, and then putting it back in. That's what we do here, but with our data.

For our problem, let’s say you used software like StatKey to generate thousands of these resampled datasets. Each dataset allows us to calculate a new difference in proportions. Collect all these differences, and voila! You've got a bootstrap distribution. The credibility of this method lies in the Law of Large Numbers – as you draw more samples, the bootstrap distribution better approximates the true sampling distribution of the statistic.

Once you have the bootstrap distribution, finding a confidence interval is pretty straightforward. If you need a 95% confidence interval, you simply locate the 2.5th and 97.5th percentiles in the distribution. These percentiles give you the range in which the actual difference in proportions lies, with a confidence level of 95%.

This method does not make any assumptions about the shape of the population distribution, making it versatile and robust, especially when the normality of the data is in question or the sample size is too small for the Central Limit Theorem to kick in effectively.

Normal Distribution

Understanding the normal distribution is crucial when diving into statistics; it's the bell-shaped curve that appears all over the natural and social sciences. This distribution is defined by two parameters - its mean and its standard deviation. The area under the curve represents probability, with the total area being equal to 1, or 100% probability.

The beauty of the normal distribution is that it allows statisticians to make inferences about populations using sample data. Why's that? Because many characteristics we measure (like test scores, height, etc.) tend to distribute normally. The great thing here is that, even if the original data isn't normally distributed, the sampling distribution (given enough sample size) of the sample mean will approximate a normal distribution, thanks to the Central Limit Theorem.

In the context of our problem, the normal distribution method is used to estimate the confidence interval of the difference in proportions. It relies on the assumption that the sampling distribution of the statistic is normal. This is reasonably safe to assume when dealing with large enough samples from the population, or when the population itself is normally distributed.

It's the familiarity and the well-established tables of probabilities that make the normal distribution method so widely used. It provides a standardized way to calculate confidence intervals, essentially saying 'given this difference in sample proportions, there's a 95% chance the true difference lies within this range,' which is incredibly useful for making decisions based on data.

Standard Error

Standard error is a concept that students sometimes mix up with standard deviation – but they serve very distinct purposes. Think of standard error as a measure of how much you can expect a sample statistic to fluctuate from one sample to another. If you were to repeat your study over and over, the statistic (like the mean or proportion) would differ slightly each time; the standard error gives you an idea of how much these differences would typically be.

Mathematically, the standard error for proportions is derived from the formula: \[\begin{equation}SE = \sqrt{ \frac{{\hat{p}_{f}(1-\hat{p}_{f})}}{{n_f}} + \frac{{\hat{p}_{m}(1-\hat{p}_{m})}}{{n_m}}} \end{equation}\],where \(\hat{p}_{f}\) and \(\hat{p}_{m}\) are the sample proportions and \(n_f\) and \(n_m\) are the sample sizes. It's an embodiment of the sample's variability and size; larger samples will typically have a smaller standard error, indicating more precise estimates.

In our problem about the difference in proportions who favor a gun control proposal, the standard error is used alongside the normal distribution to calculate the 95% confidence interval. It's essentially the 'margin of error' for estimating the difference between the two population proportions based on our samples. And remember, a smaller standard error means a tighter confidence interval, which equates to more certainty in where the true difference in proportions lies.