Question

Exercises 3.112 to 3.115 give information about the proportion of a sample that agree with a certain statement. Use StatKey or other technology to find a confidence interval at the given confidence level for the proportion of the population to agree, using percentiles from a bootstrap distribution. StatKey tip: Use "CI for Single Proportion" and then "Edit Data" to enter the sample information. Find a \(95 \%\) confidence interval if 35 agree in a random sample of 100 people.

Short answer

Unfortunately, without having actual access to StatKey or another statistical software to perform bootstrap distribution simulation, we cannot provide numerical values for the lower and upper bounds of the confidence interval. However, these steps are what you'd generally follow to obtain the 95% confidence interval given a sample proportion and the desired confidence level.

Step-by-step solution

- Calculate Sample Proportion

The first step is to find the proportion of people who agree with the given statement in the sample. This is found by dividing the number of people who agree by the total number of people in the sample. Here, \(p = 35/100 = 0.35\) where p is the sample proportion.

- Understand Confidence Level

A confidence level of 95% implies that if we were to pick many random samples from the population and calculate the proportion for each of them, 95% of the calculated proportion would be within the confidence interval. In this step, understand that we want the interval that contains the true population proportion 95% of the time.

- Utilize Bootstrap Distribution

Bootstrapping is a technique of sampling in which we draw samples with replacement from the observed dataset. In doing so, it approximates the variability in the statistic due to the sampling process. The percentiles from the bootstrap distribution give us the confidence interval. You need to use StatKey or another software to simulate many drawn samples, calculate the proportion for each, and then find the 2.5th percentile and the 97.5th percentile. Those will define the lower and upper limits of the 95% confidence interval respectively due to the symmetric nature of the normal distribution.

Key concepts

Bootstrap Distribution

Imagine you have a bag of marbles, each representing a data point from your sample. To understand what might happen if you drew many more samples, you shake the bag and draw a marble, note its color, and then put it back before drawing again. This process is called 'sampling with replacement', and if you do this enough times, recording the colors each time, you'll get a good idea of the color distribution within the bag. This is the essence of a bootstrap distribution in statistics.

For our exercise, the bootstrap distribution is created by repeatedly sampling our original 100 people's responses, with replacement, and calculating a new sample proportion each time. By doing this many, many times—think thousands or tens of thousands—we build a distribution of sample proportions. This distribution represents different potential outcomes if the survey were to be conducted repeatedly under the same conditions. Each proportion in this distribution gives us insight into the variability of sample proportions that could occur by chance alone.

Sample Proportion

Sample proportion is the fraction of the sample that displays a certain characteristic—in this case, agreeing with a statement. It's like taking a snapshot of a crowd and counting how many are wearing hats to estimate the popularity of hats among the whole population. Here, we focus on the 35 people who agree out of the 100 surveyed, giving us a sample proportion of \(0.35\).

This single statistic provides a point estimate of the true proportion in the entire population but does not give us a measure of certainty or a margin of error. That's where the bootstrap distribution comes in handy: it helps us understand the variability around our sample proportion, offering a way to construct a confidence interval.

Confidence Level

A confidence level tells you how sure you can be about your estimates. It's a lot like a weather forecast telling you there's a 95% chance of sunshine tomorrow—it's not a guarantee, but it's a strong indication you might want to wear shorts and a t-shirt. In statistics, a 95% confidence level means that if we were to take 100 different samples and calculate confidence intervals for each one, we'd expect about 95 of those intervals to contain the true population parameter.

In the context of our exercise, a confidence interval with a 95% confidence level suggests that we're 95% confident the true population proportion who agree with the statement falls within our calculated range. It does not mean that there's a 95% chance that any given interval contains the true proportion—each interval either contains it or it doesn't—but over many intervals constructed from many samples, 95% should contain the true value.

StatKey

StatKey is a digital tool designed to help us with statistical concepts, akin to a Swiss Army knife for data scientists. Think of it as a reliable assistant in our data explorations, making complex tasks more manageable. For this exercise, we use the 'CI for Single Proportion' feature in StatKey to crunch the numbers. It creates simulated bootstrap samples, calculates sample proportions, and then uses these to find the bootstrap distribution.

After entering our sample information through the 'Edit Data' option, StatKey does the heavy lifting of generating the bootstrap samples and calculates the confidence interval. It’s a clear, visual way to understand and compute statistical concepts without getting bogged down by manual calculations or programming.

Sampling with Replacement

Sampling with replacement is like picking apples from a basket, checking each one, and then putting it back before selecting again. This method ensures that each pick is independent of the others, since the same configuration of apples is possible for every selection. In statistics, sampling with replacement from a dataset means each member of the dataset has the same chance of being selected each time we draw a new sample.

In the bootstrapping method we use for our exercise, each sample drawn contributes to the bootstrap distribution. By doing this repeatedly, we accurately reflect the sample's variability, hence creating a robust simulation of the sampling process. It's a cornerstone concept for creating the bootstrap distribution and understanding how sampling variability impacts our confidence intervals.