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Chapter 3: Problem 74

Give information about the proportion of a sample that agrees with a certain statement. Use StatKey or other technology to estimate the standard error from a bootstrap distribution generated from the sample. Then use the standard error to give a \(95 \%\) confidence interval for the proportion of the population to agree with the statement. StatKey tip: Use "CI for Single Proportion" and then "Edit Data" to enter the sample information. In a random sample of 1000 people, 382 people agree, 578 disagree, and 40 are undecided.

Short Answer

Expert verified
The precise values could only be found out by performing actions in StatKey or similar statistical software. However, the sample proportion is 0.382. Then, by performing bootstrapping in StatKey or any such tool, we will find the standard error. After obtaining the standard error, the 95% confidence interval can be calculated.

Step by step solution

01

Calculating the Sample Proportion

In this exercise, the point estimate is equal to the sample proportion, since the question refers to the proportion of the population agreeing with a statement. To find it, we'll execute the following calculation: \(p = \frac{x}{n} \) where \( x \) is the number of successes (people who agree) and \( n \) is the sample size. Given \( x = 382 \) and \( n = 1000 \), the sample proportion \( p \) is \( \frac{382}{1000} = 0.382 \).
02

Generating the Bootstrap Distribution and Estimating the Standard Error

The bootstrap distribution is generated by resampling the provided sample with replacement many times, to mimic the process of obtaining new sample data. Then calculating the standard deviation of the bootstrap distribution gives an estimate for the standard error. Detailed technology steps to perform this process need to be followed. Particularly, in StatKey, the 'Bootstrap Confidence Intervals' fisher iris box is selected, then 'Bootstrapping a Stat' to generate a bootstrap distribution. Then choose the 'CI for Single Proportion' and enter the data. Finally, the estimated standard error can be found in the 'Standard Deviations of Stats' box which refer to the standard deviation of the bootstrap estimates.
03

Calculating the Confidence Interval

Once the standard error (SE) has been determined, the 95% confidence interval for the sample proportion can be calculated. The formula for the confidence interval is: \(CI = p 卤 z*SE \) where \( z \) is the z-score, which is 1.96 for a 95% confidence interval, \( p \) is the sample proportion, and \( SE \) is the standard error. Inserting the computed values in the formula will give the confidence interval.

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Key Concepts

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

Bootstrap Distribution
Understanding the bootstrap distribution is key to grasping modern statistical inference techniques. Imagine you only have a single sample to make generalizations about a population, but no other data to compare it to. Bootstrapping comes to the rescue by allowing you to simulate the process of taking more samples by reusing the data from your original sample.

In bootstrapping, you create a large number of 'resamples' by randomly selecting observations with replacement from the original sample. This creates a bootstrap distribution, which is essentially a distribution of sample statistics calculated from these resamples. By analyzing this distribution, we can estimate characteristics of the sampling distribution, such as the standard error, without requiring additional samples from the population.
Standard Error
The standard error (SE) is a statistical term that measures how precise your estimate of the population parameter is likely to be. When you're estimating the sample proportion, the standard error quantifies the variability of that estimate from sample to sample.

A smaller standard error indicates that your sample proportion is likely closer to the true population proportion. In the context of a bootstrap distribution, the standard error is estimated by calculating the standard deviation of the resampled proportions. This provides an idea of how much we would expect the sample proportion to vary if we were to take many additional samples.
Sample Proportion
The sample proportion represents a fraction of your sample that conforms to a particular criterion - in this case, the number of people agreeing with a statement. Calculating it is straightforward: divide the count of 'successful' cases by the total sample size.

Let's consider our sample where 382 out of 1000 people agreed with a statement. Here, the sample proportion is 0.382 (38.2%). This statistic is central to our analysis as it's our best point estimate of the true proportion of the population that agrees.
StatKey
StatKey is an educational tool designed to make statistical concepts more accessible and interactive. For the task at hand, StatKey provides a way to generate a bootstrap distribution and use it to estimate the standard error and confidence interval for a sample proportion without complex calculations or coding.

You begin by entering your sample data and then it will perform resampling to create a large number of simulated samples. These samples are used to build a bootstrap distribution, from which StatKey will calculate the standard error. Finally, you can use this standard error to help construct a 95% confidence interval, which offers an educated guess about where the true population proportion lies, with a certain level of assurance.

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Most popular questions from this chapter

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