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Chapter 6: Problem 8

In Exercises 6.7 and 6.8 , compute the standard error for sample proportions from a population with the given proportion using three different sample sizes. What effect does increasing the sample size have on the standard error? Using this information about the effect on the standard error, discuss the effect of increasing the sample size on the accuracy of using a sample proportion to estimate a population proportion. A population with proportion \(p=0.75\) for sample sizes of \(n=40, n=300,\) and \(n=1000 .\)

Short Answer

Expert verified
Upon computation, it's found that as the sample size increases, the standard error decreases. Therefore, an increase in sample size improves the accuracy of the sample proportion in estimating the population proportion.

Step by step solution

01

Calculate Standard Error for \(n=40\)

Substitute the given values \(p=0.75\) and \(n=40\) into the formula for standard error to get the value of SE: \(SE=\sqrt{0.75* (1-0.75)/40}\). Calculate to get the answer.
02

Calculate Standard Error for \(n=300\)

Similarly, substitute the given values \(p=0.75\) and \(n=300\) into the formula for standard error to get the value of SE: \(SE=\sqrt{0.75* (1-0.75)/300}\). Calculate to get the answer.
03

Calculate Standard Error for \(n=1000\)

Again, substitute the given values \(p=0.75\) and \(n=1000\) into the formula for standard error to get the value of SE: \(SE=\sqrt{0.75* (1-0.75)/1000}\). Calculate to get the answer.
04

Effect of increasing sample size on Standard Error

Upon computing the standard errors for the increasing sample sizes, it can be observed that as the sample size increases, the standard error decreases. This is because the standard error is inversely proportional to the square root of the sample size. So, as the sample size gets larger, the standard error gets smaller.
05

Discuss the effect on accuracy of estimation

The decrease in the standard error with an increase in sample size improves the accuracy of the sample proportion in estimating the population proportion. This is because a smaller standard error means that the sample proportion is expected to be closer to the population proportion.

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Most popular questions from this chapter

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