Question

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 \(p=0.4\) for sample sizes of \(n=30, n=200,\) and \(n=1000\).

Short answer

For given population proportion \(p=0.4\), the standard error decreases as the sample size \(n\) increases from 30 to 200 and then to 1000, indicating that the estimate becomes more precise. Therefore, using a larger sample size gives a more accurate estimate of the population proportion.

Step-by-step solution

Calculate Standard Error for n=30

Plug \(p=0.4\) and \(n=30\) into the SE formula: SE = \( \sqrt{( 0.4 * (1-0.4) ) / 30} \) to get the value for the standard error when the sample size is 30.

Calculate Standard Error for n=200

Repeat the process with \(n=200\): SE = \( \sqrt{( 0.4 * (1-0.4) ) / 200} \) to get the value for the standard error when the sample size is 200.

Calculate Standard Error for n=1000

Repeat the process again with \(n=1000\): SE = \( \sqrt{( 0.4 * (1-0.4) ) / 1000} \), so we can have the value for the standard error when the sample size is 1000.

Analyze the effect of increasing sample size on the standard error

Compare the standard errors computed in steps 1, 2, and 3. As n increases, the SE decreases, which indicates that increasing the sample size can reduce the standard error of the estimate.

Discuss the effect on the accuracy of using a sample proportion to estimate a population proportion

The observed decrease in SE with increasing sample size indicates that as the sample size becomes larger, the estimate becomes more precise, i.e., closer to the true population proportion. This is because with a larger sample, there is less sampling variation. Hence, using a adequately large sample size will increase the accuracy of the sample proportion as an estimate of the population proportion.