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Chapter 7: Problem 69

How does the value of \(\sigma_{\hat{p}}\) change as the sample size increases? Explain. Assume \(n / N \leq .05\).

Short Answer

Expert verified
The value of \(\sigma_{\hat{p}}\) decreases as the sample size (n) increases. This is because increasing the sample size provides a better estimate of the population proportion, which results in a decrease in its variability around \(p\), and hence a smaller standard error.

Step by step solution

01

Identify the formula for \(\sigma_{\hat{p}}\)

The standard error of the proportion is calculated by the formula: \(\sigma_{\hat{p}} = \sqrt{\frac{p(1 - p)}{n}}\), where \(p\) is the population proportion and \(n\) is the sample size
02

Explain the relationship between \(\sigma_{\hat{p}}\) and n

Looking at the formula for \(\sigma_{\hat{p}}\), we can see that the sample size (n) is in the denominator of the fraction under the square root. This means that as \(n\) increases, the value of the fraction decreases, which in turn means the square root of the fraction (and hence \(\sigma_{\hat{p}}\)) decreases. So, t increase in sample size (n) would cause a decrease in the standard error (\(\sigma_{\hat{p}}\)).
03

Explain the reason for this relationship

The reason behind the relationship between standard error for the population proportion and sample size is that increasing the sample size gives us a better estimate of the population proportion. Thus, the variability of \(\hat{p}\) around \(p\) decreases, which is reflected in a smaller standard error.

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