Question

Suppose that a particular candidate for public office is favored by \(48 \%\) of all registered voters in the district. A polling organization will take a random sample of 500 of these voters and will use \(\hat{p}\), the sample proportion, to estimate \(p\). a. Show that \(\sigma_{p}\), the standard deviation of \(\hat{p}\), is equal to \(0.0223 .\) b. If for a different sample size, \(\sigma_{p}=0.0500\), would you expect more or less sample-to-sample variability in the sample proportions than when \(n=500 ?\) c. Is the sample size that resulted in \(\sigma_{\rho}=0.0500\) larger than 500 or smaller than \(500 ?\) Explain your reasoning.

Short answer

a. The standard deviation of \(\hat{p}\) is 0.0223. b. For \(\sigma_p = 0.0500\), there would be more sample-to-sample variability in the sample proportions than when \(n = 500\). c. The sample size that resulted in \(\sigma_p=0.0500\) is smaller than 500.

Step-by-step solution

Calculate the standard deviation

The standard deviation of \(\hat{p}\) is calculated using the formula \(\sigma_p = \sqrt{\frac{p(1 - p)}{n}}\), where \(p\) is the proportion, and \(n\) is the size of the population. In this case, \(p = 0.48\) (from the 48% favor for the candidate) and \(n = 500\) (the sample size). Substituting these values into the formula gives \(\sigma_p = \sqrt{\frac{0.48 \times (1 - 0.48)}{500}} = 0.0223.\)

Evaluate the effect of \(\sigma_p\) on sample-to-sample variability

With a larger standard deviation, the sample-to-sample variability is expected to be higher. This means that if \(\sigma_p = 0.0500\), one would expect more sample-to-sample variability in the sample proportions compared to when \(n = 500\). This is because a larger standard deviation indicates a greater dispersion of data values.

Determine if the sample size is larger or smaller

To get a larger standard deviation, the size of the sample would have to be smaller than 500. This is because standard deviation and sample size have an inversely proportional relationship. When the sample size decreases, the standard deviation increases. Conversely, when the sample size increases, the standard deviation decreases. As such, the sample size which resulted in a larger \(\sigma_p=0.0500\) is smaller than 500.