Question

The article "Should Pregnant Women Move? Linking Risks for Birth Defects with Proximity to Toxic Waste Sites" (Chance [1992]: \(40-45)\) reported that in a large study carried out in the state of New York, approximately \(30 \%\) of the study subjects lived within 1 mile of a hazardous waste site. Let \(p\) denote the proportion of all New York residents who live within 1 mile of such a site, and suppose that \(p=0.3\). a. Would \(\hat{p}\) based on a random sample of only 10 residents have a sampling distribution that is approximately normal? Explain why or why not. b. What are the mean and standard deviation of the sampling distribution of \(\hat{p}\) if the sample size is \(400 ?\) c. Suppose that the sample size is \(n=200\) rather than \(n=\) \(400 .\) Does the change in sample size affect the mean and

Short answer

a) No, the sampling distribution would not be approximately normal for a sample of 10, as neither np nor n(1-p) would be >= 10. b) For a sample of 400, the mean would be 0.3 and the standard deviation 0.0245. c) A change in sample size doesn't affect the mean but does affect the standard deviation: reducing sample size increases standard deviation.

Step-by-step solution

Verify normality with small sample size

The sampling distribution of a proportion \(\hat{p}\) is approximately normal if \(np ≥10\) and \(n(1-p) ≥10\), where n is the size of the sample and \( p\) is the population proportion. Here, \( n=10\) and \( p=0.3\), thus \(np= 3\) and \(n(1-p)=7\). Both of these are less than 10, therefore, \(\hat{p}\) would not have a sampling distribution that is approximately normal with a sample size of 10.

Calculate Mean and Standard deviation for a sample size of 400

The mean and standard deviation of the sampling distribution of a proportion are given by \( μ_\hat{p}=p \) and \(σ_\hat{p} = \sqrt{\frac{p(1-p)}{n}}\), respectively. Here, \(n=400\) and \(p=0.3\), the mean \( μ_\hat{p}=0.3 \) and standard deviation \(σ_\hat{p} = \sqrt{\frac{0.3 * 0.7}{400}} = 0.0245\). Thus, for a sample size of 400, the mean and standard deviation are approx. 0.3 and 0.0245.

Impact of sample size on mean and standard deviation

Changing the sample size does not affect the mean of the sampling distribution, as it is equal to the population proportion \(p=0.3\), which remains constant. However, it does impact the standard deviation. The decrease in sample size will increase the standard deviation. As a result, the variation in the sampling distribution of the proportion isolated from samples will be larger.