Question

The article "Unmarried Couples More Likely to Be Interracial" (San Luis Obispo Tribune, March 13,2002 ) reported that \(7 \%\) of married couples in the United States are mixed racially or ethnically. Consider the population consisting of all married couples in the United States. a. A random sample of \(n=100\) couples will be selected from this population and \(\hat{p},\) the proportion of couples that are mixed racially or ethnically, will be calculated. What are the mean and standard deviation of the sampling distribution of \(\hat{p} ?\) b. Is the sampling distribution of \(\hat{p}\) approximately normal for random samples of size \(n=100 ?\) Explain. c. Suppose that the sample size is \(n=200\) rather than \(n=\) \(100 .\) Does the change in sample size affect the mean and standard deviation of the sampling distribution of \(\hat{p} ?\) If so, what are the new values for the mean and standard deviation? If not, explain why not. d. Is the sampling distribution of \(\hat{p}\) approximately normal for random samples of size \(n=200 ?\) Explain.

Short answer

a. The mean and standard deviation of the sampling distribution for \(n=100\) are 0.07 and 0.0256 respectively. b. The sampling distribution is approximately normal for \(n=100\). c. For \(n=200\), the mean remains 0.07 while the standard deviation is now 0.0181. d. The sampling distribution is also approximately normal for \(n=200\).

Step-by-step solution

Calculate Mean and Standard Deviation for n=100

Given that proportion of racially or ethnically mixed couples, \(p\), is 7% or 0.07. For a sample size, \(n=100\), the mean (\(μ_\hat{p}\)) of the sampling distribution is equal to \(p\) and the standard deviation (\(σ_\hat{p}\)) is given by the formula \(\sqrt{\frac{p(1-p)}{n}}\). So, mean \(μ_\hat{p} = p = 0.07\) and standard deviation \(σ_\hat{p} = \sqrt{\frac{0.07(1-0.07)}{100}} = 0.0256.\)

Determine if Sampling Distribution is Normal for n=100

The sample size is large enough to use the Central Limit Theorem if both \(np > 5\) and \(n(1-p) > 5\). In this case, \(np = 100 * 0.07 = 7 > 5\) and \(n(1-p) = 100 * (1-0.07) = 93 > 5\). Therefore, the sampling distribution of \(\hat{p}\) is approximately normal.

Calculate Mean and Standard Deviation for n=200

If the sample size increases to \(n=200\), the mean remains the same (\(p = 0.07\)), but the standard deviation decreases, given by the formula \(\sqrt{\frac{p(1-p)}{n}}\). So, standard deviation \(σ_\hat{p} = \sqrt{\frac{0.07(1-0.07)}{200}} = 0.0181.\)

Determine if Sampling Distribution is Normal for n=200

Again check if both \(np > 5\) and \(n(1-p) > 5\). Here, \(np = 200 * 0.07 = 14 > 5\) and \(n(1-p) = 200 * (1-0.07) = 186 > 5\). So, the sampling distribution of \(\hat{p}\) is still approximately normal.

Key concepts

Central Limit Theorem

Understanding the Central Limit Theorem (CLT) is crucial when dealing with the world of statistics, especially when it involves sample data. In essence, the CLT asserts that with a large sample size, the sampling distribution of the sample mean will approximate a normal distribution, regardless of the original distribution of the population.

To apply this concept to our exercise, we examine the proportion of racially or ethnically mixed couples, symbolized as \(\hat{p}\). With a substantial sample size (such as 100 or 200 couples), the distribution of \(\hat{p}\) will tend to resemble a normal distribution. This holds true under the condition that certain criteria are met – specifically, the product of the sample size \(n\) and success probability \(p\), and the product of \(n\) and failure probability \(1-p\) should both be greater than 5, guaranteeing the sample mean's normal distribution approximation.

Standard Deviation

The standard deviation of a sampling distribution, symbolized as \(\sigma_\hat{p}\), measures how much the sample proportions vary from the population proportion (\(p\)). In our example regarding mixed couples, for a given sample size (such as 100), the standard deviation can be computed using the formula \(\sigma_\hat{p} = \sqrt{\frac{p(1-p)}{n}}\).

When comparing the standard deviation for different sample sizes, as in the steps provided in the exercise, we see that as the sample size increases, the standard deviation decreases. This inversely proportional relationship implies that larger samples tend to have a tighter, more precise estimate of the population parameter, due to less variability between sample proportions. Consequently, with a sample size of 200, the standard deviation calculated for \(\sigma_\hat{p}\) will be smaller than that of a sample size of 100, leading to a more reliable estimate.

Normal Distribution

A normal distribution, often depicted as a symmetric, bell-shaped curve, is paramount in statistics for its predictability and the properties it entails. Most importantly, a considerable amount of data in natural occurrences tends to distribute normally, particularly when randomness is involved.

For the exercise at hand, we ascertain if the sampling distribution of \(\hat{p}\), the proportion of mixed couples, approximates a normal distribution. The criteria for a normal approximation rest upon the sample proportions meeting certain thresholds (as noted with the condition \(np > 5\) and \(n(1-p) > 5\)). Since these conditions are met for both sample sizes of 100 and 200, it leads to the conclusion that, indeed, the sampling distribution of \(\hat{p}\) approximates normal distribution finely. This has practical implications, such as enabling us to calculate confidence intervals and conduct hypothesis tests with greater ease and reliability.