91Ó°ÊÓ

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 computed. What are the mean and standard deviation of the sampling distribution of \(\hat{p}\) ? b. Is it reasonable to assume that the sampling distribution of \(\hat{p}\) is approximately normal for random samples of size \(n=100\) ? Explain. c. Suppose that the sample size is \(n=200\) rather than \(n=100\), as in Part (b). Does the change in sample size change 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 it reasonable to assume that the sampling distribution of \(\hat{p}\) is approximately normal for random samples of size \(n=200 ?\) Explain. e. When \(n=200\), what is the probability that the proportion of couples in the sample who are racially or ethnically mixed will be greater than 10 ?

Step by step solution

01

Calculate Mean and Standard Deviation of Sampling Distribution for \(n=100\)

Given, \(p = 0.07\) (proved proportion of interracial couples) and sample size \(n=100\). The mean of the proportion \(\hat{p}\) is equal to \(p\), so \(\mu_{\hat{p}} = p = 0.07\). The standard deviation of the proportion \(\hat{p}\) is computed as \(\sigma_{\hat{p}} = \sqrt{(p(1-p))/n} = \sqrt{(0.07 * 0.93)/100} = 0.0257\).

02

Check Normality for \(n=100\)

For the sampling distribution of \(\hat{p}\) to be approximately normal, it is necessary to check that \(np\) and \(n(1-p)\) are both greater than 10. Since \(100*0.07 = 7\) and \(100*0.93 = 93\), it is not reasonable to assume normality because the condition \(np>10\) is not satisfied.

03

Recalculate Mean and Standard Deviation for \(n=200\)

When \(n=200\), the mean remains the same, \(\mu_{\hat{p}} = p = 0.07\) as the population proportion does not depend on sample size. The standard deviation decreases, due to larger sample size: \(\sigma_{\hat{p}} = \sqrt{(p(1-p))/n} = \sqrt{(0.07 * 0.93)/200} = 0.0181\).

04

Check Normality for \(n=200\)

For the sampling distribution of \(\hat{p}\) for \(n=200\) to be approximately normal, again it is necessary that \(np\) and \(n(1-p)\) are both greater than 10. Since \(200*0.07 = 14\) and \(200*0.93 = 186\), it is reasonable to assume normality as both conditions \(np>10\) and \(n(1-p)>10\) are satisfied.

05

Calculate P(\(\hat{p}\) > 0.10) when \(n=200\)

To calculate the chance that the proportion of couples who are racially and ethnically mixed is greater than 0.10, use the z-score: \(Z = (\hat{p} - \mu_{\hat{p}}) / \sigma_{\hat{p}} = (0.10 - 0.07) / 0.0181 = 1.66\). Use a standard normal distribution table to find the probability of \(Z < 1.66\) is 0.9515. Therefore the probability \(\hat{p} > 0.10\) is \(1 - 0.9515 = 0.0485\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Proportion of Interracial Couples

The concept of the proportion of interracial couples is central to understanding how often these couples occur within a broader population. In this example, it is reported that 7% of married couples in the United States are considered mixed racially or ethnically. This percentage, denoted by the symbol \( p \), is known as the true population proportion, which serves as the basis for analyzing samples drawn from the population.
When sampling, we calculate the sample proportion \( \hat{p} \), which estimates the population proportion but is drawn from a subset of the population. Understanding how close these two values can be, especially when looking at different samples, is essential for predicting and assessing results within research and real-world applications.
The proportion \( p \) plays a crucial role, as it forms the foundation from which other calculations, like mean and standard deviation of the sample proportions, stem. By knowing the proportion, it helps guide expectations and assessments about the population dynamics.

Standard Deviation of Sample Proportion

The standard deviation of the sample proportion is a key concept in statistics as it helps us understand the expected variability in sample proportions from the true population proportion. Denoted as \( \sigma_{\hat{p}} \), this measures how much the proportion of interracial couples in a sample deviates from the known population proportion of 0.07.
To compute this standard deviation, we use the following formula:
\[ \sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} \]
For a sample size \( n = 100 \), the calculation becomes:

• \( \sigma_{\hat{p}} = \sqrt{\frac{0.07 \times 0.93}{100}} = 0.0257 \)

By understanding the standard deviation, we gain insight into the reliability and precision of our sample measures. Smaller standard deviations indicate more precise estimates of the population proportion.

Normal Approximation

When we talk about normal approximation in the context of sampling distributions, we're referring to the use of the normal distribution to model the behavior of sample proportions. For this approximation to be valid, certain conditions relating to sample size must be fulfilled.
The rule of thumb is: for the sampling distribution of \( \hat{p} \) to be approximately normal, \( np \) and \( n(1-p) \) must both be greater than 10. This condition ensures that the sample is large enough for the Central Limit Theorem to hold, making the sampling distribution resemble a normal curve.
In the exercise, for \( n = 100 \), we calculate:

• \( 100 \times 0.07 = 7 \)
• \( 100 \times 0.93 = 93 \)

Since \( 7 < 10 \), normal approximation isn't reasonable for \( n = 100 \). However, increasing the sample size to \( n = 200 \) gives:

• \( 200 \times 0.07 = 14 \)
• \( 200 \times 0.93 = 186 \)

Both values are greater than 10, making the normal approximation more applicable.

Sample Size Effect

The effect of the sample size is profound when it relates to the mean and standard deviation of the sampling distribution. As the sample size increases, the standard deviation \( \sigma_{\hat{p}} \) usually decreases, making the sample estimates of the population proportion more precise.
In our given case, moving from \( n = 100 \) to \( n = 200 \) while keeping the true population proportion consistent at 0.07 shows this effect clearly. The standard deviation changes from:

• \( \sigma_{\hat{p}} = 0.0257 \) for \( n = 100 \)
• \( \sigma_{\hat{p}} = 0.0181 \) for \( n = 200 \)

The mean, however, remains unchanged as it solely depends on the population proportion and not the sample size.
Therefore, increasing the sample size reduces variability, leading to more reliable and valid conclusions about the underlying population characteristics.