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Chapter 8: Problem 17

Marriage Obsolete? According to a study done by the Pew Research Center, \(39 \%\) of adult Americans believe that marriage is now obsolete. (a) Suppose a random sample of 500 adult Americans is asked whether marriage is obsolete. Describe the sampling distribution of \(\hat{p}\), the proportion of adult Americans who believe marriage is obsolete. (b) What is the probability that in a random sample of 500 adult Americans less than \(38 \%\) believe that marriage is obsolete? (c) What is the probability that in a random sample of 500 adult Americans between \(40 \%\) and \(45 \%\) believe that marriage is obsolete? (d) Would it be unusual for a random sample of 500 adult Americans to result in 210 or more who believe marriage is obsolete?

Short Answer

Expert verified
The sampling distribution of \( \hat{p} \) is approximately normal with \( \backslash mu_{\hat{p}} = 0.39 \) and \( \backslash sigma_{\hat{p}} \approx 0.0219 \. The probability of less than 38\% is 0.324, between 40\% and 45\% is 0.32, and 210 or more is not unusual.

Step by step solution

01

Describe the Sampling Distribution of \(\backslash hat{p}\).

The sampling distribution of the sample proportion \(\backslash hat{p}\) can be described using the Central Limit Theorem. Since the sample size (n = 500) is large enough, the distribution of \(\backslash hat{p}\) will be approximately normal. The mean of \(\backslash hat{p}\) is \( \backslash mu_{\hat{p}} = p = 0.39 \), and the standard deviation of \(\backslash hat{p}\) is \( \backslash sigma_{\hat{p}} = \backslash sqrt{\backslash frac{p(1-p)}{n}} = \backslash sqrt{\backslash frac{0.39 \times 0.61}{500}} \approx 0.0219 \).
02

Calculate the Probability of Less Than 38\% Believing Marriage is Obsolete

To find the probability that less than 38\% (0.38) believe that marriage is obsolete, we convert the sample proportion to a Z-score: \( Z = \backslash frac{\hat{p} - \backslash mu_{\hat{p}}}{\backslash sigma_{\hat{p}}} = \backslash frac{0.38 - 0.39}{0.0219} \approx -0.4575 \). Using the Z-table, the probability corresponding to Z = -0.4575 is approximately 0.324. Thus, the probability is 0.324.
03

Calculate the Probability of Between 40\% and 45\% Believing Marriage is Obsolete

To find the probability of between 40\% (0.40) and 45\% (0.45), we first convert each to Z-scores: \( Z_{0.40} = \backslash frac{0.40 - 0.39}{0.0219} \approx 0.4575 \) and \( Z_{0.45} = \backslash frac{0.45 - 0.39}{0.0219} \approx 2.737 \). Using the Z-table, the probability for Z_{0.40} = 0.4575 is approximately 0.676, and for Z_{0.45} = 2.737 it is approximately 0.996. Therefore, the probability is \( 0.996 - 0.676 = 0.32 \).
04

Determine If 210 or More Believing Marriage Is Obsolete is Unusual

To find if it is unusual, first calculate the sample proportion: \( \hat{p} = \backslash frac{210}{500} = 0.42 \), then convert this to a Z-score: \( Z = \backslash frac{0.42 - 0.39}{0.0219} = 1.37 \). Using the Z-table, the probability corresponding to Z = 1.37 is approximately 0.915. Since the probability of 210 or more is 1 - 0.915 = 0.085, which is greater than 0.05, it is not unusual.

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Key Concepts

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

Central Limit Theorem
The Central Limit Theorem (CLT) is a crucial concept in statistics. It states that, given a sufficiently large sample size from a population with a finite level of variance, the sampling distribution of the sample mean (or sample proportion) will tend to be normally distributed, regardless of the original population's distribution. In our exercise, we use the CLT to describe the sampling distribution of \(\backslash hat{p}\), the proportion of adults who believe marriage is obsolete. Since our sample size is 500, which is quite large, we can safely assume that the distribution of \(\backslash hat{p}\) is approximately normal. This allows us to use standard statistical techniques to make probability predictions.
Proportion
A proportion is a statistical measure that refers to the fraction of the total that possesses a certain attribute. In our exercise, \(p = 0.39\) represents the proportion of all adult Americans who believe marriage is obsolete. For any sample of adult Americans, we use \(\backslash hat{p}\) to represent the sample proportion who hold this belief. When we draw a sample of 500 adults, the mean of \(\backslash hat{p} \) remains \(0.39\), and the standard deviation can be found using the formula: \[ \sigma_{\backslash hat{p}} = \backslash sqrt{\backslash frac{p (1-p)}{n}} \]. For our data, substituting \(p = 0.39\) and \(n = 500\), we get \(\backslash sigma_{\backslash hat{p}} \ approx 0.0219\). This standard deviation helps us to calculate probabilities using Z-scores.
Z-score
A Z-score indicates how many standard deviations an element is from the mean. In our exercise, we use Z-scores to find probabilities for different sample proportions. For instance, when we calculate the Z-score for the proportion 0.38: \[ Z = \backslash frac{0.38 - 0.39}{0.0219} \backslash approx -0.4575 \]. This value tells us that 0.38 is roughly 0.4575 standard deviations below the mean. We can then use the Z-table to find the corresponding probability. For more complex ranges, such as between 40% and 45%, we calculate Z-scores for both proportions (0.40 and 0.45) and find the resulting probability. These calculations help us understand how likely we are to obtain certain sample proportions based on the original population proportion.

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Most popular questions from this chapter

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Explain what a sampling distribution is.
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