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Chapter 7: Problem 86

According to the Bureau of Labor Statistics, \(71.9 \%\) of young women enroll in college directly after high school graduation. Suppose a random sample of 200 female high school graduates is selected and the proportion who enroll in college is obtained. a. What value should we expect for the sample proportion? b. What is the standard error? c. Would it be surprising if only \(68 \%\) of the sample enrolled in college? Why or why not? d. What effect would increasing the sample size to 500 have on the standard error?

Short Answer

Expert verified
a. The expected value for the sample proportion is 0.719 or 71.9%. b. The standard error is 0.0343. c. No, it would not be surprising if only 68% of the sample enrolled in college as the calculated z-score, -1.13, is within the normal range. d. Increasing the sample size to 500 would decrease the standard error to 0.01974, therefore increasing the precision of the estimates.

Step by step solution

01

Find the Expected Sample Proportion

The expected value for the sample proportion is simply the population proportion, which is given to be 71.9% or 0.719.
02

Calculate the Standard Error

The standard error (SE) of the proportion can be calculated using the formula \(\sqrt{p(1-p)/n}\), where \(p\) is the population proportion and \(n\) is the sample size. So, substituting the given values, SE = \(\sqrt{(0.719)(1-0.719)/200} = 0.0343\).
03

Analyze the Surprising Enrollment Rate

To determine if 68% enrollment is surprising, compare it to the expected value and calculate how many standard errors it is away from the mean. The z-score, which measures the number of standard deviations from the mean, is (0.68-0.719)/0.0343 = -1.13. Given that it's less than two, it doesn't fall in the 'surprising' range (typically, z-scores above 2 or below -2 are considered surprising in social sciences).
04

Analyze the Effect of Sample Size on Standard Error

Increasing the sample size generally decreases the standard error. This is due to the denominator \(n\) in the formula for standard error. If we recalculate the standard error for a sample size of 500, we obtain a smaller figure, SE = \(\sqrt{(0.719)(1-0.719)/500} = 0.01974\). As sample size increases, our estimates become more precise.

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Most popular questions from this chapter

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