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Chapter 7: Problem 42

According to a 2017 Pew Research report, \(40 \%\) of millennials have a BA degree. Suppose we take a random sample of 500 millennials and find the proportion who have a driver's license. Find the probability that at most \(35 \%\) of the sample has a BA degree. Begin by verifying that the conditions for the Central Limit Theorem for Sample Proportions have been met.

Short Answer

Expert verified
The probability that at most 35% of the sample has a BA degree is 1.19%.

Step by step solution

01

Verification of the Central Limit Theorem for Sample Proportions Conditions

For the Central Limit Theorem for Sample Proportions to apply, 2 conditions must be satisfied: \(n*p* 鈮� 10\) and \(n*(1-p*) 鈮� 10\), where \(n\) is the sample size and \(p*\) is the population proportion. Using the provided numbers where \(n = 500\) and \(p* = 0.40\), checking these conditions gives \(500*0.40 = 200\) and \(500*(1-0.40) = 300\). Both these values are greater than 10, so the conditions for the theorem are met.
02

Calculating the Standard Error

After confirming both conditions are met, next calculate the standard error using the formula: \(SE = 鈭�((p*(1-p*))/n)\), where \(p* = 0.40\), and \(n = 500\). Substituting these values, the standard error works out to \(0.022\)
03

Applying Normal Distribution to Calculate the Probability

Using the standard normal distribution, the Z-score must be calculated first with the following formula: \(Z = (p - p*)/SE\), where \(p = 0.35\), \(p* = 0.40\), and \(SE = 0.022\). Substituting these values, the Z-score is \(-2.27\). Then look up this Z-score in a Z-table (or use statistical software) to find the probability, which is \(0.0119\) or 1.19%.

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Most popular questions from this chapter

In 2003 and 2017 Gallup asked Democratic voters about their views on the FBI. In \(2003,44 \%\) thought the \(\mathrm{FBI}\) did a good or excellent job. In \(2017,69 \%\) of Democratic voters felt this way. Assume these percentages are based on samples of 1200 Democratic voters. a. Can we conclude, on the basis of these two percentages alone, that the proportion of Democratic voters who think the FBI is doing a good or excellent job has increase from 2003 to \(2017 ?\) Why or why not? b. Check that the conditions for using a two-proportion confidence interval hold. You can assume that the sample is a random sample. c. Construct a \(95 \%\) confidence interval for the difference in the proportions of Democratic voters who believe the FBI is doing a good or excellent job, \(p_{1}-p_{2}\). Let \(p_{1}\) be the proportion of Democratic voters who felt this way in 2003 and \(p_{2}\) be the proportion of Democratic voters who felt this way in 2017 . d. Interpret the interval you constructed in part c. Has the proportion of Democratic voters who feel this way increased? Explain.

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According to a 2017 article in The Washington Post, \(72 \%\) of high school seniors have a driver's license. Suppose we take a random sample of 100 high school seniors and find the proportion who have a driver's license. Find the probability that more than \(75 \%\) of the sample has a driver's license. Begin by verifying that the conditions for the Central Limit Theorem for Sample Proportions have been met.

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