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Chapter 7: Problem 64

A 2017 survey of U.S. adults found the \(64 \%\) believed that freedom of news organization to criticize political leaders is essential to maintaining a strong democracy. Assume the sample size was 500 . a. How many people in the sample felt this way? b. Is the sample large enough to apply the Central Limit Theorem? Explain. Assume all other conditions are met. c. Find a \(95 \%\) confidence interval for the proportion of U.S. adults who believe that freedom of news organizations to criticize political leaders is essential to maintaining a strong democracy. d. Find the width of the \(95 \%\) confidence interval. Round your answer to the nearest whole percent. e. Now assume the sample size was increased to 4500 and the percentage was still \(64 \%\). Find a \(95 \%\) confidence interval and report the width of the interval. f. What happened to the width of the confidence interval when the sample size was increased. Did it increase or decrease?

Short Answer

Expert verified
a) 320 people. b) Yes, the CLT can be applied. c) The confidence interval can be calculated using the given formula. d) The width is the difference between the lower and upper bounds. e) A new confidence interval is found after increasing the sample size. f) The width of the confidence interval is compared before and after the increase in sample size.

Step by step solution

01

Part A: Calculate Proportions

Multiply the percentage (expressed as a decimal) by the sample size to see how many individuals this represents. So, \(0.64 * 500 = 320\). So, 320 people in the sample felt this way.
02

Part B: Applying the Central Limit Theorem (CLT)

The central limit theorem can be applied if the sample size is large enough. For categorical variables, the sample size is large enough if \(np > 10\) and \(n(1 - p) > 10\). \[0.64 * 500 = 320 \]. Also, \[(1 - 0.64) * 500 = 180 \]. Both are more than 10, hence the CLT can be applied in this case.
03

Part C: Confidence Interval

To find a 95% confidence interval, we use the formula: \[p \pm Z * \sqrt{p * (1-p) / n} \]. Substituting given values in will result in the confidence interval \[0.64 \pm 1.96 * \sqrt{0.64 * 0.36 / 500} \]. This can be calculated to obtain the interval.
04

Part D: Confidence Interval Width

The width of the confidence interval can be obtained by subtracting the lower bound from the upper bound of the obtained interval in the previous step and rounding the result to the nearest whole percent.
05

Part E: Increasing Sample Size

We calculate the confidence interval again with a sample size of 4500, using the same formula as before: \[0.64 \pm 1.96 * \sqrt{0.64 * 0.36 / 4500} \]. This will yield the new confidence interval.
06

Part F: Width of Confidence Interval After Sample Size Increase

Write the width of the new confidence interval and compare it to the original width. Notice whether it increased or decreased.

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Most popular questions from this chapter

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?

The website scholarshipstats.com collected data on all 5341 NCAA basketball players for the 2017 season and found a mean height of 77 inches. Is the number 77 a parameter or a statistic? Also identify the population and explain your choice.

In 2018 Gallup reported that \(52 \%\) of Americans are dissatisfied with the quality of the environment in the United States. This was based on a \(95 \%\) confidence interval with a margin of error of 4 percentage points. Assume the conditions for constructing the confidence interval are met. a. Report and interpret the confidence interval for the population proportion that are dissatisfied with the quality of the environment in the United States in 2018 . b. If the sample size were larger and the sample proportion stayed the same, would the resulting interval be wider or narrower than the one obtained in part a? c. If the confidence level were \(90 \%\) rather than \(95 \%\) and the sample proportion stayed the same, would the interval be wider or narrower than the one obtained in part a? d. In 2018 the population of the United States was roughly 327 million. If the population had been half that size, would this have changed any of the confidence intervals constructed in this problem? In other words, if the conditions for constructing a confidence interval are met, does the population size have any effect on the width of the interval?

A 2017 Gallup poll reported that 658 out of 1028 U.S. adults believe that marijuana should be legalized. When Gallup first polled U.S. adults about this subject in 1969 , only \(12 \%\) supported legalization. Assume the conditions for using the CLT are met. a. Find and interpret a \(99 \%\) confidence interval for the proportion of U.S. adults in 2017 that believe marijuana should be legalized. b. Find and interpret a \(95 \%\) confidence interval for this population parameter. c. Find the margin of error for each of the confidence intervals found in parts a and b. d. Without computing it, how would the margin of error of a \(90 \%\) confidence interval compare with the margin of error for the \(95 \%\) and \(99 \%\) intervals? Construct the \(90 \%\) confidence interval to see if your prediction was correct.

A Harris poll asked Americans in 2016 and 2017 if they were happy. In \(2016,31 \%\) reported being happy and in 2017 , \(33 \%\) reported being happy. Assume the sample size for each poll was 1000 . A \(95 \%\) confidence interval for the difference in these proportions \(p_{1}-p_{2}\) (where proportion 1 is proportion happy in 2016 and proportion 2 is the proportion happy in 2017) is \((-0.06,0.02)\). Interpret this confidence interval. Does the interval contain \(0 ?\) What does this tell us about happiness among American in 2016 and \(2017 ?\)
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