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Chapter 8: Problem 52

A proponent of a new proposition on a ballot wants to know the population percentage of people who support the bill. Suppose a poll is taken, and 580 out of 1000 randomly selected people support the proposition. Should the proponent use a hypothesis test or a confidence interval to answer this question? Explain. If it is a hypothesis test, state the hypotheses and find the test statistic, p-value, and conclusion. Use a \(5 \%\) significance level. If a confidence interval is appropriate, find the approximate \(95 \%\) confidence interval. In both cases, assume that the necessary conditions have been met.

Short Answer

Expert verified
The proponent should use a confidence interval, approximately \(95 \%\) confidence interval is \(0.580 \pm 1.96*\sqrt{((0.580)*(1-0.580)/1000)}\)

Step by step solution

01

Calculate the Sample Proportion

The sample proportion (\(p\)) is given by the number of successes in the sample divided by the sample size. In this case, it's 580 out of 1000, which is \(0.580\).
02

Calculate the Standard Error

The standard error for the sample proportion is the square root of \((p(1-p)/n)\), where \(n\) is the sample size. Substituting the values, the standard error is \(\sqrt{((0.580)*(1-0.580)/1000)}\).
03

Calculate the Confidence Interval

A \(95 \%\) confidence interval for the proportion is given by \(p \pm z*StandardError\), where \(z\) is the z-score that corresponds to the desired level of confidence. For a \(95 \%\) confidence level, the z-score is approximately \(1.96\). Hence, the \(95 \%\) confidence interval is \(0.580 \pm 1.96*\sqrt{((0.580)*(1-0.580)/1000)}\)

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Most popular questions from this chapter

A weight-loss diet claims that it causes weight loss by eliminating carbohydrates (breads and starches) from the diet. To test this claim, researchers randomly assign overweight subjects to two groups. Both groups eat the same amount of calories, but one group eats almost no carbs, and the other group includes carbs in their meals. After 2 months, the researchers test the claim that the no-carb diet is better than the usual diet. They record the proportion of each group that lost more than \(5 \%\) of their initial weight. They then announce that they failed to reject the null hypothesis. Which of the following are valid interpretations of the researchers' findings? a. There were no significant differences in effectiveness between the no-carb diet and the carb diet. b. The no-carb diet and the carb diet were equally effective. c. The researchers did not see enough evidence to conclude that the no-carb diet was more effective. d. The no-carb diet was less effective than the carb diet.

According to a 2015 University of Michigan poll, \(71.5 \%\) of high school seniors in the United States had a driver's license. A sociologist thinks this rate has declined. The sociologist surveys 500 randomly selected high school seniors and finds that 350 have a driver's license. a. Pick the correct null hypothesis. i. \(p=0.715\) ii. \(p=0.70\) iii. \(\hat{p}=0.715\) iv. \(\hat{p}=0.70\) b. Pick the correct alternative hypothesis. i. \(p>0.715\) ii. \(p<0.715\) iii. \(\hat{p}<0.715\) iv. \(p \neq 0.715\) c. In this context, the symbol \(p\) represents (choose one) i. the proportion of high school seniors in the entire United States that have a driver's license. ii. the proportion of high school seniors in the sociologist's random sample that have a driver's license.

In each case. choose whether the appropriate test is a one-proportion \(z\) -test or a two-proportion z-test. Name the population(s). a. A researcher takes a random sample of 4 -year-olds to find out whether girls or boys are more likely to know the alphabet. b. A pollster takes a random sample of all U.S. adult voters to see whether more than \(50 \%\) approve of the performance of the current U.S. president. c. A researcher wants to know whether a new heart medicine reduces the rate of heart attacks compared to an old medicine. d. A pollster takes a poll in Wyoming about homeschooling to find out whether the approval rate for men is equal to the approval rate for women. e. A person is studied to see whether he or she can predict the results of coin flips better than chance alone.

By establishing a small value for the significance level, are we guarding against the first type of error (rejecting the null hypothesis when it is true) or guarding against the second type of error?

According to a 2017 AAA survey, \(35 \%\) of Americans planned to take a family vacation (a vacation more than 50 miles from home involving two or more immediate family members. Suppose a recent survey of 300 Americans found that 115 planned on taking a family vacation. Carry out the first two steps of a hypothesis test to determine if the proportion of Americans planning a family vacation has changed. Explain how you would fill in the required entries in the figure for # of success, # of observations, and the value in \(\mathrm{H}_{0}\).
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