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Chapter 8: Problem 51

A proponent of a new proposition on a ballot wants to know whether the proposition is likely to pass. 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. 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
Performing a hypothesis test is appropriate. The null hypothesis is \(H_0: P = 0.5\) and the alternative hypothesis is \(H_1: P > 0.5\). The test statistic is approximately \(4\), the p-value is \(< 0.0001\), and since the p-value is less than the level of significance \(0.05\), the null hypothesis is rejected. Therefore, there is sufficient evidence to indicate that the proposition is likely to pass.

Step by step solution

01

Decision - Hypothesis Test or Confidence Interval

Given the sample proportion \(p = \frac{580}{1000} = 0.58\). The question is whether the proposition is likely to pass, i.e., whether the true population proportion \(P > 0.5\). Thus, the appropriate method is a hypothesis test.
02

Hypothesis Test - Statement of Hypotheses

The null hypothesis, \(H_0\), supposes no effect or status quo, which in this case is: \(H_0: P = 0.5\). The alternative hypothesis, \(H_1\), posits the claim we are testing for. Since we are testing if the proposition is likely to pass: \(H_1: P > 0.5\).
03

Hypothesis Test - Calculation of Test Statistic

Assuming that the necessary conditions for a z-test are satisfied, the test statistic is given by the formula: \(z = \frac{(\hat{p} - P_0)}{\sqrt{\frac{P_0(1 - P_0)}{n}}}\), where \(\hat{p}\) is the sample proportion, \(P_0\) is the proportion under the null hypothesis, and \(n\) is the sample size. Substituting the values, we get: \(z = \frac{(0.58 - 0.5)}{\sqrt{\frac{0.5(1 - 0.5)}{1000}}} = 4.\)
04

Hypothesis Test - Calculate the p-value

Next, we calculate the p-value, which is the probability of getting a test statistic as extreme or more extreme than the one calculated, assuming the null hypothesis is true. In this case, we are performing an upper-tailed test (because of the \(>\) sign in \(H_1\)), so the p-value is the area to the right of 4 under the standard normal curve. Using a standard normal table or calculator gives us the p-value \(< 0.0001\).
05

Hypothesis Test - Conclusion

Since the p-value is less than the level of significance of \(5 \%\), we reject the null hypothesis. This indicates that there is sufficient evidence to conclude that the proposition is likely to pass.

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Most popular questions from this chapter

When comparing two sample proportions with a two-sided alternative hypothesis, all other factors being equal, will you get a smaller p-value if the sample proportions are close together or if they are far apart? Explain.

A vaccine to prevent severe rotavirus gastroenteritis (diarnhea) was given to African children within the first year of life as part of a drug study. The study reported that of the 3298 children randomly assigned the vaccine, 63 got the virus. Of the 1641 children randomly assigned the placebo, 80 got the virus. (Source: Madhi et al., Effect of human rotavirus vaccine on severe diarrhea in African infants, New England Journal of Medicine, vol. \(362: 289-298\), January 28,2010 ) a. Find the sample percentage of children who caught the virus in each group. Is the sample percent lower for the vaccine group, as investigators hoped? b. Determine whether the vaccine is effective in reducing the chance of catching the virus, using a significance level of \(0.05\). Steps 1 and 2 of the hypothesis-testing procedure are given. Complete the question by doing steps 3 and 4 . Step 1: \(\mathrm{H}_{0}: p_{v}=p_{p}\left(p_{v}\right.\) is the proportion that got the virus among those who took the vaccine, and \(p_{p}\) is the proportion that got the virus among those who took the placcbo.) \(\mathrm{H}_{\mathrm{L}}: p_{e}

Shift workers, who work during the night and must sleep during the day, often become sleepy when working and have trouble sleeping during the day. In a study done at Harvard Medical School, 209 shift workers were randomly divided into two groups; one group received a new sleep medicine (modafinil, or Provigil), and the other group received a placebo. During the study, \(54 \%\) of the workers taking the placebo and \(29 \%\) of those taking the medicine reported accidents or near accidents commuting to and from work. Assume that 104 of the people were assigned the medicine and 105 were assigned the placebo. (Source: Czeisler et al., Modafinil for excessive sleepiness associated with shift-work sleep disorder, New England Joumal of Medicine, vol. 353: \(476-486\), August 4,2005 ) a. State the null and altemative hypothesis. Is the alternative hypothesis one-sided or two-sided? Explain your choice. b. Perform a statistical test to determine whether the difference in proportions is significant at the \(0.05\) level.

A magazine advertisement claims that wearing a magnetized bracelet will reduce arthritis pain in those who suffer from arthritis. A medical researcher tests this claim with 233 arthritis sufferers randomly assigned either to wear a magnetized bracelet or to wear a placebo bracelet. The researcher records the proportion of each group who report relief from arthritis pain after 6 weeks. After analyzing the data, he fails to reject the null hypothesis. Which of the following are valid interpretations of his findings? There may be more than one correct answer. a. The magnetized bracelets are not effective at reducing arthritis pain. b. There's insufficient evidence that the magnetized bracelets are effective at reducing arthritis pain. c. The magnetized bracelets had exactly the same effect as the placebo in reducing arthritis pain. d. There were no statistically significant differences between the magnetized bracelets and the placebos in reducing arthritis pain.

Group is a private conference comprising American and European political elites who make up one-third and two-thirds of the group, respectively. Suppose you are looking at conferences, each with 150 members, in the Bilderberg Group. The null hypothesis is that the probability of a European being selected into the club is \(67 \%\). a. How many Europeans would you expect in a conference of 150 people if the null hypothesis is true? b. Suppose Conference A contains 103 Europeans out of 150 and Conference B contains 108 Europeans out of 150 . Which will have a smaller p-value and why?
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