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Chapter 10: Problem 73

A number of initiatives on the topic of legalized gambling have appeared on state ballots. Suppose that a political candidate has decided to support legalization of casino gambling if he is convinced that more than twothirds of U.S. adults approve of casino gambling. USA Today (June 17,1999 ) reported the results of a Gallup poll in which 1523 adults (selected at random from households with telephones) were asked whether they approved of casino gambling. The number in the sample who approved was 1035 . Does the sample provide convincing evidence that more than two-thirds approve?

Short Answer

Expert verified
The answer depends on the computed value of the test statistic and the p-value. If the p-value is less than the elected level of significance (for example, 0.05), then there is convincing evidence that more than two-thirds of U.S. adults approve of casino gambling.

Step by step solution

01

State the Hypotheses

The null hypothesis (H0) assumes the default position, which is typically a statement of no effect or no difference. In this case, it's that two-thirds or fewer adults approve of casino gambling. The alternative hypothesis (H1) is what you might believe to be true or are testing for, here it's that more than two-thirds approve. So,\n\nNull hypothesis (H0): p 鈮� 2/3\n\nAlternative Hypothesis (H1): p > 2/3\n\nwhere p is the proportion of U.S. adults who approve of casino gambling.
02

Compute the sample proportion and Test Statistic

The sample proportion \( \hat{p} \) is calculated by dividing the number of individuals who approved (1035) by the total number in the sample (1523)\n\n\( \hat{p} = 1035 / 1523 = 0.680 \n\nThe test statistic (Z) for a test of proportion is calculated by\n\nZ = ( \( \hat{p} - p_{0} \) ) / \( \sqrt {p_{0} (1 - p_{0} ) / n } \n\nwhere \( p_{0} \) is the proportion under the null hypothesis, in this case two thirds or, 0.67 (approx), and n is the sample size.\n\nZ = (0.680 - 0.67) / \( \sqrt { (0.67 * (1 - 0.67)) / 1523 } \)
03

Find the P-value

The P-value is the probability of obtaining a test statistic as extreme as the one computed, given that the null hypothesis is true. This can be found by looking the Z statistic value up in a standard normal (Z) table or using statistical software.
04

Draw a Conclusion

Compare the p-value to your chosen significance level (for instance, 0.05) to decide whether to reject the null hypothesis. If the p-value is less than or equal to the significance level, reject the null hypothesis and conclude that the data provides convincing evidence that more than two-thirds of U.S. adults approve of casino gambling.

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Most popular questions from this chapter

Past experience has indicated that the true response rate is \(40 \%\) when individuals are approached with a request to fill out and return a particular questionnaire in a stamped and addressed envelope. An investigator believes that if the person distributing the questionnaire is stigmatized in some obvious way, potential respondents would feel sorry for the distributor and thus tend to respond at a rate higher than \(40 \%\). To investigate this theory, a distributor is fitted with an eye patch. Of the 200 questionnaires distributed by this individual, 109 were returned. Does this strongly suggest that the response rate in this situation exceeds the rate in the past? State and test the appropriate hypotheses at significance level . 05 .

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An automobile manufacturer is considering using robots for part of its assembly process. Converting to robots is an expensive process, so it will be undertaken only if there is strong evidence that the proportion of defective installations is lower for the robots than for human assemblers. Let \(\pi\) denote the true proportion of defective installations for the robots. It is known that human assemblers have a defect proportion of 02 . a. Which of the following pairs of hypotheses should the manufacturer test: $$ H_{0}: \pi=.02 \text { versus } H_{s}: \pi<.02 $$ or $$ H_{0}: \pi=.02 \text { versus } H_{a}: \pi>.02 $$ Explain your answer. b. In the context of this exercise, describe Type I and Type II errors. c. Would you prefer a test with \(\alpha=.01\) or \(\alpha=.1 ?\) Explain your reasoning.

For the following pairs, indicate which do not comply with the rules for setting up hypotheses, and explain why: a. \(H_{0}: \mu=15, H_{a}: \mu=15\) b. \(H_{0}: \pi=.4, H_{a}: \pi>.6\) c. \(H_{0}: \mu=123, H_{a}: \mu<123\) d. \(H_{0}: \mu=123, H_{a}: \mu=125\) e. \(H_{0}: p=.1, H_{a}: p=125\)
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