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Chapter 10: Problem 76

According to the article "Which Adults Do Underage Youth Ask for Cigarettes?" (American Journal of \(P u b\) lic Health [1999]: \(1561-1564\) ), \(43.6 \%\) of the 149 18- to 19 -year-olds in a random sample have been asked to buy cigarettes for an underage smoker. a. Is there convincing evidence that fewer than half of 18 to 19 -year-olds have been approached to buy cigarettes by an underage smoker? b. The article went on to state that of the 110 nonsmoking 18 - to 19 -year- olds, only \(38.2 \%\) had been approached to buy cigarettes for an underage smoker. Is there evidence that less than half of nonsmoking 18 - to 19 -year- olds have been approached to buy cigarettes?

Short Answer

Expert verified
a. There is/not enough evidence to suggest less than half of 18 to 19-year-olds have been approached to buy cigarettes for an underage smoker.\\b. There is/not enough evidence to suggest less than half of the nonsmoking 18 to 19-year-olds have been approached to buy cigarettes for an underage smoker. The exact answers will depend on the calculated z-scores and p-values.

Step by step solution

01

Hypotheses formulation

a. Formulate null and alternate hypothesis. Here, null hypothesis \(H_0: p = 0.5\) and alternate hypothesis \(H_1: p < 0.5\).\\b. Again, form null hypothesis \(H_0: p = 0.5\) and alternate hypothesis \(H_1: p < 0.5\). Here, p is the proportion of people asked to buy cigarettes for an underage smoker.
02

Calculate the test-statistic (z-score)

a. Based on the sample, the proportion is \(p = 0.436\) and sample size \(n = 149\). So, calculate the z-score using the formula:\\\(z = \frac{{\p - 0.5}}{{sqrt{(0.5(1 - 0.5) / n)}}}\) \\b. Same step is repeated for the second case with \(p = 0.382\) and \(n = 110\).
03

Use z-table to get the critical value and p-value

Based on the calculated z-score, look up the critical value in the z table. Then, calculate the p-value for this z score. Remember as the problem asked for 'less than', we only consider the tail at the left of the mean (negative).
04

Compare p-value with significance level

Next, compare p-value with 0.05 (common significance level). If p-value < 0.05, null hypothesis can be rejected concluding there's enough evidence supporting alternate hypothesis.
05

Make a conclusion

Finally, based on whether the null hypothesis was rejected or not, make conclusions for both cases a. and b.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Null Hypothesis
The null hypothesis is a crucial element in the realm of statistical hypothesis testing. Think of it as a default position that suggests there is no effect or no difference. In the context of the textbook exercise, where questions about 18 to 19-year-old individuals being asked to buy cigarettes for underage smokers are raised, the null hypothesis (\( H_0 \)) is set as \

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Most popular questions from this chapter

In a national survey of 2013 adults, 1590 responded that lack of respect and courtesy in American society is a serious problem, and 1283 indicated that they believe that rudeness is a more serious problem than in past years (Associated Press, April 3,2002 ). Is there convincing evidence that more than three-quarters of U.S. adults believe that rudeness is a worsening problem? Test the relevant hypotheses using a significance level of \(.05\).

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According to the article "Workaholism in Organizations: Gender Differences" (Sex Roles [1999]: \(333-346\) ), the following data were reported on 1996 income for random samples of male and female MBA graduates from a certain Canadian business school: \begin{tabular}{lccc} & \(\boldsymbol{N}\) & \(\overline{\boldsymbol{x}}\) & \(\boldsymbol{s}\) \\ \hline Males & 258 & \(\$ 133,442\) & \(\$ 131,090\) \\ Females & 233 & \(\$ 105,156\) & \(\$ 98,525\) \\ \hline \end{tabular} Note: These salary figures are in Canadian dollars. a. Test the hypothesis that the mean salary of male MBA graduates from this school was in excess of \(\$ 100,000\) in \(1996 .\) b. Is there convincing evidence that the mean salary for all female MBA graduates is above \(\$ 100,000 ?\) Test using \(\alpha=.10\) c. If a significance level of \(.05\) or \(.01\) were used instead of \(.10\) in the test of Part (b), would you still reach the same conclusion? Explain.

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