Question

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

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

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.

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 \(n = 110\).

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).

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.

Make a conclusion

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

Key concepts

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 \