Question

The authors of the article "Perceived Risks of Heart Disease and Cancer Among Cigarette Smokers" (Journal of the American Medical Association [1999]: \(1019-1021\) ) expressed the concem that a majority of smokers do not view themselves as being at increased risk of heart disease or cancer. A study of 737 current smokers selected at random from U.S. households with telephones found that of 737 smokers surveyed, 295 indicated that they believed they have a higher than average risk of cancer. Do these data suggest that \(\pi\), the true proportion of smokers who view themselves as being at increased risk of cancer is in fact less than .5, as claimed by the authors of the paper? Test the relevant hypotheses using \(\alpha=.05\).

Short answer

To conclude, compare the test statistic with the critical value. If 'z' (test statistic) < -1.645 (critical value) then, the null hypothesis is rejected concluding that the proportion is indeed less than 0.5.

Step-by-step solution

- Setup the Hypothesis

The null hypothesis (\(H_0\)) is that the proportion of smokers who view themselves at higher risk of cancer is 0.5. The alternate hypothesis (\(H_a\)) is that the proportion is less than 0.5.\nFormally, we can say:\n\nNull Hypothesis (\(H_0\)): \(\pi = 0.5\) \n\nAlternate Hypothesis (\(H_a\)): \(\pi < 0.5\)

- Calculate Test Statistic

In this case we have to use a one-sample proportion z-test. The test statistic (z) is given by the formula:\n\(z = \frac{(p - \pi)}{\sqrt{\frac{\pi*(1-\pi)}{n}}}\) Where:- \(p\) is the sample proportion = 295/737 = 0.4,- \(\pi\) is the proportion in the null hypothesis = 0.5- \(n\) is the number of trials = 737.Substituting the values into the formula gives us:\(z = \frac{(0.4 - 0.5)}{\sqrt{\frac{0.5*0.5}{737}}}\)

- Critical Value and Decision Rule

The decision rule for this test at the 0.05 significance level is to reject the null hypothesis if the test statistic is less than the critical value at \(\alpha=0.05\). In this case, the critical value for a one-tailed test at \(\alpha=0.05\) is approximately -1.645.

- Conclusion of Hypothesis Test

After calculating the test statistic ('z'), compare it with the critical value. If the test statistic is less than the critical value, then reject the null hypothesis, concluding there is enough evidence to say that the proportion of smokers who view themselves as at higher risk of cancer is less than 0.5.