The author’s of the article “Perceived risks of Heart Disease and Cancer Among
Cigarette Smokers” expressed the concern that a majority of smokers do not view themselves as being at increased risk of heart disease or cancer. Because of this, the authors call for a public health campaign to educate smokers about the associated risks. In support of this recommendation, the authors offered the results of a study of 737 current smokers selected at random from the U.S. households with telephones. Of the 737 smokers surveyed, 295 indicated that they believe that they have a higher then average risk for cancer. Does this data suggest that the true proportion of smokers who view themselves as being at increased risk of cancer is, in fact, less than 0.05 as claimed by the authors of the paper?
To determine whether the data suggests that the true proportion of smokers who view themselves as being at increased risk of cancer is less than 0.05, we need to perform a hypothesis test.
Hypothesis testing involves setting up a null hypothesis (H0) and an alternative hypothesis (Ha). In this case, the null hypothesis would be that the true proportion of smokers who view themselves as being at increased risk of cancer is equal to or greater than 0.05, and the alternative hypothesis would be that the true proportion is less than 0.05.
Let's calculate the test statistic and p-value based on the provided data:
Given:
Total number of smokers surveyed (n) = 737
Number of smokers who believe they have a higher than average risk for cancer = 295
First, we need to calculate the test statistic, which in this case is a z-score. The formula for the z-score of a proportion is:
z = (p - P) / sqrt((P*(1-P)) / n)
where p is the sample proportion, P is the hypothesized proportion under the null hypothesis, and n is the sample size.
The calculated sample proportion is:
p = 295 / 737
Next, we assume the null hypothesis, which states that the true proportion is equal to or greater than 0.05. So, P = 0.05.
Now, we can calculate the test statistic (z-score):
z = (p - P) / sqrt((P*(1-P)) / n)
Finally, we can calculate the p-value, which is the probability of obtaining a test statistic as extreme as the one observed, assuming the null hypothesis is true.
Using a statistical software or a z-table, we can determine the p-value based on the calculated z-score. If the p-value is less than our significance level (usually 0.05), we reject the null hypothesis in favor of the alternative hypothesis.
Thus, to fully answer the question, we need the calculated z-score and its associated p-value.