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 concern 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 found that only 295 believe they have a higher than average risk of cancer. Do these data suggest that \(p,\) the proportion of all smokers who view themselves as being at increased risk of cancer, is less than \(0.5,\) as claimed by the authors of the paper? For purposes of this exercise, assume that this sample is representative of the population of smokers. Test the relevant hypotheses using \(\alpha=0.05\)

Short answer

The answer depends on the computed Z-statistic after carrying out the calculations in Step 2 and comparing it with the critical Z-value. Based on this, a decision is made whether to reject or not reject the null hypothesis.

Step-by-step solution

Set up Hypotheses and Determine Significance Level

The null hypothesis \(H_0 : p = 0.5\) suggests that at least half of the smokers view themselves at increased risk of cancer. The alternative hypothesis \(H_a : p < 0.5\) suggests that less than half of the smokers view themselves at increased risk of cancer. The significance level \(\alpha\) is \(0.05\).

Calculate the Sample Probability and Z-statistic

Calculate the probability from the sample data. This probability \(p'\) is the number of 'successful outcomes' (295 smokers believe they have a higher risk) divided by the sample size (737 smokers), or \(p' = \frac{295}{737}\). The Z-statistic is calculated using the formula \(Z = \frac{p' - p}{\sqrt{\frac{p(1 - p)}{n}}}\) where \(n\) is the sample size.

Determine the Critical Z-Value & Make Decision

The critical Z value for a one-tailed test at the \(0.05\) level is \(-1.645\). If the computed Z-statistic is less than \(-1.645\), we reject null hypothesis, otherwise we do not reject it.

Interpret the Result

Rejecting \(H_0\) would imply that evidence supports less than half of smokers view themselves at increased risk of cancer. Failure to reject would imply that there's not enough evidence to support the claim made by the authors.

Key concepts

null hypothesis

The null hypothesis in a statistical test is a statement of no effect, no difference, or no change. It is a starting assumption made about a population parameter, such as a mean or a proportion, that is subject to testing against observed data. For example, in the exercise involving smokers' perceptions of cancer risk, the null hypothesis () states that the proportion (

) of all smokers who view themselves as being at increased risk of cancer is exactly 0.5, or in other words, half of all smokers believe they are at higher risk. It serves as the benchmark for comparing the observed sample data. If evidence suggests that the null hypothesis is unlikely, we may decide to reject it in favor of an alternative hypothesis.

It's crucial to understand that the null hypothesis is not 'proven' by a statistical test. Instead, we look for evidence to either reject it or fail to reject it based on the probability of finding our observed results if the null hypothesis were true. We must also remember that failing to reject the null hypothesis does not affirm its truth; it simply implies there isn’t enough evidence to support a contrary claim.

alternative hypothesis

The alternative hypothesis () expresses a research hypothesis that contradicts the null hypothesis; it is what a researcher wants to prove. In the context of our exercise, the alternative hypothesis posits that the proportion (

) of smokers who believe they are at an elevated risk of cancer is less than 0.5. Therefore, it represents a belief that contrasts with the status quo (null hypothesis) and is supported if the test evidence suggests the null hypothesis is likely false.

Formulating an alternative hypothesis properly is important because it provides a direction for the test. It can be two-sided (not equal to) or one-sided (greater than or less than a certain value), such as in the current example, where the hypothesis is one-sided and suggests a 'less than' relationship. After the test, if we find sufficient evidence to reject the null hypothesis, we might conclude the alternative hypothesis is likely to be true, which in this case would mean recognizing that fewer than half the smokers perceive themselves at higher cancer risk.

significance level

The significance level () in hypothesis testing is the threshold for deciding whether to reject the null hypothesis. It is the probability of making a type I error, which occurs if we wrongly reject the null hypothesis when it is actually true. Typically chosen values for () are 0.05, 0.01, or 0.10, which correspond to a 5%, 1%, or 10% risk of a type I error.

In our exercise, a significance level of 0.05 means we are willing to accept a 5% chance of incorrectly rejecting the null hypothesis. This level of significance is commonly used in many research studies, striking a balance between being too lenient and too strict. The lower we set the significance level, the more stringent and less likely we are to reject the null hypothesis. Conversely, a higher significance level increases the chance of rejecting the null hypothesis, potentially leading to more false discoveries.

Z-statistic

The Z-statistic is a measure in statistics that helps to determine how far, in standard deviation units, a data point is from the mean. It is a crucial part of hypothesis testing. In the context of our exercise, the Z-statistic is calculated to test whether the observed proportion of smokers who think they have a higher risk of cancer is significantly different from the hypothesized proportion of 0.5. The formula for the Z-statistic takes the difference between the sample proportion and the hypothesized proportion and divides it by the standard error of the proportion.

This calculated Z-value is then compared to a critical value based on the chosen significance level—it tells us how unlikely our sample result would be if the null hypothesis were true. If the Z-statistic falls within the critical region (for our example, beyond the negative Z-value corresponding to an alpha of 0.05), it indicates that the null hypothesis is unlikely, and we may decide to reject it.

sample probability

Sample probability, often denoted as (), is an estimate of the true population parameter based on sample data. It represents the observed proportion in a sample that we use to draw inferences about the population proportion with specific characteristics. For instance, in the provided exercise, the sample probability is the ratio of the number of smokers who believe they have a higher risk of cancer (295 out of 737). This sample probability is then used to compute the Z-statistic and make conclusions about the population as a whole.

The accuracy of sample probability as an estimate of the true population proportion depends on the representativeness and size of the sample. Larger sample sizes tend to yield more reliable estimates. In hypothesis testing, the sample probability is central to determining whether the observed results are statistically significant, or whether they could have occurred by random chance alone.