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When researchers want to make inferences about the population proportion based on a sample data, they use a proportion hypothesis test. In the context of the exercise involving smokers' perception of cancer risk, we're interested in whether the true proportion of smokers who believe they are at higher risk of cancer is different from what is expected under the null hypothesis.

To conduct this test effectively, we consider the entire process of hypothesis testing which includes defining null and alternative hypotheses, calculating a test statistic based on the sample data, and interpreting the p-value in relation to the significance level, \( \alpha \). It's important to make sure that the sample is large enough to use the test, generally following the rule that both the expected number of successes (\(np_0\)) and failures (\(n(1-p_0)\)) are greater than 5.

The null hypothesis (\(H_0\)) is a statement of no effect or no difference, which serves as a starting assumption for the statistical test. In our exercise, the null hypothesis is that the true proportion of smokers who view themselves at increased risk of cancer, \(p\), equals 0.5. The alternative hypothesis (\(H_1\)) is what you suspect might be true instead and is contrary to the null hypothesis. Here, we're testing the claim that actually fewer than half of smokers (\(p<0.5\)) perceive themselves at higher risk.

Defining these hypotheses correctly is foundational to the testing process, setting the stage for either rejection or failure to reject the null based on the evidence provided by our sample.

Calculating the test statistic is a core step in hypothesis testing. It's a standardized value that helps to determine how far our sample statistic (in this case, the sample proportion) deviates from the null hypothesis. Using the formula for the test statistic for a proportion \[Z = \frac{\hat{p} - p_0}{\sqrt{{p_0(1-p_0)/n}}}\], where \(\hat{p}\) is the sample proportion and \(p_0\) is the hypothesized population proportion, we compare the calculated test statistic against a critical value from the standard normal distribution.

This calculation provides a bridge between our sample data and the realm of probability, which is key to making an inference about the entire population from which the sample was drawn.

The p-value is the probability of obtaining test results at least as extreme as the results actually observed, under the assumption that the null hypothesis is correct. Interpretation of the p-value is crucial: a small p-value (typically ≤ 0.05) indicates strong evidence against the null hypothesis. Therefore, we would reject the null and accept the alternative hypothesis as more likely. Contrastingly, a large p-value suggests weak evidence against the null hypothesis, so we fail to reject it.

In the exercise, if the calculated p-value is less than or equal to \(\alpha=0.05\), it suggests that it's unlikely the observed sample came from a population where the true proportion is 0.5. Thus, supporting the research claim and leading us to reject the null hypothesis in favor of the alternative.