91Ó°ÊÓ

A study suggests that exposure to UV rays through the car window may increase the risk of skin cancer. \(^{52}\) The study reviewed the records of all 1,050 skin cancer patients referred to the St. Louis University Cancer Center in 2004\. Of the 42 patients with melanoma, the cancer occurred on the left side of the body in 31 patients and on the right side in the other 11 . (a) Is this an experiment or an observational study? (b) Of the patients with melanoma, what proportion had the cancer on the left side? (c) A bootstrap \(95 \%\) confidence interval for the proportion of melanomas occurring on the left is 0.579 to \(0.861 .\) Clearly interpret the confidence interval in the context of the problem. (d) Suppose the question of interest is whether melanomas are more likely to occur on the left side than on the right. State the null and alternative hypotheses. (e) Is this a one-tailed or two-tailed test? (f) Use the confidence interval given in part (c) to predict the results of the hypothesis test in part (d). Explain your reasoning. (g) A randomization distribution gives the p-value as 0.003 for testing the hypotheses given in part (d). What is the conclusion of the test in the context of this study? (h) The authors hypothesize that skin cancers are more prevalent on the left because of the sunlight coming in through car windows. (Windows protect against UVB rays but not UVA rays.) Do the data in this study support a conclusion that more melanomas occur on the left side because of increased exposure to sunlight on that side for drivers?

Step by step solution

01

Determine the Study Type

The study being discussed is an observational study as it records data without influencing the conditions or manipulating any variables.

02

Calculate the Proportion of Patients

The proportion of patients with melanoma who had the cancer on the left side is computed by dividing the number of patients with the cancer on the left side by the total patients with melanoma. Thus, proportion \(= \frac{31}{42} = 0.738\)

03

Interpret Confidence Interval

The \(95\%\) confidence interval for the proportion of melanomas occurring on the left side is \(0.579\) to \(0.861\). This means we are \(95\%\) sure that if the study were repeated, the true proportion of left-sided melanomas among patients would be between \(0.579\) and \(0.861\).

04

Null and Alternative Hypotheses

The null hypothesis (\(H_0\)) is that melanomas are equally likely to occur on the left side as on the right. The alternative hypothesis (\(H_1\)) is that melanomas are more likely to occur on the left side.

05

Determine the Type of Test

This is a one-tailed test because we are specifically looking for whether one side (left) is more affected than the other (right). It is not a two-tailed test since we are not checking both sides for being more affected.

06

Predict the Results of the Hypothesis Test

Given that the \(95\%\) confidence interval for the proportion of melanomas occurring on the left side is \(0.579\) to \(0.861\) and does not include \(0.5\) (which is the value assumed under the null hypothesis), we can predict that the result of the hypothesis test will reject the null hypothesis.

07

Interpret the p-value

The p-value of \(0.003\) means there is only a \(0.003\) probability of obtaining a result as or more extreme compared to the observed data, if the null hypothesis was true. Generally, a p-value less than \(0.05\) is considered statistically significant. Therefore, we reject the null hypothesis and conclude that melanomas are likely more prevalent on the left side.

08

Reason the Outcome Based on Given Hypothesis

The authors hypothesize that skin cancers are more prevalent on the left side due to sunlight exposure from car windows, and the statistical results seem to support this. However, correlation is not causation and other factors may be at play. More detailed study is needed to establish causation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Observational Study vs Experiment

Understanding the difference between observational studies and experiments is vital in statistics. An observational study, like the one conducted on UV exposure and skin cancer risk, involves monitoring and recording data without interfering in the process. Researchers observe subjects in their natural setting and do not manipulate the study variables. In contrast, an experiment involves intentionally changing one or more variables to observe the effect of these manipulations on other variables.

In the skin cancer research, the scientists reviewed medical records, which is a hallmark of an observational study, as they did not influence the exposure or the development of melanoma. This type of study is essential for identifying associations but cannot establish a cause-and-effect relationship due to potential confounding variables.

Confidence Interval Interpretation

A confidence interval provides a range of values, which likely includes the true parameter being estimated within a certain level of confidence. In the context of the skin cancer study, a 95% confidence interval for the proportion of left-side melanomas ranging from 0.579 to 0.861 translates to a very high level of certainty—95 out of 100 times, to be exact—that the true proportion will fall within this range if we were to repeat the study multiple times.

This interval indicates that the proportion of melanomas on the left side is significantly above what would be expected if there were no preference (which would be 0.5 for an equal chance), suggesting a potential association with left-side UV exposure from car windows.

Null and Alternative Hypothesis

In research, it's crucial to articulate the null and alternative hypotheses. The null hypothesis, denoted as H₀, represents a statement of no effect or no difference, serving as the benchmark for analysis. Here, the null hypothesis asserts that the likelihood of melanoma occurrence is the same on both the left and right sides of the body.

Conversely, the alternative hypothesis, denoted as H₁ or H_a, represents what the researcher aims to demonstrate, in this case, that melanomas are more frequently found on the left side. A hypothesis test determines whether the observed data provide sufficient evidence to reject the null hypothesis in favor of the alternative.

One-tailed and Two-tailed Tests

Choosing between a one-tailed and two-tailed test hinges on the research question. A one-tailed test, also known as a directional test, looks for an effect in only one direction. The skin cancer study used a one-tailed test since the hypothesis specifically targeted an increase in left-side melanoma occurrences versus the right side.

In contrast, a two-tailed test is non-directional and examines whether there is a significant difference in either direction, without specifying which direction beforehand. In scenarios where researchers are looking for any difference or when deviations in both directions have implications, a two-tailed test is appropriate.

P-value Significance

The p-value is a fundamental concept in statistics, indicating the probability of observing data at least as extreme as the sample data, under the assumption that the null hypothesis is true. If the p-value is low, it suggests that the observed data are unlikely under the null hypothesis.

In the UV exposure and skin cancer study, a p-value of 0.003 strongly implies that the observed higher proportion of left-side melanomas—which the confidence interval also supported—is unlikely due to random chance. Therefore, with a significance level typically set at 0.05, the p-value here is low enough to reject the null hypothesis and conclude that there's statistical evidence supporting more frequent left-side melanoma occurrences.

Correlation and Causation

Discerning between correlation and causation is a common issue in interpreting study outcomes. Correlation implies that two variables are related but doesn't mean that one variable causes the changes in another. For instance, in the observed study, while there's a correlation between left-side melanomas and UV exposure through car windows, it does not prove causation.

To make a causal assertion, factors that could confound the results must be controlled for, typically through a well-designed experiment. Although the statistical tests support an association, additional research with controlled experimental designs would be needed to conclusively determine that increased exposure to UV rays while driving is a causal factor for skin cancer.