Question

"Mountain Biking May Reduce Fertility in Men, Study Says" was the headline of an article appearing in the San Luis Obispo Tribune (December 3,2002 ). This conclusion was based on an Austrian study that compared sperm counts of avid mountain bikers (those who ride at least 12 hours per week) and nonbikers. Ninety percent of the avid mountain bikers studied had low sperm counts, as compared to \(26 \%\) of the nonbikers. Suppose that these percentages were based on independent samples of 100 avid mountain bikers and 100 non-bikers and that it is reasonable to view these samples as representative of Austrian avid mountain bikers and nonbikers. a. Do these data provide convincing evidence that the proportion of Austrian avid mountain bikers with low sperm count is higher than the proportion of Austrian nonbikers? b. Based on the outcome of the test in Part (a), is it reasonable to conclude that mountain biking 12 hours per week or more causes low sperm count? Explain.

Short answer

Yes, based on these data, the proportion of Austrian avid mountain bikers with low sperm count appears to be higher than the proportion of nonbikers with low sperm count. Nevertheless, it's not reasonable to conclude that mountain biking 12 hours per week or more causes low sperm count, because although there is a strong correlation, the cause-effect relationship has not been confirmed.

Step-by-step solution

Formulate the hypotheses

The null hypothesis \(H_{0}\) is that the proportion of avid mountain bikers (\(p_{1}\)) with low sperm count is equal to the proportion of non-bikers (\(p_{2}\)). The alternative hypothesis \(H_{1}\) is that \(p_{1}\) (the proportion of avid mountain bikers) is greater than \(p_{2}\) (the proportion of non-bikers), in mathematical terms: \(H_{0}: p_{1} = p_{2}\) vs \(H_{1}: p_{1} > p_{2}\).

Calculate the test statistic

First, calculate the pooled sample proportion (\(p\)). This is the total number of successes (low sperm counts) divided by the total sample size (\(n\)). Here, \(p = (0.90*100 + 0.26*100) / (100 + 100) = 0.58\). Second, calculate the standard error of the sampling distribution which is \(\sqrt{ p * ( 1 - p ) * [ (1/n_{1}) + (1/n_{2}) ] } = \sqrt{ 0.58 * ( 1 - 0.58 ) * [ (1/100) + (1/100) ] } = 0.070\). The test statistic (Z) is then \(Z = (p_{1} - p_{2}) / SE = (0.90 - 0.26) / 0.070 = 9.14\).

Find the p-value

The test is one-sided, so we only consider the right tail. The p-value is the area to the right of the test statistic on the standard normal distribution. With a Z-score of 9.14, this area is essentially zero, meaning that the p-value is practically 0 and much smaller than any usual level of significance \(\alpha\) (0.05, 0.01, etc). Therefore, we reject the null hypothesis.

Answer the second question from the problem

The data provide very strong evidence that the percentage of avid mountain bikers with low sperm count is greater than non-bikers. However, this doesn't necessarily mean that biking more causes low sperm count, only that there is a strong association between the two. This is because there could be other factors affecting sperm count not accounted for in this study. The cause can only be determined by conducting experiments.

Conclusion

The data provides evidence that the proportion of low sperm count is higher among avid mountain bikers compared to non-bikers. However, it is not enough to conclude that the increase in biking hours causes low sperm count.

Key concepts

Statistical Significance

Understanding statistical significance is essential when interpreting the results of studies like the one comparing the sperm counts of avid mountain bikers and non-bikers. To deem a finding 'statistically significant', it means there is a very low probability that the observed effect or difference happened by chance, given that the null hypothesis is true. It is typically assessed by using a threshold value, or 'alpha level', such as 0.05. If the p-value obtained in a hypothesis test is less than the alpha level, the result is considered statistically significant.

In our exercise, with a p-value near zero and well below the common alpha level of 0.05, the result is statistically significant. This suggests that the differences observed in the sperm counts between the two groups are unlikely to have occurred due to random variation alone.

Null and Alternative Hypotheses

In hypothesis testing, we start by setting up two opposing statements: the null hypothesis (\(H_0\)) and the alternative hypothesis (\(H_a\) or \(H_1\)). The null hypothesis represents a position of skepticism, indicating that there is no effect or no difference, while the alternative suggests the opposite.

In the exercise, \(H_0: p_{1} = p_{2}\) suggests that the proportion of low sperm counts among avid bikers and non-bikers is the same, whereas \(H_1: p_{1} > p_{2}\) suggests a higher proportion in avid bikers. The goal of hypothesis testing is to determine if enough evidence exists to reject the null hypothesis in favor of the alternative. In conclusion, the data led to the rejection of the null hypothesis, indicating that the proportion of low sperm count is statistically significantly higher in avid mountain bikers.

P-value

The p-value plays a central role in hypothesis testing as it measures the strength of the evidence against the null hypothesis. Specifically, the p-value represents the probability of observing test results as extreme as, or more extreme than, the results actually observed, assuming the null hypothesis is true.

In this case study, a p-value close to 0 indicates a very low probability that the significant difference in sperm count between bikers and non-bikers is due to random chance. The smaller the p-value, the stronger the evidence to reject the null hypothesis. Thus, with a p-value much lower than any conventional alpha level, the evidence strongly favors the alternative hypothesis.

Causality vs Association

Differentiating causality from association is crucial in interpreting results from observational studies. An association indicates a relationship between two variables but does not prove one causes the other. Causality, on the other hand, implies that one event is a direct result of the other.

Though our exercise revealed a strong statistical association between mountain biking and low sperm counts, this does not establish causality. There could be lurking variables or confounding factors affecting the outcome. To determine causality, controlled experiments, which can isolate variables and establish direct cause-and-effect relationships, are necessary. Until such an experiment is conducted, we can only conclude that an association exists between the variables, not that increased biking hours cause reduced sperm counts.