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 nonbikers 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, if the p-value is less than the significance level, it provides convincing evidence that the proportion of low sperm count is higher in Austrian avid mountain bikers than in Austrian non-bikers. However, a significant test doesn't establish a causality, it just shows an association.

Step-by-step solution

Defining the Null and Alternative Hypotheses

Define the null hypothesis (\(H_0\)) as there is no difference in the proportion of low sperm counts between avid mountain bikers (\(p_1\)) and nonbikers (\(p_2\)): \(H_0: p_1 - p_2 = 0\). The alternative hypothesis (\(H_A\)) is that the proportion of low sperm counts is higher for bikers: \(H_A: p_1 - p_2 > 0\).

Calculating the Test Statistic

First, calculate the pooled proportion (\(p\)) as the total number of successes (low sperm counts) divided by the total number of trials (total number of bikers and nonbikers). Then, calculate the standard error (\(SE\)) using the formula \(SE = \sqrt{p(1 - p)(1/n_1 + 1/n_2)}\). Finally, calculate the test statistic (\(Z\)) as the observed difference between the sample proportions minus the difference under the null hypothesis, divided by the standard error, that is, \(Z = (p_1 - p_2 - 0) / SE\).

Finding the P-Value

The p-value is the probability of observing a test statistic as extreme as the one calculated under the null hypothesis. In this case, as the alternative hypothesis is greater than type, use a Z table/ standard normal distribution table or software to find the probability greater than the calculated Z score which is the p-value.

Conclusion

If the p-value is less than the chosen significance level (generally 0.05), then reject the null hypothesis and conclude that there is convincing evidence that the proportion of Austrian avid mountain bikers with low sperm count is significantly higher than the proportion of Austrian non bikers.

Interpretation

Even if the test in part (a) is significant, it doesn't establish a causal relationship. It simply reveals an association between mountain biking and low sperm count, which might be due to confounding factors.