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"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.

Step by step solution

01

Null and alternative hypothesis

State the null hypothesis \(H_0\) and the alternative hypothesis \(H_a\) for the problem. In this case, \(H_0: p_1 = p_2\), the proportion of avid mountain bikers with a low sperm count is equal to the proportion of nonbikers. The alternative hypothesis \(H_a: p_1 > p_2\), the proportion of avid bikers with low sperm count is greater than nonbikers. Here, \(p_1\) and \(p_2\) represent the proportion of low sperm count for bikers and nonbikers, respectively.

02

Test statistic

Calculate the test statistic for the difference in proportions. The test statistic, z, is given by \[ z = \frac{{(p_1 - p_2) - 0}}{{\sqrt{\bar{p}(1 - \bar{p}) \left(\frac{1}{n_1} + \frac{1}{n_2}\right)}}}\] where \(n_1\) and \(n_2\) are the sample sizes for bikers and nonbikers respectively, and \(\bar{p}\) is the combined sample proportion \(\bar{p} = \frac{{x_1 + x_2}}{{n_1 + n_2}}\) where \(x_1\) and \(x_2\) are the numbers of successes (low sperm count) in each sample.

03

P-value

Use the calculated test statistic to find the P-value. The P-value is the probability of observing a test statistic as extreme or more extreme given that the null hypothesis is true. Because there's a greater than alternative hypothesis in this test, the P-value is the probability that a standard normal random variable is greater than the calculated test statistic.

04

Conclusion

Based on the P-value, decide whether to reject or fail to reject the null hypothesis. If the P-value is smaller than the significance level, usually 0.05, then reject the null hypothesis. Otherwise, fail to reject the null hypothesis.

05

Interpretation

The test results can indicate a significant difference in sperm counts between the two groups. However, statistical significance does not mean causation. Although there's a significant association between being an avid biker and low sperm count, it does not necessarily mean that biking 12 hours per week causes low sperm count. There can be other lurking variables that may be causing this association.

06

Causation

For causation to be established, the study would need to be experimental, not observational, with properly assigned treatment and control groups. Therefore, based on this study's outcome, it would be misleading to conclude that biking 12 hours or more per week causes low sperm counts. Further, controlled, and preferably randomized, experiments are needed to possibly establish causation.

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Key Concepts

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

Difference in Proportions

In studies involving two groups, such as avid mountain bikers and nonbikers, it is common to compare proportions to see if one group differs from another in some significant way. In this study, the focus is on the proportion of individuals with low sperm count.We calculate the difference in proportions to determine if a statistically significant disparity exists. Here, the proportions are represented as \(p_1\) for the bikers and \(p_2\) for the nonbikers. By comparing these proportions, researchers seek to find out if more mountain bikers have a low sperm count compared to nonbikers.The calculation of the difference is followed by a statistical test to see if the observed difference is due to random chance or is truly significant. If a significant difference is found, researchers can then explore potential associations or influencing factors more closely.

Null Hypothesis

The null hypothesis is a crucial component in statistical hypothesis testing, serving as the default or starting assumption in any test. In this scenario, the null hypothesis \(H_0\) states that there is no difference in the proportion of low sperm counts between avid mountain bikers and nonbikers.Formally, it is expressed as \( p_1 = p_2 \), where both group proportions are presumed equal. The alternative hypothesis \(H_a\) posits that the proportion of bikers with low sperm count is greater than that of non-bikers, represented as \( p_1 > p_2 \).This hypothesis structure serves as the foundation to gather evidence and make decisions. It's important because any evidence against the null hypothesis must be strong enough to reject it. Otherwise, we'd assume any observed differences might be due to random sampling variations.

P-value

The P-value is a probability that helps determine the significance of the test results. It quantifies the likelihood of observing our data, or something more extreme, if the null hypothesis were true.In this context, researchers calculate the P-value from the test statistic to gauge whether to reject the null hypothesis. A low P-value (usually less than 0.05) suggests that the observed data is unlikely under the null hypothesis. Therefore, a P-value under the threshold leads to rejecting the null hypothesis.On the other hand, a high P-value indicates that the observed result could reasonably occur by random chance under the null hypothesis, thus leading to a decision not to reject \(H_0\). This critical value helps in making data-driven decisions about whether any apparent differences are statistically significant or not.

Causation vs Correlation

In interpreting statistical results, differentiating between causation and correlation is key. Correlation implies a relationship between two variables, while causation indicates that one event is the result of the occurrence of the other. In this study, even if a significant correlation is found between mountain biking and low sperm count, it does not prove causation. Observational studies, like this one, can reveal patterns or associations but can't establish that mountain biking causes low sperm counts. To prove causation, further experimental research is essential where variables can be controlled and manipulated. Only then, with monitoring over time and elimination of other variables, can causation potentially be established. Until more definitive studies are conducted, any claim of causation would be speculative, reminding us that correlation does not imply causation.