Researchers believe that an increase in lean body mass is associated with an increase in maximal oxygen uptake. A scatterplot of measurements from 18 randomly selected college athletes showed a strong positive linear relationship between the two variables. A significance test for the null hypothesis that the slope of the regression line is zero versus the alternative that the slope is greater than zero yielded a p-value of 0.04. Which statement is an appropriate conclusion for the test? (A) The p-value of 0.04 does not indicate that 4% of the variation in maximal oxygen uptake for college athletes can be explained by the amount of lean body mass. (B) The p-value of 0.04 does not indicate that 16% of the variation in maximal oxygen uptake for college athletes can be explained by the amount of lean body mass. (C) The strong positive linear relationship displayed in the scatterplot, along with a p-value of less than 0.05, indicates that college athletes with higher lean body mass tend to have higher maximal oxygen uptake. (D) The strong positive linear relationship displayed in the scatterplot, along with a p-value of less than 0.05, does not prove that an increase in lean body mass causes an increase in maximal oxygen uptake for college athletes. (E) A p-value of less than 0.05 does not indicate that the relationship displayed in the scatterplot is likely due to chance, and there is statistical evidence of a relationship between lean body mass and maximal oxygen uptake for college athletes. The correct statement, given the information provided and if choosing an option based on the correct understanding of p-value and statistical evidence, would be (C). This statement accurately conveys that there is statistical evidence to suggest a positive relationship between lean body mass and maximal oxygen uptake, without implying causation or explaining the variation in terms of the percentage (which is a common misconstruction of the p-value).

The correct conclusion from the given options would be (C): The strong positive linear relationship displayed in the scatterplot, along with a p-value of less than 0.05, indicates that college athletes with higher lean body mass tend to have higher maximal oxygen uptake. This statement addresses the statistical significance of the observed relationship based on the p-value and the positive trend seen in the scatterplot, but it does not assert causation or interpret the p-value as the proportion of variance explained.

Extra: The concept of a p-value is fundamental in statistics. It measures the strength of the evidence against the null hypothesis. In this context, the null hypothesis is that there is no relationship between lean body mass and maximal oxygen uptake (i.e., the slope of the regression line is zero). If the p-value is less than a predetermined significance level (often 0.05), then we can reject the null hypothesis in favor of the alternative hypothesis, which in this case, is that the slope is greater than zero.

The p-value itself does not indicate the proportion of variation explained by the independent variable (lean body mass, in this case). This is instead measured by the coefficient of determination, denoted as R², which is not provided in the question.

Also, it's important to understand that a significant p-value indicates there's statistical evidence of an association but does not imply causation. There might be other factors at play that influence both lean body mass and maximal oxygen uptake, or there could be a third variable affecting both. Establishing causation often requires controlled experimental studies or additional analytical techniques that are able to account for potential confounding variables.

Lastly, a p-value simply tells us that if the null hypothesis were true, there is a low probability of observing data as extreme as or more than what was actually observed. A low p-value indicates that either a rare event has occurred, or the null hypothesis does not hold—researchers typically conclude the latter. However, this does not prove the null hypothesis false; it just provides evidence against it.