Question

Competitive Running Is the number of years of competitive running experience related to a runner's distance running performance? The data on nine runners, obtained from a study by Scott Powers and colleagues, are shown in the table: $$\begin{array}{crc}\hline & \text { Years of Competitive } & \text { 10-Kilometer Finish } \\\\\text { Runner } & \text { Running } & \text { Time (minutes) } \\\\\hline 1 & 9 & 33.15 \\\2 & 13 & 33.33 \\\3 & 5 & 33.50 \\\4 & 7 & 33.55 \\\5 & 12 & 33.73 \\\6 & 6 & 33.86 \\\7 & 4 &33.90 \\\8 & 5 & 34.15 \\\9 & 3 & 34.90 \\\\\hline\end{array}$$ a. Calculate the rank correlation coefficient between years of competitive running and a runner's finish time in the 10 -kilometer race. b. Do the data provide evidence to indicate a significant rank correlation between the two variables? Test using \(\alpha=.05\)

Short answer

Answer: Yes, there is a significant relationship between years of competitive running and finish times in a 10-kilometer race, as the rank correlation coefficient was found to be approximately -0.72 and the hypothesis test led to the rejection of the null hypothesis in favor of the alternative hypothesis, indicating a significant rank correlation between the two variables.

Step-by-step solution

Calculate the ranks for years of competitive running and finish times

First, we need to replace the numbers of years of competitive running and finish times with their respective ranks. Rank 1 would be given to the smallest number, and so on. Years of competitive running: [9, 13, 5, 7, 12, 6, 4, 5, 3] Ranks for years of competitive running: [5, 9, 3, 4, 8, 6, 2, 3, 1] Finish times: [33.15, 33.33, 33.5, 33.55, 33.73, 33.86, 33.9, 34.15, 34.9] Ranks for finish times: [1, 2, 3, 4, 5, 6, 7, 8, 9] Remember to average the ranks, where there is more than one data point with the same value.

Calculate the rank correlation coefficient

Now we need to calculate the rank correlation coefficient (\(r_s\)) using the formula: $$ r_s = 1 - \frac{6 \sum d_i^2}{n(n^2-1)} $$ Where, \(d_i\) is the difference between ranks of each pair and \(n\) is the number of pairs. Calculate the differences between ranks \((d_i)\): [5-1, 9-2, 3-3, 4-4, 8-5, 6-6, 2-7, 3-8, 1-9] [4, 7, 0, 0, 3, 0, -5, -5, -8] Calculate the squared differences \((d_i^2)\): [16, 49, 0, 0, 9, 0, 25, 25, 64] Now, we can plug these values into the formula: $$ r_s = 1 - \frac{6(16 + 49 + 0 + 0 + 9 + 0 + 25 + 25 + 64)}{9(9^2-1)} \\ r_s = 1 - \frac{1116}{648} \\ r_s \approx -0.72 $$ The rank correlation coefficient is approximately -0.72.

Hypothesis testing

We will use a hypothesis test to determine if there is a significant rank correlation between the two variables. Our null hypothesis (\(H_0\)) will be that there is no significant rank correlation between the two variables, and our alternative hypothesis (\(H_a\)) will be that there is a significant rank correlation. \(H_0: r_s = 0\) \(H_a: r_s \neq 0\) We will use a significance level (\(\alpha\)) of 0.05. Since we have a small sample size (n=9), we will use a t-distribution: $$ t = \frac{r_s \sqrt{n - 2}}{\sqrt{1 - r_s^2}} \\ t = \frac{-0.72 \sqrt{9 - 2}}{\sqrt{1 - (-0.72)^2}} \\ t \approx -3.154 \\ $$ Next, find the critical values for a two-tailed test with 7 degrees of freedom (n-2) and a significance level of 0.05. Using a t-table, we find that the critical values are approximately -2.365 and 2.365.

Compare and make a decision

Since our calculated t-value (-3.154) is less than the critical value (-2.365), we reject the null hypothesis in favor of the alternative hypothesis. This means that there is evidence to indicate a significant rank correlation between years of competitive running and a runner's finish time in the 10-kilometer race.

Key concepts

Hypothesis Testing

When examining the relationship between two variables, statisticians use hypothesis testing as a formal procedure to evaluate statistical evidence. With this method, we begin by assuming that there is no association between the variables—this is known as the null hypothesis ().

The alternative hypothesis () represents what we want to establish. In the context of competitive running performance, if we are trying to determine whether more years of running results in faster race times, our null hypothesis states there is no correlation between the years of running and running times, and the alternative hypothesis posits that a correlation does exist.

To test the hypothesis, we compute a test statistic that, in this case, measures how far our sample correlation is from the hypothesized value under the null hypothesis, usually zero. We then compare this value to a critical value from an appropriate distribution, such as the t-distribution for small samples, to decide whether to reject the null hypothesis. If our test statistic falls into the extreme tails of the distribution, it suggests that the observed correlation is unlikely to occur by random chance alone, leading us to reject the null hypothesis. In our exercise, a significant rank correlation coefficient supported the alternative hypothesis, indicating a relationship between years of competitive running and 10-Kilometer race performance.

Competitive Running Performance

Competitive running performance can be influenced by numerous factors including training intensity, diet, genetics, and also the experience of the runner. The number of years someone has been running competitively can indicate a higher level of skill or endurance, potentially leading to better race times.

When studying these effects, researchers gather data from different participants and look for patterns. In our exercise, the goal was to investigate whether the years of competitive running experience correlated with runners' finish times in a 10-Kilometer race. Spearman's rank correlation coefficient was used because it measures the strength and direction of association between two ranked variables. With a negative correlation found, it suggests that, generally, more years of experience may lead to lower (faster) finish times, potentially attributable to better technique, efficiency, and pacing strategies honed over years of running.

Spearman's Rank Correlation

Spearman's rank correlation, denoted by ), is a non-parametric measure of statistical dependence between two variables. It assesses how well the relationship between two variables can be described by a monotonic function. Simply put, if one variable increases, does the other one also increase or decrease consistently?

To calculate it, each element in the data set is ranked. If there are duplicated data points, they receive a rank equal to the average of their positions. Differences in these ranks are squared and summed to calculate Spearman's rank correlation coefficient, following the formula:).

In our example, the calculated coefficient of approximately -0.72 pointed to a strong negative correlation between years of running and finish times—suggesting that as the years of running increase, the race times decrease. Given that this method does not require the data to be normally distributed, it's particularly useful when dealing with ordinal data or non-linear relationships.