Question

A researcher was interested in a hockey player's ability to make a fast start from a stopped position. \({ }^{16}\) In the experiment, each skater started from a stopped position and skated as fast as possible over a 6-meter distance. The correlation coefficient \(r\) between a skater's stride rate (number of strides per second) and the length of time to cover the 6 -meter distance for the sample of 69 skaters was -.37 . a. Do the data provide sufficient evidence to indicate a correlation between stride rate and time to cover the distance? Test using \(\alpha=.05 .\) b. Find the approximate \(p\) -value for the test. c. What are the practical implications of the test in part a?

Short answer

Answer: Yes, there is sufficient evidence to indicate a correlation between a skater's stride rate and the time to cover a 6-meter distance, as the p-value (0.0011) is less than the significance level (0.05).

Step-by-step solution

State the null and alternative hypotheses

The null hypothesis \(H_0\) is that there is no correlation between stride rate and time to cover the distance, meaning that the population correlation coefficient is equal to 0: $$H_0: \rho = 0$$ The alternative hypothesis \(H_a\) is that there is a correlation between stride rate and time to cover the distance, meaning that the population correlation coefficient is not equal to 0: $$H_a: \rho \neq 0$$

Calculate the test statistic

We are given the sample correlation coefficient, \(r = -0.37\), the sample size, \(n = 69\), and the significance level, \(\alpha = 0.05\). Before we compute the test statistic, let's compute the degrees of freedom: $$df = n - 2 = 69 - 2 = 67$$ We will use Pearson's correlation coefficient formula to compute the test statistic, which is as follows: $$t = r\sqrt{\frac{n-2}{1-r^{2}}} = -0.37\sqrt{\frac{67}{1-(-0.37)^{2}}}$$ Now compute the test statistic: $$t \approx -3.46$$

Determine the critical value and p-value

Using the t-distribution table or a calculator with the degrees of freedom (67) and the significance level (0.05, two-tailed), we find the critical value for this test: $$t_{critical} \approx \pm1.997$$ Now, we'll find the p-value. Using a calculator or software that computes the cumulative distribution function (CDF) of the t-distribution, we find: $$p = P(T_{67} < (-3.46)) * 2 \approx 0.0011$$

Make a decision based on the p-value

Since the p-value (\(0.0011\)) is less than the significance level (\(0.05\)), we reject the null hypothesis \(H_0\). This means that there is sufficient evidence to indicate a correlation between stride rate and time to cover the 6-meter distance.

Interpret the results

The practical implications of these results are that a hockey player's stride rate is negatively correlated with the time it takes to cover the 6-meter distance. A higher stride rate can lead to a faster time covering the distance. This information can be useful for coaches and trainers in designing training programs to improve the players' speed and performance on the ice.

Key concepts

Hypothesis Testing

Hypothesis testing is a systematic method used in statistics to determine whether there is enough evidence in a sample to infer a particular condition or relationship for the entire population. When a researcher wants to check if an observed effect, such as a correlation between two variables, is likely due to chance or represents a true effect, they conduct a hypothesis testing.

In the context of the exercise, the researcher seeks to determine if there's a meaningful correlation between stride rate and time taken to cover a distance. To do this, they start by setting up two hypotheses: the null hypothesis (\(H_0\)) which asserts that there is no correlation, and the alternative hypothesis (\(H_a\)) which suggests there is a correlation. Rejecting the null hypothesis paves the way to conclude there may be a significant correlation. The decision to accept or reject the null hypothesis is made by comparing the p-value to a predefined significance level, typically \(\alpha\), which is often set at 0.05.

Pearson's Correlation Coefficient

Pearson's correlation coefficient, denoted as \(r\), is a measure of the linear correlation between two variables. The value of \(r\) ranges from -1 to 1, where 1 indicates a perfect positive correlation, -1 indicates a perfect negative correlation, and 0 indicates no correlation at all.

For our exercise, the Pearson's correlation coefficient \(r = -0.37\) suggests a moderate negative correlation; as stride rate increases, the time to cover the distance decreases. Calculating this coefficient involves assessing how much two variables change together relative to their independent variations. In hypothesis testing, \(r\) is used to calculate the test statistic which is then compared against critical values from the t-distribution to make decisions about the null hypothesis.

P-Value

The p-value in statistics is the probability of obtaining test results at least as extreme as the ones observed, under the assumption that the null hypothesis is true. It measures the strength of the evidence against the null hypothesis. A low p-value, typically less than 0.05, indicates that the observed data are unlikely under the null hypothesis and suggests rejecting \(H_0\).

In the exercise provided, the calculated p-value is approximately 0.0011, which is much lower than the significance level of 0.05. This small p-value indicates a strong evidence against the null hypothesis, leading to its rejection and the acceptance of the alternative hypothesis that there is a meaningful correlation between stride rate and the time to cover the distance.

Statistical Significance

Statistical significance refers to the likelihood that a relationship between two or more variables is caused by something other than random chance. It's assessed using the p-value in the context of a defined significance level. In most social science research, the threshold for declaring statistical significance is a p-value less than 0.05.

If the p-value is below the chosen threshold, the results are considered statistically significant, meaning that they are unlikely to be the product of random variation. In our exercise, because the p-value is 0.0011, which is well below the alpha level of 0.05, we can conclude that the data provide statistically significant evidence of a correlation between stride rate and time to cover the distance for the hockey players.