Question

Refer to Exercise \(11 .\) The sample correlation coefficient \(r\) for the stride rate and the average acceleration rate for the 69 skaters was . \(36 .\) Do the data provide sufficient evidence to indicate a correlation between stride rate and average acceleration for the skaters? Use the \(p\) -value approach.

Short answer

Answer: Based on the calculated p-value and comparison to the significance level (α), we can make a decision about whether there is enough evidence to indicate a correlation between stride rate and average acceleration rate for the skaters. If we rejected the null hypothesis, we would conclude that there is a correlation between stride rate and average acceleration rate for the skaters. If we failed to reject the null hypothesis, we would not have enough evidence to conclude a correlation between stride rate and average acceleration rate for the skaters.

Step-by-step solution

State the hypotheses

The null hypothesis (\(H_0\)) and alternative hypothesis (\(H_a\)) for testing the correlation: \(H_0 : \rho = 0\), meaning there is no correlation between stride rate and average acceleration rate. \(H_a: \rho \neq 0\), meaning a correlation exists between stride rate and average acceleration rate.

Calculate the test statistic (t)

We will use the formula for the t-test statistic for testing correlation: \(t = \frac{r\sqrt{n-2}}{\sqrt{1-r^2}}\) where \(r\) is the sample correlation coefficient, and \(n\) is the sample size. Using the given information (\(r=0.36\) and \(n=69\)), we get: \(t = \frac{0.36\sqrt{69-2}}{\sqrt{1-(0.36)^2}}\)

Determine the p-value

To find the p-value, we will use the t-distribution with \(df = n-2 = 69-2=67\) degrees of freedom. \(p-value = 2 × P(T > |t|)\), where T is a t-distribution with \(df = 67\) and t is the calculated test statistic. We can use a statistical calculator or table to determine the probability.

Compare the p-value to the significance level (α)

Usually, the significance level is chosen as \(α=0.05\). Now, we compare the calculated p-value to \(α\): If \(p-value < α\), then we have enough evidence to reject the null hypothesis and conclude that there is a correlation. If \(p-value \geq α\), we do not have enough evidence to reject the null hypothesis, and cannot conclude a correlation.

Interpret the results

Based on the calculated p-value and comparison to the significance level, we can make a decision about whether there is enough evidence to indicate a correlation between stride rate and average acceleration rate for the skaters. If we rejected the null hypothesis, we would conclude that there is a correlation between stride rate and average acceleration rate for the skaters. If we failed to reject the null hypothesis, we would not have enough evidence to conclude a correlation between stride rate and average acceleration rate for the skaters.

Key concepts

Null and Alternative Hypotheses

The null and alternative hypotheses are the starting point of hypothesis testing. In essence, the null hypothesis (\(H_0\)) suggests that there's no effect or no relationship, while the alternative hypothesis (\(H_a\)) posits that an effect or relationship does indeed exist.

In the context of this exercise, \(H_0 : \rho = 0\) postulates that there is no correlation between stride rate and average acceleration rate among the skaters. On the contrary, the alternative hypothesis, \(H_a: \rho eq 0\), supports the existence of a correlation. Forming these hypotheses is crucial as it sets the stage for what we're expecting to find and what would be sufficient evidence to change our current understanding.

T-test Statistic Calculation

To test the hypotheses, we calculate the t-test statistic which quantifies the difference between the observed sample statistic and the null hypothesis. The formula \(t = \frac{r\sqrt{n-2}}{\sqrt{1-r^2}}\) is used when dealing with correlation coefficients.

Here, \(r\) is the sample correlation coefficient, and \(n\) is the sample size. Plugging in the values from our problem, \(r=0.36\) and \(n=69\), we derive the t-statistic, which will be used in conjunction with a t-distribution to find the p-value. This is a pivotal step in the hypothesis testing process as it converts the sample statistic into a standard form that can be used to ascertain the level of significance of our result.

P-value Approach

The p-value approach in hypothesis testing assesses how compatible our sample data is with the null hypothesis. It is the probability of observing a test statistic as extreme as, or more extreme than, the one observed if the null hypothesis were true.

In our scenario, the calculated p-value is twice the probability of observing a t-value greater than the absolute value of our test statistic \(t\) given \(df = 67\) degrees of freedom. A smaller p-value suggests stronger evidence against the null hypothesis, indicating that we may consider the alternative hypothesis to be more plausible. This number can be retrieved through statistical software or a t-distribution table.

Significance Level

The significance level, typically denoted as \(\alpha\), is a critical value in hypothesis testing that defines the threshold for rejecting the null hypothesis. Common practice often uses \(\alpha = 0.05\), which implies a 5% risk of concluding that a correlation exists when there is none (false positive).

By comparing the calculated p-value to our \(\alpha\), we determine our test's outcome. If the p-value is less than \(\alpha\), it leads us to reject the null hypothesis, suggesting significant evidence for a correlation. Conversely, if the p-value is greater or equal to \(\alpha\), we fail to reject \(H_0\), indicating insufficient evidence to support a correlation.