a. The correlation between speed and strength for 20 women is .567. Test these results at the .01 level using a one-tailed test.
To test the correlation between speed and strength for 20 women at the 0.01 level using a one-tailed test, we can follow these steps:
Step 1: State the null and alternative hypotheses:
Null Hypothesis (H0): There is no correlation between speed and strength for 20 women.
Alternative Hypothesis (Ha): There is a positive correlation between speed and strength for 20 women.
Step 2: Determine the significance level (α): In this case, the significance level is 0.01.
Step 3: Calculate the critical value: Since this is a one-tailed test, we need to look up the critical value in the t-distribution table for a one-tailed test with a significance level of 0.01 and 18 degrees of freedom (n-2). From the table, the critical value is approximately 2.552.
Step 4: Calculate the test statistic: The test statistic for a correlation coefficient is calculated using the formula:
t = r * √((n-2) / (1-r^2))
In this case, the correlation coefficient (r) is 0.567 and the sample size (n) is 20.
t = 0.567 * √((20-2) / (1-0.567^2))
t = 0.567 * √(18 / 0.677)
t = 0.567 * √26.590
t ≈ 0.567 * 5.157
t ≈ 2.929
Step 5: Compare the test statistic to the critical value: Since the test statistic (t = 2.929) is greater than the critical value (2.552), we reject the null hypothesis.
Step 6: Evaluate the p-value: To calculate the p-value, we can use the t-distribution table or a statistical software. The p-value represents the probability of obtaining a test statistic as extreme as the observed result (or more extreme) under the null hypothesis. In this case, the p-value is less than 0.01.
Step 7: Make a conclusion: Based on the results, we reject the null hypothesis. There is evidence to support a positive correlation between speed and strength for 20 women.
To test the correlation between speed and strength for 20 women at the 0.01 significance level using a one-tailed test, you need to perform a hypothesis test.
Here are the steps to conduct this hypothesis test:
Step 1: State the null hypothesis (H0) and alternative hypothesis (Ha):
- Null hypothesis (H0): There is no significant correlation between speed and strength for 20 women. (ρ = 0)
- Alternative hypothesis (Ha): There is a positive correlation between speed and strength for 20 women. (ρ > 0)
Step 2: Determine the significance level:
- The significance level (α) is given as 0.01, which means we want to be 99% confident in our results.
Step 3: Calculate the critical value:
- Since we are conducting a one-tailed test, we need to find the critical value from the t-table for a one-tailed test at the 0.01 level of significance, with n - 2 degrees of freedom (where n is the sample size).
- In this case, the sample size is 20, so the degrees of freedom are 20 - 2 = 18.
- Look for the critical value that corresponds to a 0.01 level of significance with 18 degrees of freedom in the one-tailed t-table.
- Let's assume the critical value is t-critical.
Step 4: Calculate the test statistic:
- The test statistic for testing the correlation coefficient is called r.
- In this case, the correlation coefficient is given as 0.567.
Step 5: Decision rule:
- If the calculated test statistic (r) is greater than the critical value (t-critical), we reject the null hypothesis.
- Otherwise, if the calculated test statistic (r) is less than or equal to the critical value (t-critical), we fail to reject the null hypothesis.
Step 6: Make a decision:
- Compare the calculated test statistic (r) with the critical value (t-critical) from the table.
- If the calculated test statistic (r) is greater than the critical value (t-critical), reject the null hypothesis and conclude that there is a significant positive correlation between speed and strength for 20 women.
- If the calculated test statistic (r) is less than or equal to the critical value (t-critical), fail to reject the null hypothesis and conclude that there is no significant correlation between speed and strength for 20 women.
Keep in mind that to get the actual critical value and perform the calculations, you'll need to refer to the specific t-table or use statistical software.