Determine whether a correlation coefficient of r = 0.471 is significant at the 5% level for a sample size of 20.

To determine whether a correlation coefficient of r = 0.471 is significant at the 5% level for a sample size of 20, we need to perform a hypothesis test.

The hypothesis test for the correlation coefficient involves testing the null hypothesis (H0) that the true population correlation coefficient is equal to zero, against the alternative hypothesis (Ha) that it is not equal to zero.

Here's how we can do it:

Step 1: State the hypotheses:
H0: The true population correlation coefficient (ρ) is equal to zero.
Ha: The true population correlation coefficient (ρ) is not equal to zero.

Step 2: Determine the significance level and the test statistic:
The significance level is given as 5%, which corresponds to an alpha level (α) of 0.05.
For a sample size of 20, we can use the t-distribution to calculate the test statistic.

Step 3: Calculate the critical values:
Since we are performing a two-tailed test (because Ha is "not equal to zero"), we need to find the critical t-values for a significance level of 0.05 and a sample size of 20.
Using a t-table or a statistical software, we find the critical t-value at α/2 (0.025) with degrees of freedom (df) equal to n-2 (20-2 = 18). The critical t-value is approximately 2.101.

Step 4: Calculate the test statistic:
The formula to calculate the t-test statistic for correlation coefficient is given by:

t = r * sqrt((n-2)/(1-r^2))

Substituting the given values, we have:
t = 0.471 * sqrt((20-2)/(1-0.471^2))

Calculating this value, we find t ≈ 2.227.

Step 5: Make a decision:
If the absolute value of the calculated test statistic is greater than the critical t-value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

In this case, the calculated test statistic (t ≈ 2.227) is greater than the critical t-value (2.101), so we reject the null hypothesis.

Therefore, we can conclude that there is a significant correlation at the 5% level for a sample size of 20, with a correlation coefficient of r = 0.471.