Testing the Significance of the Correlation Coefficient (Optional) Quiz

PDF page 665: The correlation coefficient, r, tells us about the strength and direction of the linear relationship between x and y. However, the reliability of the linear model also depends on how many observed data points are in the sample. We need to look at both the correlation coefficient r and the sample size n, together. We perform a hypothesis test of the significance of the correlation coefficient to decide whether the linear relationship in the sample data is strong enough to use to model the relationship in the population. The sample data are used to compute r, the correlation coefficient for the sample. If we had data for the entire population, we could find the population correlation coefficient. But, because we have only sample data, we cannot calculate the population correlation coefficient. The sample correlation coefficient, r, is our estimate of the unknown population correlation coefficient. The symbol for the population correlation coefficient is ρ, the Greek letter rho. ρ = population correlation coefficient (unknown). r = sample correlation coefficient (known; calculated from sample data). The hypothesis test lets us decide whether the value of the population correlation coefficient ρ is close to zero or significantly different from zero. We decide this based on the sample correlation coefficient r and the sample size n. If the test concludes the correlation coefficient is significantly different from zero, we say the correlation coefficient is significant.

PDF page 666: • Null hypothesis: H0: ρ = 0.

PDF page 666: • Alternate hypothesis: Ha: ρ ≠ 0.

PDF page 666: In this chapter, we will always use a significance level of 5 percent, α = 0.05.

PDF page 666: significant linear relationship (correlation) between x and y in the population. Drawing a Conclusion: There are two methods to make a conclusion. The two methods are equivalent and give the same result.

PDF page 666: • Method 1: Use the p-value.

PDF page 666: • Method 2: Use a table of critical values.

PDF page 666: If the p-value is less than the significance level (α = 0.05):

PDF page 666: • Decision: Reject the null hypothesis.

PDF page 666: because the correlation coefficient is not significantly different from zero. You will use technology to calculate the p-value, but it is useful to know that the p-value is calculated using a t distribution with n – 2 degrees of freedom and that the p-value is the combined area in both tails. An alternative way to calculate the p-value (p) given by LinRegTTest is the command 2*tcdf(abs(t),10^99, n–2) in 2nd DISTR. Third Exam vs. Final Exam Example: p-value Method

PDF page 666: because the correlation coefficient is not significantly different from zero. You will use technology to calculate the p-value, but it is useful to know that the p-value is calculated using a t distribution with n – 2 degrees of freedom and that the p-value is the combined area in both tails. An alternative way to calculate the p-value (p) given by LinRegTTest is the command 2*tcdf(abs(t),10^99, n–2) in 2nd DISTR. Third Exam vs. Final Exam Example: p-value Method

PDF page 667: • The p-value is 0.026 (from LinRegTTest on a calculator or from computer software).

PDF page 667: score (x) and the final exam score (y) because the correlation coefficient is significantly different from zero. Because r is significant and the scatter plot shows a linear trend, the regression line can be used to predict final exam scores. METHOD 2: Using a Table of Critical Values to Make a Decision The 95 Percent Critical Values of the Sample Correlation Coefficient Table (Table 12.9) can be used to give you a good idea of whether the computed value of r is significant. Use it to find the critical values using the degrees of freedom, df = n – 2. The table has already been calculated with α = 0.05. The table tells you the positive critical value, but you should also make that number negative to have two critical values. If r is not between the positive and negative critical values, then the correlation coefficient is significant. If r is significant, then you may use the line for prediction. If r is not significant (between the critical values), you should not use the line to make predictions.

PDF page 667: score (x) and the final exam score (y) because the correlation coefficient is significantly different from zero. Because r is significant and the scatter plot shows a linear trend, the regression line can be used to predict final exam scores. METHOD 2: Using a Table of Critical Values to Make a Decision The 95 Percent Critical Values of the Sample Correlation Coefficient Table (Table 12.9) can be used to give you a good idea of whether the computed value of r is significant. Use it to find the critical values using the degrees of freedom, df = n – 2. The table has already been calculated with α = 0.05. The table tells you the positive critical value, but you should also make that number negative to have two critical values. If r is not between the positive and negative critical values, then the correlation coefficient is significant. If r is significant, then you may use the line for prediction. If r is not significant (between the critical values), you should not use the line to make predictions.

PDF page 667: Using Table 12.9 with df = 8, we find that the critical value is 0.632. This means the critical values are really ±0.632. Since r = 0.801 and 0.801 > 0.632, r is significant and the line may be used for prediction. If you view this example on a number line, it will help you to see that r is not between the two critical values.