Testing the Significance of the Correlation Coefficient Quiz

PDF page 699: 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 value of 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 have 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 that the correlation coefficient is significantly different from zero, we say that the

PDF page 700: • Null Hypothesis: H0: ρ = 0

PDF page 700: In this chapter of this textbook, we will always use a significance level of 5%, α = 0.05 NOTE Using the p-value method, you could choose any appropriate significance level you want; you are not limited to using α = 0.05. But the table of critical values provided in this textbook assumes that we are using a significance level of 5%, α = 0.05. (If we wanted to use a different significance level than 5% with the critical value method, we would need different tables of critical values that are not provided in this textbook.)

PDF page 701: because the correlation coefficient is NOT significantly different from zero." You will use technology to calculate the p-value. The following describes the calculations to compute the test statistics and the p-value: The p-value is calculated using a t-distribution with n - 2 degrees of freedom. r n − 2 The formula for the test statistic is t = . The value of the test statistic, t, is shown in the computer or 2

PDF page 701: 1 − r calculator output along with the p-value. The test statistic t has the same sign as the correlation coefficient r. 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 701: If the p-value is less than the significance level (α = 0.05):

PDF page 701: • Decision: Reject the null hypothesis.

PDF page 700: In this chapter of this textbook, we will always use a significance level of 5%, α = 0.05 NOTE Using the p-value method, you could choose any appropriate significance level you want; you are not limited to using α = 0.05. But the table of critical values provided in this textbook assumes that we are using a significance level of 5%, α = 0.05. (If we wanted to use a different significance level than 5% with the critical value method, we would need different tables of critical values that are not provided in this textbook.)

PDF page 701: • The line of best fit is: ŷ = -173.51 + 4.83x with r = 0.6631 and there are n = 11 data points.

PDF page 701: • The line of best fit is: ŷ = -173.51 + 4.83x with r = 0.6631 and there are n = 11 data points.

PDF page 701: Suppose you computed r = 0.801 using n = 10 data points.df = n - 2 = 10 - 2 = 8. The critical values associated

PDF page 701: with df = 8 are -0.632 and + 0.632. If r < negative critical value or r > positive critical value, then r issignificant. Since r = 0.801 and 0.801 > 0.632, r is significant and the line may be usedfor prediction. If you view this example on a number line, it will help you.

PDF page 702: Suppose you computed r = 0.776 and n = 6. df = 6 – 2 = 4. The critical values are –0.811 and 0.811. Since –0.811

PDF page 702: < 0.776 < 0.811, r is not significant, and the line should not be used for prediction.

PDF page 703: d. r = 0 and the sample size, n, is five. No matter what the dfs are, r = 0 is between the two critical values so r