10-1 Correlation 515 Using the value of r = 0.947 for the nine pairs of data in Table 10-1 and using a significance level of 0.05, is there sufficient evidence to support a claim that there is a linear correlation between Powerball jackpot amounts and numbers of tickets sold? CP EXAMPLE 4 Is There a Linear Correlation? SOLUTION REQUIREMENT CHECK The data are a simple random sample, so the first requirement is satisfied. The second requirement of a scatterplot showing a straight-line pattern is satisfied. See the scatterplot in Figure 10-1 on page 509. The scatterplot of Figure 10-1 also shows that the third requirement of no outliers is satisfied. We can base our conclusion about correlation on either the P-value obtained from technology or the critical value found in Table A-6. (See the criteria for “Interpreting the Linear Correlation Coefficient r” given in the preceding Key Elements box.) ■ Using P-Value from Technology to Interpret r: Use the P-value and significance level a as follows: P@value … a: Supports the claim of a linear correlation. P@value 7 a: Does not support the claim of a linear correlation. The Statdisk display shows that the P-value is 0.0001. Because that P-value is less than or equal to the significance level of 0.05, we conclude that there is sufficient evidence to support the conclusion that there is a linear correlation between Powerball lottery jackpot amounts and numbers of tickets sold. Statdisk ■ Using Table A-6 to Interpret r: Consider critical values from Table A-6 as being both positive and negative, and draw a graph similar to Figure 10-3 on the next page. For the 9 pairs of data in Table 10-1, Table A-6 yields a critical value of r = 0.666; technology yields a critical value of r = 0.666. We can now compare the computed value of r = 0.947 to the critical values of r = {0.666, as shown in Figure 10-3. Correlation If the computed linear correlation coefficient r lies in the left or right tail region at or beyond the critical value for that tail, conclude that there is sufficient evidence to support the claim of a linear correlation. No Correlation If the computed linear correlation coefficient lies between the two critical values, conclude that there is not sufficient evidence to support the claim of a linear correlation. Because Figure 10-3 shows that the computed value of r = 0.947 lies beyond the upper critical value, we conclude that there is sufficient evidence to support the claim of a linear correlation between Powerball jackpot amounts and numbers of lottery tickets sold. continued Teacher Evaluations Correlate with Grades Student evaluations of faculty are often used to measure teaching effectiveness. Many studies reveal a correlation, with higher student grades being associated with higher faculty evaluations. One study at Duke University involved student evaluations collected before and after final grades were assigned. The study showed that “grade expectations or received grades caused a change in the way students perceived their teacher and the quality of instruction.” It was noted that with student evaluations, “the incentives for faculty to manipulate their grading policies in order to enhance their evaluations increase.” It was concluded that “the ultimate consequence of such manipulations is the degradation of the quality of education in the United States.” (See “Teacher Course Evaluations and Student Grades: An Academic Tango,” by Valen Johnson, Chance, Vol. 15, No. 3.) i i hhi h
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