Correlation and Regression Elementary Statistics Correlation A relationship

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Correlation and Regression Elementary Statistics

Correlation and Regression Elementary Statistics

Correlation A relationship between two variables Explanatory (Independent) Variable x Hours of Training Shoe

Correlation A relationship between two variables Explanatory (Independent) Variable x Hours of Training Shoe Size Cigarettes smoked per day Score on SAT Height Response (Dependent) Variable y Number of Accidents Height Lung Capacity Grade Point Average IQ What type of relationship exists between the two variables and is the correlation significant?

Scatter Plots and Types of Correlation x = hours of training y = number

Scatter Plots and Types of Correlation x = hours of training y = number of accidents 60 Accidents 50 40 30 20 10 0 0 2 4 6 8 10 12 14 16 18 20 Hours of Training Negative Correlation–as x increases, y decreases

Scatter Plots and Types of Correlation GPA x = SAT score y = GPA

Scatter Plots and Types of Correlation GPA x = SAT score y = GPA 4. 00 3. 75 3. 50 3. 25 3. 00 2. 75 2. 50 2. 25 2. 00 1. 75 1. 50 300 350 400 450 500 550 600 650 700 750 800 Math SAT Positive Correlation–as x increases, y increases

Scatter Plots and Types of Correlation IQ x = height y = IQ 160

Scatter Plots and Types of Correlation IQ x = height y = IQ 160 150 140 130 120 110 100 90 80 60 64 68 72 Height No linear correlation 76 80

Correlation Coefficient A measure of the strength and direction of a linear relationship between

Correlation Coefficient A measure of the strength and direction of a linear relationship between two variables The range of r is from – 1 to 1. – 1 If r is close to – 1 there is a strong negative correlation. 0 If r is close to 0 there is no linear correlation. 1 If r is close to 1 there is a strong positive correlation.

Application Final Grade Final Absences Grade 95 90 85 80 75 70 65 60

Application Final Grade Final Absences Grade 95 90 85 80 75 70 65 60 55 50 45 40 0 2 4 6 8 10 Absences X 12 14 16 x 8 2 5 12 15 9 6 y 78 92 90 58 43 74 81

Computation of r n 1 2 3 4 5 6 7 y 78 92

Computation of r n 1 2 3 4 5 6 7 y 78 92 90 58 43 74 81 xy 624 184 450 696 645 666 486 x 2 64 4 25 144 225 81 36 y 2 6084 8464 8100 3364 1849 5476 6561 57 516 3751 579 39898 x 8 2 5 12 15 9 6

Hypothesis Test for Significance r is the correlation coefficient for the sample. The correlation

Hypothesis Test for Significance r is the correlation coefficient for the sample. The correlation coefficient for the population is (rho). For a two tail test for significance: (The correlation is not significant) (The correlation is significant) For left tail and right tail to test negative or positive significance: The sampling distribution for r is a t-distribution with n – 2 d. f. Standardized test statistic

Test of Significance You found the correlation between the number of times absent and

Test of Significance You found the correlation between the number of times absent and a final grade r = – 0. 975. There were seven pairs of data. Test the significance of this correlation. Use = 0. 01. 1. Write the null and alternative hypothesis. (The correlation is not significant) (The correlation is significant) 2. State the level of significance. = 0. 01 3. Identify the sampling distribution. A t-distribution with 5 degrees of freedom

Rejection Regions Critical Values ± t 0 t – 4. 032 0 4. 032

Rejection Regions Critical Values ± t 0 t – 4. 032 0 4. 032 4. Find the critical value. 5. Find the rejection region. 6. Find the test statistic.

t – 4. 032 0 – 4. 032 7. Make your decision. t =

t – 4. 032 0 – 4. 032 7. Make your decision. t = – 9. 811 falls in the rejection region. Reject the null hypothesis. 8. Interpret your decision. There is a significant correlation between the number of times absent and final grades.

The Line of Regression Once you know there is a significant linear correlation, you

The Line of Regression Once you know there is a significant linear correlation, you can write an equation describing the relationship between the x and y variables. This equation is called the line of regression or least squares line. The equation of a line may be written as y = mx + b where m is the slope of the line and b is the y-intercept. The line of regression is: The slope m is: The y-intercept is:

(xi, yi) = a data point = a point on the line with the

(xi, yi) = a data point = a point on the line with the same x-value = a residual 260 revenue 250 240 230 220 210 200 190 180 1. 5 2. 0 Ad $ 2. 5 3. 0

x 1 2 3 4 5 6 7 xy y 8 2 5 12

x 1 2 3 4 5 6 7 xy y 8 2 5 12 15 9 6 78 92 90 58 43 74 81 624 184 450 696 645 666 486 57 516 3751 x 2 64 4 25 144 225 81 36 y 2 6084 8464 8100 3364 1849 5476 6561 579 39898 The line of regression is: Write the equation of the line of regression with x = number of absences and y = final grade. Calculate m and b. = – 3. 924 x + 105. 667

The Line of Regression Final Grade m = – 3. 924 and b =

The Line of Regression Final Grade m = – 3. 924 and b = 105. 667 The line of regression is: 95 90 85 80 75 70 65 60 55 50 45 40 0 2 4 6 8 10 12 14 16 Absences Note that the point = (8. 143, 73. 714) is on the line.

Predicting y Values The regression line can be used to predict values of y

Predicting y Values The regression line can be used to predict values of y for values of x falling within the range of the data. The regression equation for number of times absent and final grade is: = – 3. 924 x + 105. 667 Use this equation to predict the expected grade for a student with (a) 3 absences (b) 12 absences (a) = – 3. 924(3) + 105. 667 = 93. 895 (b) = – 3. 924(12) + 105. 667 = 58. 579

The Coefficient of Determination The coefficient of determination, r 2, is the ratio of

The Coefficient of Determination The coefficient of determination, r 2, is the ratio of explained variation in y to the total variation in y. The correlation coefficient of number of times absent and final grade is r = – 0. 975. The coefficient of determination is r 2 = (– 0. 975)2 = 0. 9506. Interpretation: About 95% of the variation in final grades can be explained by the number of times a student is absent. The other 5% is unexplained and can be due to sampling error or other variables such as intelligence, amount of time studied, etc.

The Standard Error of Estimate, se, is the standard deviation of the observed yi

The Standard Error of Estimate, se, is the standard deviation of the observed yi values about the predicted value.

The Standard Error of Estimate 1 2 3 4 5 6 7 x y

The Standard Error of Estimate 1 2 3 4 5 6 7 x y 8 2 5 12 15 9 6 78 92 90 58 43 74 81 74. 275 97. 819 86. 047 58. 579 46. 807 70. 351 82. 123 13. 8756 33. 8608 15. 6262 0. 3352 14. 4932 13. 3152 1. 2611 92. 767 Calculate for each x. = 4. 307

Prediction Intervals Given a specific linear regression equation and x 0, a specific value

Prediction Intervals Given a specific linear regression equation and x 0, a specific value of x, a c-prediction interval for y is: where The point estimate is and E is the maximum error of estimate. Use a t-distribution with n – 2 degrees of freedom.

Application Construct a 90% confidence interval for a final grade when a student has

Application Construct a 90% confidence interval for a final grade when a student has been absent 6 times. 1. Find the point estimate: The point (6, 82. 123) is the point on the regression line with xcoordinate of 6.

Application Construct a 90% confidence interval for a final grade when a student has

Application Construct a 90% confidence interval for a final grade when a student has been absent 6 times. 2. Find E, At the 90% level of confidence, the maximum error of estimate is 9. 438.

Application Construct a 90% confidence interval for a final grade when a student has

Application Construct a 90% confidence interval for a final grade when a student has been absent 6 times. 3. Find the endpoints. – E = 82. 123 – 9. 438 = 72. 685 + E = 82. 123 + 9. 438 = 91. 561 72. 685 < y < 91. 561 When x = 6, the 90% confidence interval is from 72. 685 to 91. 586.