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Correlation and Regression

Correlation and Regression. Elementary Statistics. Correlation. A relationship between two variables. Explanatory (Independent) Variable. Response (Dependent) Variable. y. x. Hours of Training. Number of Accidents. Shoe Size. Height . Cigarettes smoked per day. Lung Capacity.

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Correlation and Regression

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  1. Correlation and Regression Elementary Statistics

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

  3. Scatter Plots and Types of Correlation x = hours of training y = number of accidents 60 50 40 Accidents 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

  4. Scatter Plots and Types of Correlation x = SAT score y = GPA 4.00 3.75 3.50 3.25 GPA 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

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

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

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

  8. Computation of r xy n x y y2 x2 1 8 78 2 2 92 3 5 90 4 12 58 5 15 43 6 9 74 7 6 81 624 184 450 696 645 666 486 64 4 25 144 225 81 36 6084 8464 8100 3364 1849 5476 6561 57 516 3751 579 39898

  9. 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

  10. 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

  11. 4.032 –4.032 Rejection Regions Critical Values ± t0 t 0 4. Find the critical value. 5. Find the rejection region. 6. Find the test statistic.

  12. t 0 –4.032 –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.

  13. 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:

  14. (xi,yi) = a data point = a point on the line with the same x-value = a residual 260 250 240 230 revenue 220 210 200 190 180 1.5 2.0 2.5 3.0 Ad $

  15. xy xy x2 y2 Write the equation of the line of regression with x = number of absences and y = final grade. 1 8 78 2 2 92 3 5 90 4 12 58 5 15 43 6 9 74 7 6 81 624 184 450 696 645 666 486 64 4 25 144 225 81 36 6084 8464 8100 3364 1849 5476 6561 Calculate m and b. 516 3751 579 39898 57 The line of regression is: = –3.924x + 105.667

  16. The Line of Regression 0 2 4 6 8 10 12 14 16 m = –3.924 and b = 105.667 The line of regression is: 95 90 85 Grade 80 75 70 65 Final 60 55 50 45 40 Absences Note that the point = (8.143, 73.714) is on the line.

  17. 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: = –3.924x + 105.667 Use this equation to predict the expected grade for a student with (a) 3 absences (b) 12 absences = –3.924(3) + 105.667 = 93.895 (a) = –3.924(12) + 105.667 = 58.579 (b)

  18. The Coefficient of Determination The coefficient of determination, r2,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 r2 = (–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.

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

  20. The Standard Error of Estimate x y 1 8 78 74.275 13.8756 2 2 92 97.819 33.8608 3 5 90 86.047 15.6262 4 12 58 58.579 0.3352 5 15 43 46.807 14.4932 6 9 74 70.351 13.3152 7 6 81 82.123 1.2611 92.767 Calculate for each x. = 4.307

  21. Prediction Intervals Given a specific linear regression equation and x0, a specific value of x, a c-prediction interval for y is: where The point estimate is andEis the maximum error of estimate. Use a t-distribution with n – 2 degrees of freedom.

  22. 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 x-coordinate of 6.

  23. 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.

  24. 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.

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