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Correlation and Regression

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

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Correlation measures and describes the strength and direction of the relationship Bivariate techniques requires two variable scores from the same individuals (dependent and independent variables) Multivariate when more than two independent variables (e.g effect of advertising and prices on sales) Variables must be ratio or interval scale

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Negative Correlation–as x increases, y decreases x = hours of training (horizontal axis) y = number of accidents (vertical axis) Scatter Plots and Types of Correlation 60 50 40 30 20 10 0 02468 1214161820 Hours of Training Accidents

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Positive Correlation–as x increases, y increases x = SAT score y = GPA GPA Scatter Plots and Types of Correlation 4.00 3.75 3.50 3.00 2.75 2.50 2.25 2.00 1.50 1.75 3.25 300350400450500550600650700750800 Math SAT

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No linear correlation x = height y = IQ Scatter Plots and Types of Correlation 160 150 140 130 120 110 100 90 80 606468727680 Height IQ

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Strong, negative relationship but non-linear! Scatter Plots and Types of Correlation

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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. If r is close to 1 there is a strong positive correlation. If r is close to –1 there is a strong negative correlation. If r is close to 0 there is no linear correlation. –1 0 1

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Outliers..... Outliers are dangerous Here we have a spurious correlation of r=0.68 without IBM, r=0.48 without IBM & GE, r=0.21

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x y 8 78 2 92 5 90 12 58 15 43 9 74 6 81 Absences Final Grade Application 95 90 85 80 75 70 65 60 55 45 40 50 0246810121416 Final Grade X Absences

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6084 8464 8100 3364 1849 5476 6561 624 184 450 696 645 666 486 57516375157939898 1 8 78 2 2 92 3 5 90 4 12 58 5 15 43 6 9 74 7 6 81 64 4 25 144 225 81 36 xy x 2 y2y2 Computation of r x y

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

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A t-distribution with 5 degrees of freedom Test of Significance 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. 2. State the level of significance. 3. Identify the sampling distribution. (The correlation is not significant) (The correlation is significant) = 0.01

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t 0 4.032 –4.032 Rejection Regions Critical Values ± t 0 4. Find the critical value. 5. Find the rejection region. 6. Find the test statistic. df\p0.400.250.100.050.0250.010.0050.0005 10.3249201.0000003.0776846.31375212.7062031.8205263.65674636.6192 20.2886750.8164971.8856182.9199864.302656.964569.9248431.5991 30.2766710.7648921.6377442.3533633.182454.540705.8409112.9240 40.2707220.7406971.5332062.1318472.776453.746954.604098.6103 50.2671810.7266871.4758842.0150482.570583.364934.032146.8688

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t 0 –4.032 t = –9.811 falls in the rejection region. Reject the null hypothesis. There is a significant negative correlation between the number of times absent and final grades. 7. Make your decision. 8. Interpret your decision.

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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: Regression indicates the degree to which the variation in one variable X, is related to or can be explained by the variation in another variable Y 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 Line of Regression

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180 190 200 210 220 230 240 250 260 1.52.02.53.0 Ad $ = a residual (xi,yi)(xi,yi) = a data point revenue = a point on the line with the same x-value Best fitting straight line

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Calculate m and b. Write the equation of the line of regression with x = number of absences and y = final grade. The line of regression is:= –3.924x + 105.667 6084 8464 8100 3364 1849 5476 6561 624 184 450 696 645 666 486 57 516375157939898 1 8 78 2 2 92 3 5 90 4 12 58 5 15 43 6 9 74 7 6 81 64 4 25 144 225 81 36 xy x 2 y2y2 x y

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0246810121416 40 45 50 55 60 65 70 75 80 85 90 95 Absences Final Grade m = –3.924 and b = 105.667 The line of regression is: Note that the point = (8.143, 73.714) is on the line. The Line of Regression

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

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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. Strength of the Association The coefficient of determination, r 2, measures the strength of the association and is the ratio of explained variation in y to the total variation in y.

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