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Bivariate Data and Scatter Plots Bivariate Data: The values of two different variables that are obtained from the same population element. While the variables may be either categorical or quantitative, we will focus on cases where they are both quantitative. Can we predict values of one variable from values of the other variable? Do the values of one variable cause the values of the other variable? Section 3.1, Page 59 1 Scatter Plot Example TI-83 Scatter Plots always have and explanatory variable and a response variable. The choice is arbitrary. The explanatory variable is always plotted on the x-axis, and the response variable is always plotted on the y axis. STAT – EDIT – ENTER; Enter x data in L1, and y in L2 2nd STAT PLOT – ENTER -1: Plot 1 Highlight ON Type: Highlight first icon XList: 2nd L1 YList 2nd L2 ZOOM 9: ZoomStat TRACE; Use arrows to move to points and display values. Section 3.1, Page 60 2 Linear Correlation Linear Correlation: A measure of the strength of a linear relationship between two variables. The closer to a straight line the dots are, the stronger the relationship. If there correlation, then we say the two variables are associated. Changes in the value of one variable are associated with changes in the value of the other variable. Section 3.1, Page 61 3 Coefficient of Correlation Measure of Strength Zx Zy (x x ) (y y ) r where Z x ; Zy n 1 sx sy 1 r 1; r 1 perfect straight line negative slope no relationship at all r0 r 1 perfect straight line with positive slope Also known as the Pearson Correlation Coefficient. Section 3.2, Page 62 4 Problems Problems, Page 71 5 Correlation Coefficient TI-83 Add-In Program Finding r. STAT – EDIT – ENTER: Enter data in L1 and L2 PRGM-CORRELTN 2nd LI – Comma – 2nd L2 SCATTER PLOT? – 1=YES; (Displays scatter plot) ENTER; (Displays: r=.8394) This is a moderately strong positive relationship. Section 3.2, Page 62 6 Association and Causality Elementary School Students Reading Scores 8 Grade Level 4 1 1 4 Shoe Size 8 Is this a reasonable association? Does giving students bigger shoes cause reading scores to improve? What explains this association? Lurking Variable: A third variable, often unexpressed, that has an effect on either or both x and y variables making it appear they are related. Association alone can never establish causality! Section 3.2, Page 63 7 Problems Problems, Page 71 8 Problems Problems, Page 72 9 Problems Problems, Page 72 10 Linear Regression Line of Best Fit If a straight line model seems appropriate, the best fit straight line is found by using the method of least squares. Suppose that yˆ a bx is the equation of a straight line, where yˆ (read “y-hat) represents the predicted value of y that corresponds to a particular value of x. The least that we find the squares criteria requires 2 constants, a and b suchthat (y yˆ ) is as small as possible. yˆ a bx Section 3.3, Page 65 11 Line of Best Fit The best line will be the one where the sum of the squares of the “misses” is at a minimum. Calculus procedures are used to find the coefficients, a and b such that the line ŷ = a + bx has the least squares. br sy sx r is the correlation coefficient, sy is the standard deviation of y-values and sx is the standard deviation of the x values Section 3.3, Page 66 12 Linear Regression TI-83 Add-In Program a. For the above data, make a scatter plot, and comment on the suitability of the data for regression analysis. STAT – EDIT; Enter Height in L1, and Weight in L2. PRGN – REGBASIC X LIST=2ND L1; Y LIST=2ND L2 SCATTER PLOT: 1=YES The pattern looks positive, linear, and no outliers which could cause problems. Scatter Plot Section 3.3, Page 68 13 Linear Regression TI-83 Add-In Program b. Find the regression equation and r. ENTER; The program is paused to view graph, hitting ENTER moves the program along. The equation is: yˆ =-186.4706 + 4.7059x r, the coefficient of correlation = .7979, a relatively strong relationship. c. Check the plot of the regression line versus the scatter plot. ENTER – 1=YES Section 3.3, Page 68 14 Linear Regression TI-83 Add-In Program d. What is the value of the slope of the line, and what does it mean? b = 4.7095 is the slope of the line. It indicates the number of units change in the y value for every one unit increase in the x value. In this problem, for each one inch increase in height, weight increases by 4.7095 lbs. Its units are lbs/inch. e. What is the value of the intercept of the line, and what does it mean? a = -186.4706 is the y intercept. It has no meaning in this problem. It would be the weight of a person of zero height. f. What is the value of r2? It is called the index of determination. It measures the strength of the model, 1 being perfect and 0 being useless. r2 = .6367 indicating a relative strong positive correlation. Section 3.3, Page 68 15 Linear Regression TI-83 Add-In Program g. Check the residual plot and explain what it means ENTER; 1 = YES The horizontal line represents the regression line. For each actual value of x, the residual is the actual y-value – predicted y-value. The dots show the “misses” or residuals. If the residuals show some kind of a pattern, it means that the linear regression model is not appropriate for the data, so other model, i.e. quadratic, may be better. Since there is not pattern is this plot, the linear model is appropriate for this data. Section 3.3, Page 68 16 Linear Regression TI-83 Add-In Program h. Use the model to predict the weight of a woman who is 65 inches tall. PREDICTED Y: 1 = YES X=65 Answer: 119.4 lbs i. Use the model to predict the weight of a woman who is 77 inches tall. ENTER: 1 = YES X=77 Answer 175.9 lbs. Notice that the range of the x values is from 61 to 69 inches. 77 inches is too far above the actual values used to develop the model. While the result is mathematically correct, the result is not valid in the context of the problem. Section 3.3, Page 68 17 Problems Problems, Page 72 18 Problems a. b. c. d. e. f. g. h. i. Construct a scatter diagram. Does the pattern appear linear? Find the equation of best fit. What is the value of r and what does it mean? What is the slope? What are its units? Interpret its meaning. What is the y-intercept value? What does it mean? What does the residual plot show? What does it mean? Estimate the the stride rate for a speed of 19.2 ft/sec. Is the estimate reliable? Why? Estimate the stride rate for a speed of 31 ft/sec. Is the estimate reliable? Why? Problems, Page 73 19 Problems c. d. e. f. g. h. What is the value of r and what does it mean? What is the slope? What are its units? Interpret its meaning. What is the y-intercept value? What does it mean? What does the residual plot show? What does it mean? Estimate the # of intersections for a state with 450 miles. Is the estimate reliable? Why? Estimate the # of intersections for a state with 950 miles. Is the estimate reliable? Why? Problems, Page 73 20 Problems a. b. c. d. e. f. g. h. Construct a scatter diagram. What does it indicate to you? Find the equation of best fit. What is the value of r and what does it mean? What is the slope? What are its units? Interpret its meaning. What is the y-intercept value? What does it mean? What does the residual plot show? What does it mean? Estimate the price of an 8 year old car. Is the estimate reliable? Why? Estimate price of a 22 year old car. Is the estimate reliable? Why? Problems, Page 73 21