Name: Date: Period: Chapter 2: Take Home TEST Bivariate Data Part 1: Multiple Choice. (2.5 points each) Hand write the letter corresponding to the best answer in space provided on page 6. 1. In a statistics course, a linear regression equation was computed to predict the final exam score from the score on the first test. The equation was ŷ = 10 +.9x where y is the final exam score and x is the score on the first test. Carla scored 95 on the first test. What is the predicted value of her score on the final exam? (a) 95 (b) 85.5 (c) 90 (d) 95.5 (e) none of the above 2. Refer to the previous problem. On the final exam Carla scored 98. What is the value of her residual? (a) 98 (b) 2.5 (c) 2.5 (d) 0 (e) none of the above 3. A study of the fuel economy for various automobiles plotted the fuel consumption (in liters of gasoline used per 100 kilometers traveled) vs. speed (in kilometers per hour). A least squares line was fit to the data. Here is the residual plot from this least squares fit. What does the pattern of the residuals tell you about the linear model? (a) The evidence is inconclusive. (b) The residual plot confirms the linearity of the fuel economy data. (c) The residual plot does not confirm the linearity of the data. (d) The residual plot clearly contradicts the linearity of the data. (e) none of the above 4. All but one of the following statements contains a blunder. Which statement is correct? (a) There is a correlation of 0.54 between the position a football player plays and their weight. (b) The correlation between planting rate and yield of corn was found to be r = 0.23. (c) The correlation between the gas mileage of a car and its weight is r = 0.71 MPG. (d) We found a high correlation (r = 1.09) between the height and age of children. (e) We found a correlation of r =.63 between gender and political party preference. 5. In regression, the residuals are which of the following? (a) Those factors unexplained by the data (b) The difference between the observed responses and the values predicted by the regression line (c) Those data points which were recorded after the formal investigation was completed (d) Possible models unexplored by the investigator (e) None of the above Mrs. Poyner/Mr. Page Chapter 3 page 1
6. What does the square of the correlation (r 2 ) measure? (a) The slope of the least squares regression line (b) The intercept of the least squares regression line (c) The extent to which cause and effect is present in the data (d) The fraction of the variation in the values of y that is explained by least-squares regression on the other 7. A researcher finds that the correlation between the personality traits greed and superciliousness is.40. What percentage of the variation in greed can be explained by the relationship with superciliousness? (a) 0% (b) 16% (c) 20% (d) 40% (e) 60% 8. Suppose the following information was collected, where X = diameter of tree trunk in inches, and Y = tree height in feet. X 4 2 8 6 10 6 Y 8 4 18 22 30 8 If the LSRL equation is y = 3.6 + 3.1x, what is your estimate of the average height of all trees having a trunk diameter of 7 inches? (a) 18.1 (b) 19.1 (c) 20.1 (d) 21.1 (e) 22.1 9. The following are resistant: (a) Least squares regression line (b) Correlation coefficient (c) Both the least square line and the correlation coefficient (d) Neither the least square line nor the correlation coefficient (e) It depends 10. If dataset A of (x,y) data has correlation coefficient r = 0.65, and a second dataset B has correlation r = 0.65, then (a) The points in A exhibit a stronger linear association than B. (b) The points in B exhibit a stronger linear association than A. (c) Neither A nor B has a stronger linear association. (d) You can t tell which dataset has a stronger linear association without seeing the data or seeing the scatterplots. 11. A set of data relates the amount of annual salary raise and the performance rating. The least squares regression equation is ŷ = 1,400 + 2,000x where y is the estimated raise and x is the performance rating. Which of the following statements is not correct? (a) For each increase of one point in performance rating, the raise will increase on average by $2,000. (b) This equation produces predicted raises with an average error of 0. (c) A rating of 0 will yield a predicted raise of $1,400. (d) The correlation for the data is positive. (e) All of the above are true. Mrs. Poyner/Mr. Page Chapter 3 page 2
12. There is a linear relationship between the number of chirps made by the striped ground cricket and the air temperature. A least squares fit of some data collected by a biologist gives the model y ˆ = 25.2 + 3.3x, 9 < x < 25, where x is the number of chirps per minute and y ˆ is the estimated temperature in degrees Fahrenheit. What is the estimated increase in temperature that corresponds to an increase in 5 chirps per minute? (a) 3.3 F (b) 16.5 F (c) 25.2 F (d) 28.5 F (e) 41.7 F 13. A simple random sample of 35 world-ranked chess players provides the following statistics: Number of hours of study per day: x = 6. 2 s = 1.3 Yearly winnings: y = $208, 000, x, s y = $42,000 Correlation r = 0.15 Based on this data, what is the resulting linear regression equation? (a) Winnings = 178,000 + 4850 Hours (b) Winnings = 169,000 + 6300 Hours (c) Winnings = 14,550 + 31,200 Hours (d) Winnings = 7750 + 32,300 Hours (e) Winnings = -52,400 + 42,000 Hours 14. Suppose the correlation is negative. Given two points from the scatterplot, which of the following is possible? I. The first point has a larger x-value and a smaller y-value than the second point. II. The first point has a larger x-value and a larger y-value than the second point. III. The first point has a smaller x-value and a larger y-value than the second point. (a) I and II (b) I and III (c) II and III (d) I, II and III (e) None of the above gives the complete set of true responses. 15. Suppose the regression line for a set of data, y = 3x + b, passes through point (2, 5). If x and y are the sample means of the x- and y-values, respectively, then y = (a) x (b) x - 2 (c) x + 5 (d) 3 x (e) 3 x - 1 16. Consider the set of points {(2, 5), (3, 7), (4, 9), (5, 12), (10, n)}. What should the value for n be so that the correlation between the x- and y-values is 1? (a) 21 (b) 24 (c) 25 (d) A value different from any of the above choices. (e) No value for n can make r = 1. Mrs. Poyner/Mr. Page Chapter 3 page 3
17. Extra study sessions were offered to students after the midterm to help improve their understanding of statistics. Student scores on the midterm and the final exam were recorded. The following scatterplot shows final test scores against the midterm test scores. Collection 1 100 Scatter Plot 95 90 Score_On_Final 85 80 75 70 65 60 30 40 50 60 70 80 90 100 Score_On_Midterm Which of the following statements correctly interprets the scatterplot? (a) All students have shown significant improvements in the final exam scores as a result of the extra study sessions. (b) The extra study sessions were of no help. Each student s final exam score was about the same as his or her score on the midterm. (c) The extra study session further confuses students. All student scores decreased from midterm to final exam. (d) Students that scored below 55 on the midterm showed considerable improvement on the final exam; those who scored between 55 and 80 on the midterm showed minimal improvement on the final exam; and those who scored above 80 on the midterm showed almost no improvement on the final exam. (e) Students that scored below 55 on the midterm showed minimal improvement on the final exam; those who scored between 55 and 80 on the midterm showed moderate improvement on the final exam; and those who scored above 80 on the midterm showed considerable improvement on the final exam. 18. Consider the following computer software printout: Summary of Fit 1250 RSquare 0.770503 RSquare Adj 0.76582 1200 Root Mean Square Error 32.11888 1150 1100 Analysis of Variance Mean of Response 1067.098 Observations (or Sum Wgts) 51 Source DF Sum of Squares 1050 Model 1 169713.02 1000 Error 49 50549.49 C. Total 50 220262.51 Total SAT 2002 950 0 10 20 30 40 50 60 70 80 90 Percent 2002 Parameter Estimates Term Estimate Std Error Intercept 1145.1989 7.57007 Percent 2002-2.046837 0.159583 In the context of the problem, what is the equation of the least squares regression line? (a) Total SAT 2002 = -2.046837 + 1145.1989(Percent 2002) (b) Total SAT 2002 = 1145.1989-2.046837(Percent 2002) (c) Percent 2002 = -2.046837 + 1145.1989(Total SAT 2002) (d) Percent 2002 = 1145.1989-2.046837(Total SAT 2002) (e) Intercept = -2.046837 + 1145.1989(Percent 2002) Mrs. Poyner/Mr. Page Chapter 3 page 4
19. Which of the following statements about residuals are true? I. The mean of the residuals is always zero II. The regression line for a residual plot is a horizontal line. III. A definite pattern in the residual plot is an indicator that a nonlinear model will show a better fit to the data than the straight regression line. (f) I and II (g) I and III (h) II and III (i) I, II and III (j) None of the above gives the complete set of true responses. 20. Data are obtained for a group of college freshmen examining their SAT scores (math plus verbal) from their senior year of high school (x) and their GPAs during their first year of college (y). The resulting regression equation is ŷ = 0.00161x + 1.35 with r = 0.632 What percentage of the variation in GPAs can be explained by looking at SAT scores? (a) 0.0161% (b) 16.1% (c) 39.9% (d) 63.2% (e) The value cannot be computed from the information given. Mrs. Poyner/Mr. Page Chapter 3 page 5
Name: PART I answers Date: Period: 1. 2. 3. 4. 5. 6. 7. 8. 9 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. PART II - Answer completely, but be concise. Write sequentially and show all steps. Show all your work. Indicate clearly the methods you used, because you will be graded on the correctness of your methods as well as on the accuracy of your results and explanations. YOU MUST WRITE ANSWERS IN BY HAND you need not print out the first 5 pages. 11. Given the following observations of quantitative variables X and Y: X 0 1 2 3 15 Y 3 4 6 10 0 (a) Make a scatterplot of the data on the axes. (6 points) Circle the most influential observation. (2 points) (b) Determine the LSRL of Y on X.(4 points) Draw this line carefully on your scatterplot.(2 points) Indicate how you plotted the line (hint: calculate actual points on line).(2 points) (c) What is a regression outlier? (3 points) (d) Which data point is the biggest regression outlier? (2 points) (e) What is the residual at the point identified in part (d), justify numerically? (3 points) Mrs. Poyner/Mr. Page Chapter 3 page 6
(f) Construct a residual plot for these data. (6 points) (g) Interpret your residual plot. I.e., what does the residual plot tell you? (4 points) 2. Each of 25 adult women was asked to provide her own height (y), in inches, and the height (x), in inches, of her father. The scatterplot below displays the results. Only 22 of the 25 pairs are distinguishable because some of the (x, y) pairs were the same. The equation of the least squares regression line is ŷ = 35.1 + 0.427x. (16 points) (a) Draw the least squares regression line on the scatterplot above. (b) One father s height was x = 67 inches and his daughter s height was y = 61 inches. Circle the point on the scatterplot above that represents this pair and draw the segment on the scatterplot that corresponds to the residual for it. Give a numberical value for the residual. Mrs. Poyner/Mr. Page Chapter 3 page 7
(c) Suppose the point x = 84, y = 71 is added to the data set. Would the slope of the least squares regression line increase, decrease, or remain about the same? Explain. (Note: No calculations are necessary to answer this question.) (d) Would the correlation increase, decrease, or remain about the same? Explain. (Note: No calculations are necessary to answer this question.) I pledge that the answers to the questions on this test have been formulated by myself and that I can explain and reproduce all the answers on my own if asked: (2 pts) 102 points total/100 Mrs. Poyner/Mr. Page Chapter 3 page 8