3. For the accompanying data set, (a) draw a scatter diagram of the data, (b) compute the correlation coefficient, and (c) determine whether there is a linear relation between x and y. 1 Click the icon to view the data set. 2 Click the icon to view the critical values table. (a) Draw a scatter diagram of the data. Choose the correct graph below. A. y 10 0 B. 0 y 10 x 10 0 C. y 10 x 0 0 10 D. y 10 x 0 0 10 x 0 10 (b) Compute the correlation coefficient. The correlation coefficient is r = . (Round to three decimal places as needed.) (c) Determine whether there is a linear relation between x and y. Because the correlation coefficient is (1) and the absolute value of the correlation coefficient, , is (2) than the critical value for this data set, relation exists between x and y. (Round to three decimal places as needed.) , (3) 1: Data set x 7 6 6 7 9 y 3 7 6 9 5 2: critical values for the correlation coecient Critical Values for Correlation Coefficient linear n 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 (1) negative positive (2) not greater greater (3) 0.997 0.950 0.878 0.811 0.754 0.707 0.666 0.632 0.602 0.576 0.553 0.532 0.514 0.497 0.482 0.468 0.456 0.444 0.433 0.423 0.413 0.404 0.396 0.388 0.381 0.374 0.367 0.361 no a negative a positive 4. A pediatrician wants to determine the relation that may exist between a child's height and head circumference. She randomly selects 8 children, measures their height and head circumference, and obtains the data shown in the table. Head Circumference Height (inches) (inches) 27.75 17.6 25 17.2 26.25 17.2 3 Click here to see the Table of Critical Values for Correlation 25.25 17 Coefficient. 27.25 17.6 26.25 17.1 26 17.2 26.75 17.4 (a) If the pediatrician wants to use height to predict head circumference, determine which variable is the explanatory variable and which is the response variable. The explanatory variable is height and the response variable is head circumference. The explanatory variable is head circumference and the response variable is height. (b) Draw a scatter diagram. Which of the following represents the data? B. 16.9 25 28 Circ. (in.) C. 17.6 28 25 16.9 17.6 Circ. (in.) Circ. (in.) 28 D. Circ. (in.) 17.6 Height (in.) Height (in.) A. 16.9 25 28 25 16.9 Height (in.) 17.6 Height (in.) (c) Compute the linear correlation coefficient between the height and head circumference of a child. r= (Round to three decimal places as needed.) (d) Does a linear relation exist between height and head circumference? A. Yes, there appears to be a negative linear association because r is negative and is less than the negative of the critical value. B. Yes, there appears to be a positive linear association because r is positive and is less than the critial value. C. Yes, there appears to be a positive linear association because r is positive and is greater than the critical value. D. No, there is no linear association since r is positive and is less than the critical value. 3: Data Table Critical Values for Correlation Coefficient n 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 0.997 0.950 0.878 0.811 0.754 0.707 0.666 0.632 0.602 0.576 0.553 0.532 0.514 0.497 0.482 0.468 0.456 0.444 0.433 0.423 0.413 0.404 0.396 0.388 0.381 0.374 0.367 0.361 5. Researchers wondered whether the size of a person's brain was related to the individual's mental capacity. They selected a sample of 3 females and 3 males and measured their MRI counts and IQ scores. The data is reported on the right. Females MRI IQ 857,781 135 991,305 138 856,473 141 Males MRI 965,355 935,493 1,001,121 Treat the MRI count as the explanatory variable. Compute the linear correlation coefficient between MRI count and IQ for both the males and the females. Do you believe that MRI count and IQ are linearly related? The linear correlation coefficient for females is . IQ 132 144 141 The linear correlation coefficient for males is . (Round to three decimal places as needed.) Are MRI count and IQ linearly related? The linear correlation coefficient for females is close to (1) so (2) linear relation exists between MRI count and IQ for females. The linear correlation coefficient for males is close to (3) so (4) linear relation exists between MRI count and IQ for males. (1) (2) 1, a negative (3) 0, (4) a positive 0, a positive 1, no 1, no 1, a negative 6. For the data set below, (a) Determine the leastsquares regression line. (b) Graph the leastsquares regression line on the scatter diagram. x y 5 8 6 9 7 10 8 14 10 18 (a) Determine the leastsquares regression line. y= x+( ) (Round to four decimal places as needed.) (b) Choose the correct graph below. A. y 19 0 B. 0 y 11 x 11 0 C. 0 y 11 x 19 0 D. x 0 y 19 19 0 x 0 11 7. For the data set below, (a) Determine the leastsquares regression line. (b) Compute the sum of the squared residuals for the leastsquares regression line. x y 30 83 40 66 50 61 60 63 70 50 (a) Determine the leastsquares regression line. y= x+ (Round to four decimal places as needed.) (b) The sum of the squared residuals is (Round to two decimal places as needed.) . 8. A student at a junior college conducted a survey of 20 randomly selected fulltime students to determine the relation between the number of hours of video game playing each week, x, and gradepoint average, y. She found that a linear relation exists between the two variables. The leastsquares regression line that describes this relation is y = 0.0582x + 2.9422. (a) Predict the gradepoint average of a student who plays video games 8 hours per week. The predicted gradepoint average is (Round to the nearest hundredth as needed.) . (b) Interpret the slope. For each additional hour that a student spends playing video games in a week, the gradepoint average will (1) by points, on average. (c) If appropriate, interpret the yintercept. A. The gradepoint average of a student who does not play video games is 2.9422. B. The average number of video games played in a week by students is 2.9422. C. It cannot be interpreted without more information. (d) A student who plays video games 7 hours per week has a gradepoint average of 2.65. Is the student's gradepoint average above or below average among all students who play video games 7 hours per week? The student's gradepoint average is (2) (1) decrease increase (2) above below average for those who play video games 7 hours per week. 9. An author of a book discusses how statistics can be used to judge both a baseball player's potential and a team's ability to win games. One aspect of this analysis is that a team's onbase percentage is the best predictor of winning percentage. The onbase percentage is the proportion of time a player reaches a base. For example, an onbase percentage of 0.3 would mean the player safely reaches bases 3 times out of 10, on average. For a certain baseball season, winning percentage, y, and onbase percentage, x, are linearly related by the leastsquares regression equation y = 2.94x 0.4873. Complete parts (a) through (d). (a) Interpret the slope. Choose the correct answer below. A. For each percentage point increase in onbase percentage, the winning percentage will decrease by 2.94 percentage points, on average. B. For each percentage point increase in onbase percentage, the winning percentage will increase by 2.94 percentage points, on average. C. For each percentage point increase in winning percentage, the onbase percentage will decrease by 2.94 percentage points, on average. D. For each percentage point increase in winning percentage, the onbase percentage will increase by 2.94 percentage points, on average. (b) For this baseball season, the lowest onbase percentage was 0.310 and the highest onbase percentage was 0.366. Does it make sense to interpret the yintercept? Yes No (c) Would it be a good idea to use this model to predict the winning percentage of a team whose onbase percentage was 0.220? No, it would be a bad idea. Yes, it would be a good idea. (d) A certain team had an onbase percentage of 0.324 and a winning percentage of 0.546. What is the residual for that team? How would you interpret this residual? The residual for the team is . (Round to four decimal places as needed.) How would you interpret this residual? A. This residual indicates that the winning percentage of the team is below average for teams with an onbase percentage of 0.324. B. This residual indicates that the winning percentage of the team is above average for teams with a winning percentage of 0.546. C. This residual indicates that the winning percentage of the team is above average for teams with an onbase percentage of 0.324. D. This residual indicates that the winning percentage of the team and the onbase percentage of other teams do not vary. 10. Because colas tend to replace healthier beverages and colas contain caffeine and phosphoric acid, researchers wanted to know whether consumption of cola is associated with lower bone mineral density in women. The data shown in the accompanying table represent the typical number of cans of soda consumed in a week and the bone mineral density of the femoral neck for a sample of 15 women. The data were collected through a prospective cohort study. Complete parts (a) through (f). 4 Click the icon to view the data table. 5 Click the icon to view a table of critical values for the correlation coefficient. (a) Find the leastsquares regression line treating cola consumption per week as the explanatory variable. Choose the correct answer below. A. The leastsquares regression line is y = 0.0032x 0.8877. B. The leastsquares regression line is y = 0.8877x 0.0032. C. The leastsquares regression line is y = 0.0032x + 0.8877. D. The leastsquares regression line is y = 0.8877x + 0.0032. (b) Interpret the slope. For each additional cola consumed per week, bone mineral density will (1) by 2 (2) g / cm , on average. (c) Interpret the intercept. Choose the correct answer below. A. For each additional cola consumed per week, bone mineral density will decrease by 0.0032 g/cm2 , on average. B. For a woman who does not drink cola, bone mineral density will be 0.0032 g/cm2 . C. For a woman who does not drink cola, bone mineral density will be 0.8877 g/cm2 . D. It is not appropriate to interpret the yintercept. It is outside the scope of the model. (d) Predict the bone mineral density of the femoral neck of a woman who consumes four colas per week. The predicted value of the bone mineral density of the femoral neck of this woman is four decimal places as needed.) 2 g / cm . (Round to 2 (e) The researchers found a woman who consumed four colas per week to have a bone mineral density of 0.873 g/cm . Is this woman's bone mineral density above or below average among all women who consume four colas per week? Above average Below average (f) Would you recommend using the model found in part (a) to predict the bone mineral density of a woman who consumes two cans of cola per day? No Yes 4: Data Table Full data set Number of Colas per Week Bone Mineral Density (g/cm ) Number of Colas per Week Density (g/cm ) 0 0.897 4 0.873 0 0.886 5 0.875 1 0.891 5 0.871 1 0.881 6 0.867 2 0.888 7 0.862 2 0.871 7 0.872 3 0.868 8 0.865 3 0.876 2 5: Critical Values for Correlation Coecient Bone Mineral 2 (1) decrease increase (2) 0.0032 1.0054 0.5423 0.8877 11. The given data represent the total compensation for 10 randomly selected CEOs and their company's stock performance in 2009. Analysis of this data reveals a correlation coefficient of r = 0.2110. What would be the predicted stock return for a company whose CEO made $15 million? What would be the predicted stock return for a company whose CEO made $25 million? 6 Click the icon to view the compensation and stock performance data. 7 Click the icon to view a table of critical values for the correlation coefficient. What would be the predicted stock return for a company whose CEO made $15 million? % (Type an integer or decimal rounded to one decimal place as needed.) What would be the predicted stock return for a company whose CEO made $25 million? % (Type an integer or decimal rounded to one decimal place as needed.) 6: CEO Compensation and Stock Performance Compensation (millions of dollars) 7: Critical Values for Correlation Coecient Stock Return (%) 26.84 6.13 12.86 30.26 19.07 32.19 13.08 79.89 11.76 8.18 12.13 2.34 26.22 4.17 14.98 10.46 17.67 4.01 13.86 11.35 12. Analyze the residual plot below and identify which, if any, of the conditions for an adequate linear model is not met. Residuals 4 0 2 5 15 25 Explanatory Which of the conditions below might indicate that a linear model would not be appropriate? Outlier None Constant error variance Patterned residuals 13. The following data represent the time between eruptions and the length of eruption for 8 randomly selected geyser eruptions. Complete parts (a) and (b) below. Click here to view a scatter plot of the data.8 Click here to view a residual plot of the data.9 Time, x 12.16 11.78 11.96 12.15 Length, y 1.86 1.77 1.84 1.88 Time, x 11.49 11.57 12.09 11.32 Length, y 1.70 1.75 1.83 1.66 (a) Does the residual plot confirm that the relation between time between eruptions and length of eruption is linear? A. Yes. The plot of the residuals shows no discernible pattern, so a linear model is appropriate. B. Yes. The plot of the residuals shows a discernible pattern, implying that the explanatory and response variables are linearly related. C. No. The plot of the residuals shows that the spread of the residuals is increasing or decreasing, violating the requirements of a linear model. D. No. The plot of the residuals shows no discernible pattern, implying that the explanatory and response variables are not linearly related. (b) The coefficient of determination is found to be 95.9%. Choose the best interpretation below. A. The leastsquares regression equation explains 95.9% of the variation in time between eruptions. B. The leastsquares regression equation explains 95.9% of the variation in length of eruption. C. The leastsquares regression equation does not explain 95.9% of the variation in length of eruption. D. The leastsquares regression equation does not explain 95.9% of the variation in time between eruptions. 8: Scatter plot of eruption data. 2 y 1.9 1.8 1.7 1.6 x 1.5 11 11.4 11.8 12.2 12.6 9: Residual plot of eruption data. 0.2 y 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2 11 x 11.4 11.8 12.2 12.6 14. The time it takes for a planet to complete its orbit around a particular star is called the planet's sidereal year. The sidereal year of a planet is related to the distance the planet is from the star. The accompanying data show the distances of the planets from a particular star and their sidereal years. Complete parts (a) through (e). 10 Click the icon to view the data table. (a) Draw a scatter diagram of the data treating distance from the star as the explanatory variable. Choose the correct graph below. A. B. C. Sidereal Year Distance (millions of miles) Sidereal Year 250 4000 250 125 2000 125 0 0 0 0 2000 4000 0 Distance (millions of miles) 125 250 0 Sidereal Year 2000 4000 Distance (millions of miles) (b) Determine the correlation between distance and sidereal year. The correlation between distance and sidereal year is (Round to three decimal places as needed.) . Does this imply a linear relation between distance and sidereal year? Yes No (c) Compute the leastsquares regression line. Choose the correct answer below. y = 0.0655x + 132.467 y = 0.0655x + 12.503 y = 0.0655x + 12.503 y = 0.0655x 12.503 (d) Plot the residuals against the distance from the star. Choose the correct graph below. A. B. Residual 500 20 0 C. Residual Residual 20 250 20 0 2000 4000 Distance (millions of miles) 0 0 20 0 2000 4000 Distance (millions of miles) (e) Do you think the leastsquares regression line is a good model? Yes No 0 2000 4000 Distance (millions of miles) 10: Data Table Planet Distance from the Star, x Sidereal Year, y (millions of miles) Planet 1 36 0.24 Planet 2 67 0.62 Planet 3 93 1.00 Planet 4 142 1.88 Planet 5 483 11.8 Planet 6 887 29.3 Planet 7 1,785 84.0 Planet 8 2,797 163.0 Planet 9 3,675 248.0