1. Suppose a least-squares regression line is given by y = 4.302x 3.293. What is the mean value of the response variable if x = 20? y 20 = (Round to one decimal place as needed.) 2. In the least-squares regression model, y = x + + , is a random error term with mean i 1 i 0 i i i = . In the least-squares regression model, yi = 1 xi + 0 + i, i is a random error term with mean (1) standard deviation = (2) i (1) and standard deviation 0 (2) . 0. 1 1. . and 3. For the data set shown below, complete parts (a) through (d) below. x 3 y 4 4 5 6 7 7 12 8 15 (a) Find the estimates of 0 and 1 . 0 b 0 = 1 b 1 = (Round to three decimal places as needed.) (Round to three decimal places as needed.) (b) Compute the standard error, the point estimate for . se = (Round to four decimal places as needed.) (c) Assuming the residuals are normally distributed, determine s b . 1 sb = 1 (Round to three decimal places as needed.) (d) Assuming the residuals are normally distributed, test H 0 : 1 = 0 versus H 1 : 1 0 at the = 0.05 level of significance. Use the P-value approach. The P-value for this test is . (Round to three decimal places as needed.) Make a statement regarding the null hypothesis and draw a conclusion for this test. Choose the correct answer below. A. Reject H 0 . There is not sufficient evidence at the = 0.05 level of significance to conclude that a linear relation exists between x and y. B. Reject H 0 . There is sufficient evidence at the = 0.05 level of significance to conclude that a linear relation exists between x and y. C. Do not reject H 0 . There is sufficient evidence at the = 0.05 level of significance to conclude that a linear relation exists between x and y. D. Do not reject H 0 . There is not sufficient evidence at the = 0.05 level of significance to conclude that a linear relation exists between x and y. 4. For the data set shown below, complete parts (a) through (d) below. x y 20 30 40 98 95 93 50 60 85 72 (a) Use technology to find the estimates of 0 and 1 . 0 b 0 = 1 b 1 = (Round to two decimal places as needed.) (Round to two decimal places as needed.) (b) Use technology to compute the standard error, the point estimate for . se = (Round to four decimal places as needed.) (c) Assuming the residuals are normally distributed, use technology to determine s b . 1 sb = 1 (Round to four decimal places as needed.) (d) Assuming the residuals are normally distributed, test H 0 : 1 = 0 versus H 1 : 1 0 at the = 0.05 level of significance. Use the P-value approach. Determine the P-value for this hypothesis test. P-value = (Round to three decimal places as needed.) Which of the following conclusions is correct? A. Do not reject H 0 and conclude that a linear relation exists between x and y. B. Do not reject H 0 and conclude that a linear relation does not exist between x and y. C. Reject H 0 and conclude that a linear relation exists between x and y. D. Reject H 0 and conclude that a linear relation does not exist between x and y. 5. The data in the accompanying table represent the rate of return of a certain company stock for 11 months, compared with the rate of return of a certain index of 500 stocks. Both are in percent. Complete parts (a) through (d) below. 1 Click the icon to view the data table. (a) Treating the rate of return of the index as the explanatory variable, x, use technology to determine the estimates of 0 and 1 . The estimate of 0 is . (Round to four decimal places as needed.) The estimate of 1 is . (Round to four decimal places as needed.) (b) Assuming the residuals are normally distributed, test whether a linear relation exists between the rate of return of the index, x, and the rate of return for the company stock, y, at the = 0.10 level of significance. Choose the correct answer below. State the null and alternative hypotheses. A. H0 : 1 = 0 H 1 : 1 > 0 B. H0 : 1 = 0 H 1 : 1 0 C. H0 : 0 = 0 H 1 : 0 > 0 D. H0 : 0 = 0 H 1 : 0 0 Determine the P-value for this hypothesis test. P-value = (Round to three decimal places as needed.) State the appropriate conclusion at the = 0.10 level of significance. Choose the correct answer below. A. Do not reject H 0 . There is sufficient evidence to conclude that a linear relation exists between the rate of return of the index and the rate of return of the company stock. B. Do not reject H 0 . There is not sufficient evidence to conclude that a linear relation exists between the rate of return of the index and the rate of return of the company stock. C. Reject H 0 . There is not sufficient evidence to conclude that a linear relation exists between the rate of return of the index and the rate of return of the company stock. D. Reject H 0 . There is sufficient evidence to conclude that a linear relation exists between the rate of return of the index and the rate of return of the company stock. (c) Assuming the residuals are normally distributed, construct a 90% confidence interval for the slope of the true least-squares regression line. Lower bound: (Round to four decimal places as needed.) Upper bound: (Round to four decimal places as needed.) (d) What is the mean rate of return for the company stock if the rate of return of the index is 3.25%? The mean rate of return for the company stock if the rate of return of the index is 3.25% is (Round to three decimal places as needed.) %. 1: Rate of Return Month Rates of return of the index, x Rates of return of the company stock, y Apr-07 4.23 3.28 May-07 3.35 5.09 Jun-07 1.78 0.54 Jul-07 3.20 2.88 Aug-07 1.29 2.69 Sept-07 3.58 7.41 Oct-07 1.48 4.83 Nov-07 4.40 2.38 Dec-07 0.86 2.37 Jan-08 6.12 4.27 Feb-08 3.48 3.77 6. A doctor wanted to determine whether there is a relation between a male's age and his HDL (so-called good) cholesterol. The doctor randomly selected 17 of his patients and determined their HDL cholesterol. The data obtained by the doctor is the in the data table below. Complete parts (a) through (f) below. 2 Click the icon to view the data obtained by the doctor. (a) Draw a scatter diagram of the data, treating age as the explanatory variable. What type of relation, if any, appears to exist between age and HDL cholesterol? A. The relation appears to be linear. B. The relation appears to be nonlinear. C. There does not appear to be a relation. (b) Determine the least-squares regression equation from the sample data. y= x+ (Round to three decimal places as needed.) (c) Are there any outliers or influential observations? No Yes (d) Assuming the residuals are normally distributed, test whether a linear relation exists between age and HDL cholesterol levels at the = 0.01 level of significance. What are the null and alternative hypotheses? A. H0 : 1 = 0; H1 : 1 0 B. H0 : 1 = 0; H1 : 1 < 0 C. H0 : 1 = 0; H1 : 1 > 0 Use technology to compute the P-value. Use the Tech Help button for further assistance. The P-value is . (Round to three decimal places as needed.) What conclusion can be drawn at = 0.01 level of significance? A. Reject the null hypothesis because the P-value is less than = 0.01. B. Reject the null hypothesis because the P-value is greater than = 0.01. C. Do not reject the null hypothesis because the P-value is less than = 0.01. D. Do not reject the null hypothesis because the P-value is greater than = 0.01. (e) Assuming the residuals are normally distributed, construct a 95% confidence interval about the slope of the true least-squares regression line. Lower Bound = Upper Bound = (Round to three decimal places as needed.) (f) For a 42-year-old male patient who visits the doctor's office, would using the least-squares regression line obtained in part (b) to predict the HDL cholesterol of this patient be recommended? If the null hypothesis was rejected, that means that this least-squares regression line can accurately predict the HDL cholesterol of a patient. If the null hypothesis was not rejected, that means the least-squares regression line cannot accurately predict the HDL cholesterol of a patient. Should this least-squares regression line be used to predict the patient's HDL cholesterol? Choose the correct answer below. A. Yes, because the null hypothesis was rejected. B. No, because the null hypothesis was rejected. C. No, because the null hypothesis was not rejected. D. Yes, because the null hypothesis was not rejected. A good estimate for the HDL cholesterol of this patient is . (Round to two decimal places as needed.) 2: Age vs. HDL Cholesterol data Age, x HDL Cholesterol, y Age, x HDL Cholesterol, y 37 44 46 30 56 53 62 60 26 56 54 32 58 35 40 41 39 45 39 68 30 49 27 50 51 41 44 61 55 38 46 38 57 27 7. What do the y-coordinates on the least-squares regression line represent? Choose the correct answer below. A. The y-coordinates represent the mean value of the response variable for any given value of the explanatory variable. B. The y-coordinates represent the minimum expected value of the response variable for any given value of the explanatory variable. C. The y-coordinates represent the values of the explanatory variable. D. The y-coordinates represent the maximum expected value of the response variable for any given value of the explanatory variable. 8. Suppose a multiple regression model is given by y = 0.09x + 7.55x + 36.27. What would an interpretation of the 1 2 coefficient of x 1 be? Fill in the blank below. An interpretation of the coefficient of x 1 would be, "if x 1 decreases by 1 unit, then the response variable will increase by units, on average, while holding x 2 constant." 9. The multiple regression equation y = 10 x 4x is obtained from a set of sample data. Complete parts (a) through (e). 1 2 (a) Interpret the slope coefficient for x 1 . Choose the correct answer below. A. The slope coefficient of x is 1. This indicates that y will decrease 1 unit, for every one unit increase 1 in x 1 , x 2 remains constant. B. The slope coefficient of x is 1. This indicates that y will increase 1 unit, for every one unit increase in 1 x 1 , x 2 remains constant. C. The slope coefficient of x is 10. This indicates that y will increase 10 units, for every one unit increase 1 in x 1 , x 2 remains constant. Interpret the slope coefficient for x2 . Choose the correct answer below. A. The slope coefficient of x is 10. This indicates that y will increase 10 units, for every one unit increase 1 in x 1 , x 2 remains constant. B. The slope coefficient of x is 4. This indicates that y will decrease 1 / 4 units, for every one unit 2 increase in x 2 , x 1 remains constant. C. The slope coefficient of x is 4. This indicates that y will decrease 4 units, for every one unit increase 2 in x 2 , x 1 remains constant. (b) Determine the regression equation with x 1 = 10. Choose the correct answer below. y = 30 4x 2 y = 0 4x 2 y = 10 4x 1 y = 10 4x 2 Graph the regression equation with x 1 = 10. Choose the correct graph below. A. B. C. y y 80 x -80 80 -80 y 80 80 x -80 80 x -80 -80 (c) Determine the regression equation with x 1 = 15. Choose the correct answer below. y = 32 4x 2 y = 10 4x 2 y = 5 4x 2 80 -80 y = 10 4x 1 Graph the regression equation with x 1 = 15. Choose the correct graph below. y y 80 y 80 80 x -80 80 x -80 -80 80 x -80 -80 A. 80 -80 B. C. (d) Determine the regression equation with x 1 = 20. Choose the correct answer below. y = 10 4x 2 y = 10 4x 2 y = 10 4x 1 y = 45 4x 2 Graph the regression equation with x 1 = 20. Choose the correct graph below. A. B. C. y y 80 x -80 y 80 80 -80 80 x -80 80 -80 x -80 80 -80 (e) What is the effect of changing the value x 1 on the graph of the regression equation? Changes in the slope No changes Changes in the y-intercept 10. Determine if there is a linear relation among air temperature x , wind speed x , and wind chill y. The following data show 1 2 the measured values for various days. 0 15 5 15 25 45 35 15 25 50 25 x 2 20 y 22 44 16 48 32 4 52 31 10 x1 15 10 15 10 15 3 5 0 5 40 5 10 10 15 20 15 22 11 22 35 0 (a) Find the least-squares regression equation y = b 0 + b 1 x 1 + b 2 x 2 , where x 1 is air temperature and x 2 is wind speed, and y is the response variable, wind chill. y= + x1 + x 2 (Round to three decimal places as needed.) (b) Draw residual plots to assess the adequacy of the model. Create the residual plot for air temperature. Choose the correct graph below. A. B. C. y y 5 y 5 x -20 D. 20 x -20 -5 y 5 20 5 x -20 -5 20 x -20 -5 20 -5 Create the residual plot for wind speed. Choose the correct graph below. A. B. C. y y y 5 5 -5 5 x 60 0 -5 y 5 x 0 D. x 60 0 x 60 -5 0 60 -5 What might you conclude based on the plot of residuals against wind speed? Choose the correct answer below. A. The plot shows no discernable pattern, so the linear model is appropriate. B. The plot shows a discernable pattern, so the linear model is appropriate. C. The plot shows a discernable pattern, so the linear model is inappropriate. D. The plot shows no discernable pattern, so the linear model is inappropriate