Solved: The weight and systolic blood pressure of 26

An article in Concrete Research [Near Surface Characteristics of Concrete: Intrinsic Permeability (Vol. 41, 1989)] presented data on compressive strength x and intrinsic permeability y of various concrete mixes and cures. Summary quantities are n 14, gyi 572, gy2 i 23,530, g xi 43, g xi 2 (e) G157.42, and g xiyi 1697.80. Assume that the two variables are related according to the simple linear regression model. (a) Calculate the least squares estimates of the slope and intercept. Estimate 2 . Graph the regression line. (b) Use the equation of the fitted line to predict what permeability would be observed when the compressive strength is x 4.3. (c) Give a point estimate of the mean permeability when compressive strength is x 3.7. (d) Suppose that the observed value of permeability at x 3.7 is y 46.1. Calculate the value of the corresponding residual.

Regression methods were used to analyze the data from a study investigating the relationship between roadway surface temperature (x) and pavement deflection ( y). Summary quantities were n 20, g yi 12.75, g yi 2 8.86, g xi 1478, 143,215.8, and g xiyi 1083.67. (a) Calculate the least squares estimates of the slope and intercept. Graph the regression line. Estimate 2 . (b) Use the equation of the fitted line to predict what pavement deflection would be observed when the surface temperature is 85 F. (c) What is the mean pavement deflection when the surface temperature is 90 F? (d) What change in mean pavement deflection would be expected for a 1 F change in surface temperature? 11-3.

The following table presents data on the ratings of quarterbacks for the 2008 National Football League season (source: The Sports Network). It is suspected that the rating (y) is related to the average number of yards gained per pass attempt (x). (a) Calculate the least squares estimates of the slope and intercept. What is the estimate of ? Graph the regression model. (b) Find an estimate of the mean rating if a quarterback averages 7.5 yards per attempt. (c) What change in the mean rating is associated with a decrease of one yard per attempt? (d) To increase the mean rating by 10 points, how much increase in the average yards per attempt must be generated?(e) Given that yards, find the fitted value of y and the corresponding residual.

An article in Technometrics by S. C. Narula and J. F. Wellington [Prediction, Linear Regression, and a Minimum Sum of Relative Errors (Vol. 19, 1977)] presents data on the selling price and annual taxes for 24 houses. The data are shown in the following table. Taxes Sale (Local, School), Price/1000 County)/1000 25.9 4.9176 29.5 5.0208 27.9 4.5429 25.9 4.5573 29.9 5.0597 29.9 3.8910 30.9 5.8980 28.9 5.6039 35.9 5.8282 31.5 5.3003 31.0 6.2712 30.9 5.9592 Taxes Sale (Local, School), Price/1000 County)/1000 30.0 5.0500 36.9 8.2464 41.9 6.6969 40.5 7.7841 43.9 9.0384 37.5 5.9894 37.9 7.5422 44.5 8.7951 37.9 6.0831 38.9 8.3607 36.9 8.1400 45.8 9.1416 (a) Assuming that a simple linear regression model is appropriate, obtain the least squares fit relating selling price to taxes paid. What is the estimate of 2 ? (b) Find the mean selling price given that the taxes paid are x 7.50. (c) Calculate the fitted value of y corresponding to x 5.8980. Find the corresponding residual. (d) Calculate the fitted for each value of xi used to fit the model. Then construct a graph of versus the corresponding observed value yi and comment on what this plot would look like if the relationship between y and x was a deterministic (no random error) straight line. Does the plot actually obtained indicate that taxes paid is an effective regressor variable in predicting selling price?

The number of pounds of steam used per month by a chemical plant is thought to be related to the average ambient temperature (in F) for that month. The past years usage and temperature are shown in the following table: y i y i Month Temp. Usage/1000 Jan. 21 185.79 Feb. 24 214.47 Mar. 32 288.03 Apr. 47 424.84 May 50 454.58 June 59 539.03 Month Temp. Usage/1000 July 68 621.55 Aug. 74 675.06 Sept. 62 562.03 Oct. 50 452.93 Nov. 41 369.95 Dec. 30 273.98 (a) Assuming that a simple linear regression model is appropriate, fit the regression model relating steam usage (y) to the average temperature (x). What is the estimate of 2 ? Graph the regression line. (b) What is the estimate of expected steam usage when the average temperature is 55 F? (c) What change in mean steam usage is expected when the monthly average temperature changes by 1 F? (d) Suppose the monthly average temperature is 47 F. Calculate the fitted value of y and the corresponding residual.

The following table presents the highway gasoline mileage performance and engine displacement for DaimlerChrysler vehicles for model year 2005 (source: U.S. Environmental Protection Agency). (a) Fit a simple linear model relating highway miles per gallon ( y) to engine displacement (x) in cubic inches using least squares. (b) Find an estimate of the mean highway gasoline mileage performance for a car with 150 cubic inches engine displacement. (c) Obtain the fitted value of y and the corresponding residual for a car, the Neon, with an engine displacement of 122 cubic inches. Engine Displacement MPG Carline (in3 ) (highway) 300C/SRT-8 215 30.8 CARAVAN 2WD 201 32.5 CROSSFIRE ROADSTER 196 35.4 DAKOTA PICKUP 2WD 226 28.1 DAKOTA PICKUP 4WD 226 24.4 DURANGO 2WD 348 24.1 GRAND CHEROKEE 2WD 226 28.5 GRAND CHEROKEE 4WD 348 24.2 LIBERTY/CHEROKEE 2WD 148 32.8 LIBERTY/CHEROKEE 4WD 226 28 NEON/SRT-4/SX 2.0 122 41.3 PACIFICA 2WD 215 30.0 PACIFICA AWD 215 28.2 PT CRUISER 148 34.1 RAM 1500 PICKUP 2WD 500 18.7 RAM 1500 PICKUP 4WD 348 20.3 SEBRING 4-DR 165 35.1 STRATUS 4-DR 148 37.9 TOWN & COUNTRY 2WD 148 33.8 VIPER CONVERTIBLE 500 25.9 WRANGLER/TJ 4WD 148 26.4

An article in the Tappi Journal (March, 1986) presented data on green liquor Na2S concentration (in grams per liter) and paper machine production (in tons per day). The data (read from a graph) are shown as follows: y 40 42 49 46 44 48 x 825 830 890 895 890 910 y 46 43 53 52 54 57 58 x 915 960 990 1010 1012 1030 1050 (a) Fit a simple linear regression model with y green liquor Na2S concentration and x production. Find an estimate of 2 . Draw a scatter diagram of the data and the resulting least squares fitted model. (b) Find the fitted value of y corresponding to x 910 and the associated residual. (c) Find the mean green liquor Na2S concentration when the production rate is 950 tons per day. 1

An article in the Journal of Sound and Vibration (Vol. 151, 1991, pp. 383394) described a study investigating the relationship between noise exposure and hypertension. The following data are representative of those reported in the article. y 1012514 6 2 3 x 60 63 65 70 70 70 80 90 80 80 y 5468457 9 7 6 x 85 89 90 90 90 90 94 100 100 100 (a) Draw a scatter diagram of y (blood pressure rise in millimeters of mercury) versus x (sound pressure level in decibels). Does a simple linear regression model seem reasonable in this situation? (b) Fit the simple linear regression model using least squares. Find an estimate of 2 . (c) Find the predicted mean rise in blood pressure level associated with a sound pressure level of 85 decibels.

An article in Wear (Vol. 152, 1992, pp. 171181) presents data on the fretting wear of mild steel and oil viscosity. Representative data follow, with x oil viscosity and y wear volume ( cubic millimeters). 10 4 y 110 113 75 94 x 35.5 43.0 40.5 33.0 y 240 181 193 155 172 x 1.6 9.4 15.5 20.0 22.0 (a) Construct a scatter plot of the data. Does a simple linear regression model appear to be plausible? (b) Fit the simple linear regression model using least squares. Find an estimate of 2 . (c) Predict fretting wear when viscosity x 30. (d) Obtain the fitted value of y when x 22.0 and calculate the corresponding residual. 11-

An article in the Journal of Environmental Engineering (Vol. 115, No. 3, 1989, pp. 608619) reported the results of a study on the occurrence of sodium and chloride in surface streams in central Rhode Island. The following data are chloride concentration y (in milligrams per liter) and roadway area in the watershed x (in percentage). y 4.4 6.6 9.7 10.6 10.8 10.9 x 0.19 0.15 0.57 0.70 0.67 0.63 y 11.8 12.1 14.3 14.7 15.0 17.3 x 0.47 0.70 0.60 0.78 0.81 0.78 y 19.2 23.1 27.4 27.7 31.8 39.5 x 0.69 1.30 1.05 1.06 1.74 1.62 (a) Draw a scatter diagram of the data. Does a simple linear regression model seem appropriate here? (b) Fit the simple linear regression model using the method of least squares. Find an estimate of 2 . (c) Estimate the mean chloride concentration for a watershed that has 1% roadway area. (d) Find the fitted value corresponding to x 0.47 and the associated residual.

A rocket motor is manufactured by bonding together two types of propellants, an igniter and a sustainer. The shear strength of the bond y is thought to be a linear function of the age of the propellant x when the motor is cast. Twenty observations are shown in the following table. (a) Draw a scatter diagram of the data. Does the straight-line regression model seem to be plausible? (b) Find the least squares estimates of the slope and intercept in the simple linear regression model. Find an estimate of 2 . (c) Estimate the mean shear strength of a motor made from propellant that is 20 weeks old. (d) Obtain the fitted values that correspond to each observed value yi . Plot versus yi and comment on what this plot would look like if the linear relationship between shear strength and age were perfectly deterministic (no error). Does this plot indicate that age is a reasonable choice of regressor variable in this model?

An article in the Journal of the American Ceramic Society [Rapid Hot-Pressing of Ultrafine PSZ Powders (1991, Vol. 74, pp. 15471553)] considered the microstructure of the ultrafine powder of partially stabilized zirconia as a function of temperature. The data are shown below: x Temperature ( C): 1100 1200 1300 1100 1500 1200 1300 y Porosity (%): 30.8 19.2 6.0 13.5 11.4 7.7 3.6 (a) Fit the simple linear regression model using the method of least squares. Find an estimate of (b) Estimate the mean porosity for a temperature of 1400 C. (c) Find the fitted value corresponding to and the associated residual. (d) Draw a scatter diagram of the data. Does a simple linear regression model seem appropriate here? Explain.

An article in the Journal of the Environmental Engineering Division [Least Squares Estimates of BOD Parameters (1980, Vol. 106, pp. 11971202)] took a sample from the Holston River below Kingport, Tennessee, during August 1977. The biochemical oxygen demand (BOD) test is conducted over a period of time in days. The resulting data are shown below: Time (days): 1 2 4 6 8 10 12 14 16 18 20 BOD (mg/liter): 0.6 0.7 1.5 1.9 2.1 2.6 2.9 3.7 3.5 3.7 3.8 (a) Assuming that a simple linear regression model is appropriate, fit the regression model relating BOD (y) to the time (x). What is the estimate of ? (b) What is the estimate of expected BOD level when the time is 15 days? (c) What change in mean BOD is expected when the time changes by three days? (d) Suppose the time used is six days. Calculate the fitted value of y and the corresponding residual. (e) Calculate the fitted for each value of xi used to fit the model. Then construct a graph of versus the corresponding observed values yi and comment on what this plot would look like if the relationship between y and x was a deterministic (no random error) straight line. Does the plot actually obtained indicate that time is an effective regressor variable in predicting BOD?

An article in Wood Science and Technology [Creep in Chipboard, Part 3: Initial Assessment of the Influence of Moisture Content and Level of Stressing on Rate of Creep and Time to Failure (1981, Vol. 15, pp. 125144)] studied the deflection (mm) of particleboard from stress levels of relative humidity. Assume that the two variables are related according to the simple linear regression model. The data are shown below: x Stress level (%): 54 54 61 61 68 y Deflection (mm): 16.473 18.693 14.305 15.121 13.505 x Stress level (%): 68 75 75 75 y Deflection (mm): 11.640 11.168 12.534 11.224 (a) Calculate the least square estimates of the slope and intercept. What is the estimate of ? Graph the regression model and the data. (b) Find the estimate of the mean deflection if the stress level can be limited to 65%. (c) Estimate the change in the mean deflection associated with a 5% increment in stress level. (d) To decrease the mean deflection by one millimeter, how much increase in stress level must be generated? (e) Given that the stress level is 68%, find the fitted value of deflection and the corresponding residual.

In an article in Statistics and Computing [An Iterative Monte Carlo Method for Nonconjugate Bayesian Analysis (1991, pp. 119128)] Carlin and Gelfand investigated the age (x) and length (y) of 27 captured dugongs (sea cows).x 1.0, 1.5, 1.5, 1.5, 2.5, 4.0, 5.0, 5.0, 7.0, 8.0, 8.5, 9.0, 9.5, 9.5, 10.0, 12.0, 12.0, 13.0, 13.0, 14.5, 15.5, 15.5, 16.5, 17.0, 22.5, 29.0, 31.5 y 1.80, 1.85, 1.87, 1.77, 2.02, 2.27, 2.15, 2.26, 2.47, 2.19, 2.26, 2.40, 2.39, 2.41, 2.50, 2.32, 2.32, 2.43, 2.47, 2.56, 2.65, 2.47, 2.64, 2.56, 2.70, 2.72, 2.57 (a) Find the least squares estimates of the slope and the intercept in the simple linear regression model. Find an estimate of . (b) Estimate the mean length of dugongs at age 11. (c) Obtain the fitted values that correspond to each observed value yi . Plot versus yi , and comment on what this plot would look like if the linear relationship between length and age were perfectly deterministic (no error). Does this plot indicate that age is a reasonable choice of regressor variable in this model?

Consider the regression model developed in Exercise 11-2. (a) Suppose that temperature is measured in C rather than F. Write the new regression model. (b) What change in expected pavement deflection is associated with a 1 C change in surface temperature?

onsider the regression model developed in Exercise 11-6. Suppose that engine displacement is measured in cubic centimeters instead of cubic inches. yi yi 2 (a) Write the new regression model. (b) What change in gasoline mileage is associated with a 1 cm3 change is engine displacement?

Show that in a simple linear regression model the point ( ) lies exactly on the least squares regression line.

Consider the simple linear regression model Y 0  1x  . Suppose that the analyst wants to use z x as the regressor variable. (a) Using the data in Exercise 11-11, construct one scatter plot of the ( ) points and then another of the xi ( ) points. Use the two plots to intuitively explain how the two models, Y 0 1x and , are related. (b) Find the least squares estimates of and in the model . How do they relate to the least squares estimates and ?

Suppose we wish to fit a regression model for which the true regression line passes through the point (0, 0). The appropriate model is Y x  . Assume that we have n pairs of data (x1, y1), (x2, y2), p , (xn, yn). (a) Find the least squares estimate of . (b) Fit the model Y x   to the chloride concentrationroadway area data in Exercise 11-10. Plot the fitted model on a scatter diagram of the data and comment on the appropriateness of the model.

Consider the computer output below. The regression equation is Y 12.9  2.34 x Predictor Coef SE Coef T P Constant 12.857 1.032 ? ? X 2.3445 0.1150 ? ? S 1.48111 R Sq 98.1% R Sq(adj) 97.9% Analysis of Variance Source DF SS MS F P Regression 1 912.43 912.43 ? ? Residual Error 8 17.55 ? Total 9 929.98 (a) Fill in the missing information. You may use bounds for the P-values. (b) Can you conclude that the model defines a useful linear relationship? (c) What is your estimate of 2 ? 11-

Consider the computer output below. The regression equation is Y = 26.8  1.48 x Predictor Coef SE Coef T P Constant 26.753 2.373 ? ? X 1.4756 0.1063 ? ? S 2.70040 R Sq 93.7% R-Sq (adj) 93.2% Analysis of Variance Source DF SS MS F P Regression 1 ? ? ? ? Residual Error ? 94.8 7.3 Total 15 1500.0 (a) Fill in the missing information. You may use bounds for the P-values. (b) Can you conclude that the model defines a useful linear relationship? (c) What is your estimate of 2 ? 11

Consider the data from Exercise 11-1 on x compressive strength and y intrinsic permeability of concrete. (a) Test for significance of regression using  0.05. Find the P-value for this test. Can you conclude that the model specifies a useful linear relationship between these two variables? (b) Estimate 2 and the standard deviation of (c) What is the standard error of the intercept in this model? 11

Consider the data from Exercise 11-2 on x roadway surface temperature and y pavement deflection. (a) Test for significance of regression using  0.05. Find the P-value for this test. What conclusions can you draw? (b) Estimate the standard errors of the slope and intercept. 11

Consider the National Football League data in Exercise 11-3. (a) Test for significance of regression using . Find the P-value for this test. What conclusions can you draw? (b) Estimate the standard errors of the slope and intercept. (c) Test versus with . Would you agree with the statement that this is a test of the hypothesis that a one-yard increase in the average yards per attempt results in a mean increase of 10 rating points?

Consider the data from Exercise 11-4 on y sales price and x taxes paid. (a) Test H0: 1 0 using the t-test; use  0.05. (b) Test H0: 1 0 using the analysis of variance with  0.05. Discuss the relationship of this test to the test from part (a). H0: (c) Estimate the standard errors of the slope and intercept. (d) Test the hypothesis that 0 0.

Consider the data from Exercise 11-5 on y steam usage and x average temperature. (a) Test for significance of regression using  0.01. What is the P-value for this test? State the conclusions that result from this test. (b) Estimate the standard errors of the slope and intercept. (c) Test the hypothesis H0: 1 10 versus H1: 1 10 using  0.01. Find the P-value for this test. (d) Test H0: 0 0 versus H1: 0 0 using  0.01. Find the P-value for this test and draw conclusions. 11-28. Consi

Consider the data from Exercise 11-6 on y highway gasoline mileage and x engine displacement. (a) Test for significance of regression using  0.01. Find the P-value for this test. What conclusions can you reach? (b) Estimate the standard errors of the slope and intercept. (c) Test H0: 1 0.05 versus H1: 1  0.05 using  0.01 and draw conclusions. What is the P-value for this test? (d) Test the hypothesis H0: 0 0 versus H1: 0 0 using  0.01. What is the P-value for this test? 11-29. Cons

Consider the data from Exercise 11-7 on y green liquor Na2S concentration and x production in a paper mill. (a) Test for significance of regression using  0.05. Find the P-value for this test. (b) Estimate the standard errors of the slope and intercept. (c) Test H0: 0 0 versus H1: 0 0 using  0.05. What is the P-value for this test? 11-30.

Consider the data from Exercise 11-8 on y blood pressure rise and x sound pressure level. (a) Test for significance of regression using  0.05. What is the P-value for this test? (b) Estimate the standard errors of the slope and intercept. (c) Test H0: 0 0 versus H1: 0 0 using  0.05. Find the P-value for this test. 11-31.

Consider the data from Exercise 11-11, on y shear strength of a propellant and x propellant age. (a) Test for significance of regression with  0.01. Find the P-value for this test. (b) Estimate the standard errors of and (c) Test H0: 1 30 versus H1: 1 30 using  0.01. What is the P-value for this test? (d) Test H0: 0 0 versus H1: 0 0 using  0.01. What is the P-value for this test? (e) Test H0: 0 2500 versus H1: 0 2500 using  0.01. What is the P-value for this test? 11-32. Consider

Consider the data from Exercise 11-10 on y chloride concentration in surface streams and x roadway area. (a) Test the hypothesis H0: 1 0 versus H1: 1 0 using the analysis of variance procedure with  0.01. (b) Find the P-value for the test in part (a). (c) Estimate the standard errors of and 0 . 1 (d) Test H0: 0 0 versus H1: 0 0 using 0.01. What conclusions can you draw? Does it seem that the model might be a better fit to the data if the intercept were removed?

Consider the data in Exercise 11-13 on and . (a) Test for significance of regression using . Find the P-value for this test. What conclusions can you draw? (b) Estimate the standard errors of the slope and intercept. (c) Test the hypothesis that

Consider the data in Exercise 11-14 on and . (a) Test for significance of regression using . What is the P-value for this test? State the conclusions that result from this test. (b) Does this model appear to be adequate? (c) Estimate the standard errors of the slope and intercept.

An article in The Journal of Clinical Endocrinology and Metabolism [Simultaneous and Continuous 24-Hour Plasma and Cerebrospinal Fluid Leptin Measurements: Dissociation of Concentrations in Central and Peripheral Compartments (2004, Vol. 89, pp. 258265)] studied the demographics of simultaneous and continuous 24-hour plasma and cerebrospinal fluid leptin measurements. The data follow: y BMI (kg/m2 ): 19.92 20.59 29.02 20.78 25.97 20.39 23.29 17.27 35.24 x Age (yr): 45.5 34.6 40.6 32.9 28.2 30.1 52.1 33.3 47.0 (a) Test for significance of regression using . Find the P-value for this test. Can you conclude that the model specifies a useful linear relationship between these two variables? (b) Estimate and the standard deviation of . (c) What is the standard error of the intercept in this model? 1

Suppose that each value of xi is multiplied by a positive constant a, and each value of yi is multiplied by another positive constant b. Show that the t-statistic for testing H0: 1 0 versus H1: 1 0 is unchanged in value. 11-

The type II error probability for the t-test for H0: 1 1,0 can be computed in a similar manner to the t-tests of Chapter 9. If the true value of 1 is 1, the value is calculated and used as the horizontal scale factor on the operating characteristic curves for the t-test (Appendix Charts VIIe through VIIh) and the type II error probability is read from the vertical scale using the curve for n 2 degrees of freedom. Apply this procedure to the football data of Exercise 11-3, using  5.5 and 1 12.5, where the hypotheses are H0: 1 10 versus H1: 1 10. 11-38. Con

Consider the no-intercept model Y x   with the s NID(0, 2 ). The estimate of 2 is s 2 gn i1 and V 2 gn i1 (a) Devise a test statistic for H0: 0 versus H1: 0. (b) Apply the test in (a) to the model from Exercise 11-20. x 2 i 1 .

Refer to the data in Exercise 11-1 on y intrinsic permeability of concrete and x compressive strength. Find a 95% confidence interval on each of the following: (a) Slope (b) Intercept (c) Mean permeability when x 2.5 (d) Find a 95% prediction interval on permeability when x 2.5. Explain why this interval is wider than the interval in part (c). 11-

Exercise 11-2 presented data on roadway surface temperature x and pavement deflection y. Find a 99% confi- dence interval on each of the following: (a) Slope (b) Intercept (c) Mean deflection when temperature (d) Find a 99% prediction interval on pavement deflection when the temperature is .

Refer to the NFL quarterback ratings data in Exercise 11-3. Find a 95% confidence interval on each of the following: (a) Slope (b) Intercept (c) Mean rating when the average yards per attempt is 8.0 (d) Find a 95% prediction interval on the rating when the average yards per attempt is 8.0.

Refer to the data on y house selling price and x taxes paid in Exercise 11-4. Find a 95% confidence interval on each of the following: (a) 1 (b) 0 (c) Mean selling price when the taxes paid are x 7.50 (d) Compute the 95% prediction interval for selling price when the taxes paid are x 7.50. 11-

Exercise 11-5 presented data on y steam usage and x monthly average temperature. 90F x 85F (a) Find a 99% confidence interval for 1. (b) Find a 99% confidence interval for 0. (c) Find a 95% confidence interval on mean steam usage when the average temperature is . (d) Find a 95% prediction interval on steam usage when temperature is . Explain why this interval is wider than the interval in part (c). 11

Exercise 11-6 presented gasoline mileage performance for 21 cars, along with information about the engine displacement. Find a 95% confidence interval on each of the following: (a) Slope (b) Intercept (c) Mean highway gasoline mileage when the engine displacement is x 150 in3 (d) Construct a 95% prediction interval on highway gasoline mileage when the engine displacement is x 150 in3 .

Consider the data in Exercise 11-7 on y green liquor Na2S concentration and x production in a paper mill. Find a 99% confidence interval on each of the following: (a) 1 (b) 0 (c) Mean Na2S concentration when production x 910 tonsday (d) Find a 99% prediction interval on Na2S concentration when x 910 tonsday. 11-46

Exercise 11-8 presented data on y blood pressure rise and x sound pressure level. Find a 95% confidence interval on each of the following: (a) 1 (b) 0 (c) Mean blood pressure rise when the sound pressure level is 85 decibels (d) Find a 95% prediction interval on blood pressure rise when the sound pressure level is 85 decibels. 5

Refer to the data in Exercise 11-9 on y wear volume of mild steel and x oil viscosity. Find a 95% confi- dence interval on each of the following: (a) Intercept (b) Slope (c) Mean wear when oil viscosity x 30 1

Exercise 11-10 presented data on chloride concentration y and roadway area x on watersheds in central Rhode Island. Find a 99% confidence interval on each of the following: (a) 1 (b) 0 (c) Mean chloride concentration when roadway area x 1.0% (d) Find a 99% prediction interval on chloride concentration when roadway area x 1.0%. 11-

Refer to the data in Exercise 11-11 on rocket motor shear strength y and propellant age x. Find a 95% confidence interval on each of the following: (a) Slope 1 (b) Intercept 0 (c) Mean shear strength when age x 20 weeks (d) Find a 95% prediction interval on shear strength when age x 20 weeks.

Refer to the data in Exercise 11-12 on the microstructure of zirconia. Find a 95% confidence interval on each of the following: (a) Slope (b) Intercept (c) Mean length when (d) Find a 95% prediction interval on length when Explain why this interval is wider than the interval in part (c).

Refer to the data in Exercise 11-13 on oxygen demand. Find a 99% confidence interval on each of the following: (a) (b) (c) Find a 95% confidence interval on mean BOD when the time is 8 days.

Refer to the compressive strength data in Exercise 11-1. Use the summary statistics provided to calculate R2 and provide a practical interpretation of this quantity.

Refer to the NFL quarterback ratings data in Exercise 11-3. (a) Calculate R2 for this model and provide a practical interpretation of this quantity. (b) Prepare a normal probability plot of the residuals from the least squares model. Does the normality assumption seem to be satisfied? (c) Plot the residuals versus the fitted values and against x. Interpret these graphs

Refer to the data in Exercise 11-4 on house selling price y and taxes paid x. (a) Find the residuals for the least squares model. (b) Prepare a normal probability plot of the residuals and interpret this display. (c) Plot the residuals versus and versus x. Does the assumption of constant variance seem to be satisfied? (d) What proportion of total variability is explained by the regression model?

Refer to the data in Exercise 11-5 on y steam usage and x average monthly temperature. (a) What proportion of total variability is accounted for by the simple linear regression model? (b) Prepare a normal probability plot of the residuals and interpret this graph. (c) Plot residuals versus and x. Do the regression assumptions appear to be satisfied?

Refer to the gasoline mileage data in Exercise 11-6. (a) What proportion of total variability in highway gasoline mileage performance is accounted for by engine displacement? (b) Plot the residuals versus and y x, and comment on the graphs.(c) Prepare a normal probability plot of the residuals. Does the normality assumption appear to be satisfied?

Exercise 11-9 presents data on wear volume y and oil viscosity x. (a) Calculate R2 for this model. Provide an interpretation of this quantity. (b) Plot the residuals from this model versus and versus x. Interpret these plots. (c) Prepare a normal probability plot of the residuals. Does the normality assumption appear to be satisfied?

Refer to Exercise 11-8, which presented data on blood pressure rise y and sound pressure level x. (a) What proportion of total variability in blood pressure rise is accounted for by sound pressure level? (b) Prepare a normal probability plot of the residuals from this least squares model. Interpret this plot. (c) Plot residuals versus and versus x. Comment on these plots

Refer to Exercise 11-10, which presented data on chloride concentration y and roadway area x. (a) What proportion of the total variability in chloride concentration is accounted for by the regression model? (b) Plot the residuals versus and versus x. Interpret these plots. (c) Prepare a normal probability plot of the residuals. Does the normality assumption appear to be satisfied?

An article in the Journal of the American Statistical Association [Markov Chain Monte Carlo Methods for y y y Computing Bayes Factors: A Comparative Review (2001, Vol. 96, pp. 11221132)] analyzed the tabulated data on compressive strength parallel to the grain versus resin-adjusted density for specimens of radiata pine. (a) Fit a regression model relating compressive strength to density. (b) Test for significance of regression with . (c) Estimate for this model. (d) Calculate R2 for this model. Provide an interpretation of this quantity. (e) Prepare a normal probability plot of the residuals and interpret this display. (f ) Plot the residuals versus and versus x. Does the assumption of constant variance seem to be satisfied?

Consider the rocket propellant data in Exercise 11-11. (a) Calculate R2 for this model. Provide an interpretation of this quantity. (b) Plot the residuals on a normal probability scale. Do any points seem unusual on this plot? (c) Delete the two points identified in part (b) from the sample and fit the simple linear regression model to the remaining 18 points. Calculate the value of R2 for the new model. Is it larger or smaller than the value of R2 computed in part (a)? Why? (d) Did the value of change dramatically when the two points identified above were deleted and the model fit to the remaining points? Why?

Consider the data in Exercise 11-7 on y green liquor Na2S concentration and x paper machine production. Suppose that a 14th sample point is added to the original data, where y14 59 and x14 855. (a) Prepare a scatter diagram of y versus x. Fit the simple linear regression model to all 14 observations. (b) Test for significance of regression with  0.05. (c) Estimate 2 for this model. (d) Compare the estimate of 2 obtained in part (c) above with the estimate of 2 obtained from the original 13 points. Which estimate is larger and why? (e) Compute the residuals for this model. Does the value of e14 appear unusual? (f) Prepare and interpret a normal probability plot of the residuals. (g) Plot the residuals versus and versus x. Comment on these graphs. 11-

Consider the rocket propellant data in Exercise 11-11. Calculate the standardized residuals for these data. Does this provide any helpful information about the magnitude of the residuals?

Studentized Residuals. Show that the variance of the ith residual is V1ei2 2 c 1 a 1 n  1xi x2 2 Sxx b dHint: The ith studentized residual is defined as (a) Explain why ri has unit standard deviation. (b) Do the standardized residuals have unit standard deviation? (c) Discuss the behavior of the studentized residual when the sample value xi is very close to the middle of the range of x. ri ei B 2 c 1 a 1 n 1xi x 2 2 Sxx b d cov1Yi , Y i2 2 c 1 n 1xi x2 2 Sxx d . (d) Discuss the behavior of the studentized residual when the sample value xi is very near one end of the range of x.

Show that an equivalent way to define the test for significance of regression in simple linear regression is to base the test on R2 as follows: to test H0: 1 0 versus H1: 1 0, calculate and to reject H0: 1 0 if the computed value f0 f,1,n 2. Suppose that a simple linear regression model has been fit to n 25 observations and R2 0.90. (a) Test for significance of regression at  0.05. (b) What is the smallest value of R2 that would lead to the conclusion of a significant regression if  0.05? F0 R21n

Suppose data is obtained from 20 pairs of (x, y) and the sample correlation coefficient is 0.8. (a) Test the hypothesis that against with . Calculate the P-value. (b) Test the hypothesis that against with . Calculate the P-value. (c) Construct a 95% two-sided confidence interval for the correlation coefficient. Explain how the questions in parts (a) and (b) could be answered with a confidence interval.

Suppose data are obtained from 20 pairs of (x, y) and the sample correlation coefficient is 0.75. (a) Test the hypothesis that against with . Calculate the P-value. (b) Test the hypothesis that against with . Calculate the P-value. (c) Construct a 95% one-sided confidence interval for the correlation coefficient. Explain how the questions in parts (a) and (b) could be answered with a confidence interval.

A random sample of n 25 observations was made on the time to failure of an electronic component and the temperature in the application environment in which the component was used. (a) Given that r 0.83, test the hypothesis that  0, using  0.05. What is the P-value for this test? (b) Find a 95% confidence interval on . (c) Test the hypothesis H0:  0.8 versus H1:  0.8, using  0.05. Find the P-value for this test. 11-69.

A random sample of 50 observations was made on the diameter of spot welds and the corresponding weld shear strength. (a) Given that r 0.62, test the hypothesis that  0, using  0.01. What is the P-value for this test? (b) Find a 99% confidence interval for . (c) Based on the confidence interval in part (b), can you conclude that  0.5 at the 0.01 level of significance? 11-

The following data gave X the water content of snow on April 1 and Y the yield from April to July (in inches) on the Snake River watershed in Wyoming for 1919 to 1935. (The data were taken from an article in Research Notes, Vol. 61, 1950, Pacific Northwest Forest Range Experiment Station, Oregon.) xyxy 23.1 10.5 37.9 22.8 32.8 16.7 30.5 14.1 31.8 18.2 25.1 12.9 32.0 17.0 12.4 8.8 30.4 16.3 35.1 17.4 24.0 10.5 31.5 14.9 39.5 23.1 21.1 10.5 24.2 12.4 27.6 16.1 52.5 24.9 (a) Estimate the correlation between Y and X. (b) Test the hypothesis that 0, using 0.05. (c) Fit a simple linear regression model and test for signifi- cance of regression using 0.05. What conclusions can you draw? How is the test for significance of regression related to the test on in part (b)? (d) Analyze the residuals and comment on model adequacy.

The final test and exam averages for 20 randomly selected students taking a course in engineering statistics and a course in operations research follow. Assume that the final averages are jointly normally distributed. (a) Find the regression line relating the statistics final average to the OR final average. Graph the data. (b) Test for significance of regression using  0.05. (c) Estimate the correlation coefficient. (d) Test the hypothesis that  0, using  0.05. (e) Test the hypothesis that  0.5, using  0.05. (f) Construct a 95% confidence interval for the correlation coefficient. Statistics OR Statistics OR Statistics OR 86 80 86 81 83 81 75 81 71 76 75 70 69 75 65 72 71 73 75 81 84 85 76 72 90 92 71 72 84 80 94 95 62 65 97 98 83 80 90 93 JWCL

The weight and systolic blood pressure of 26 randomly selected males in the age group 25 to 30 are shown in the following table. Assume that weight and blood pressure are jointly normally distributed.Systolic Subject Weight BP 1 165 130 2 167 133 3 180 150 4 155 128 5 212 151 6 175 146 7 190 150 8 210 140 9 200 148 10 149 125 11 158 133 12 169 135 13 170 150 Systolic Subject Weight BP 14 172 153 15 159 128 16 168 132 17 174 149 18 183 158 19 215 150 20 195 163 21 180 156 22 143 124 23 240 170 24 235 165 25 192 160 26 187 159 (a) Find a regression line relating systolic blood pressure to weight. (b) Test for significance of regression using 0.05. (c) Estimate the correlation coefficient. (d) Test the hypothesis that 0, using 0.05. (e) Test the hypothesis that 0.6, using 0.05. (f) Construct a 95% confidence interval for the correlation coefficient

In an article in IEEE Transactions on Instrumentation and Measurement (2001, Vol. 50, pp. 986990), researchers studied the effects of reducing current draw in a magnetic core by electronic means. They measured the current in a magnetic winding with and without the electronics in a paired experiment. Data for the case without electronics are provided in the following table. Current Without Supply Voltage Electronics (mA) 0.66 7.32 1.32 12.22 1.98 16.34 2.64 23.66 3.3 28.06 3.96 33.39 4.62 34.12 3.28 39.21 5.94 44.21 6.6 47.48(a) Graph the data and fit a regression line to predict current without electronics to supply voltage. Is there a significant regression at ? What is the P-value? (b) Estimate the correlation coefficient. (c) Test the hypothesis that against the alternative with . What is the P-value? (d) Compute a 95% confidence interval for the correlation coefficient.

The monthly absolute estimate of global (land and ocean combined) temperature indexes (degrees C) in 2000 and 2001 are (source: http://www.ncdc.noaa.gov/ oa/climate/): 2000: 12.28, 12.63, 13.22, 14.21, 15.13, 15.82, 16.05, 16.02, 15.29, 14.29, 13.16, 12.47 2001: 12.44, 12.55, 13.35, 14.22, 15.28, 15.99, 16.23, 16.17, 15.44, 14.52, 13.52, 12.61 (a) Graph the data and fit a regression line to predict 2001 temperatures from those in 2000. Is there a significant regression at ? What is the P-value? (b) Estimate the correlation coefficient. (c) Test the hypothesis that against the alternative with . What is the P-value? (d) Compute a 95% confidence interval for the correlation coefficient.

Refer to the NFL quarterback ratings data in Exercise 11-3. (a) Estimate the correlation coefficient between the ratings and the average yards per attempt. (b) Test the hypothesis versus using . What is the P-value for this test? (c) Construct a 95% confidence interval for . (d) Test the hypothesis versus using . Find the P-value for this test

Consider the following (x, y) data. Calculate the correlation coefficient. Graph the data and comment on the relationship between x and y. Explain why the correlation coefficient does not detect the relationship between x and y.

Determine if the following models are intrinsically linear. If yes, determine the appropriate transformation to generate the linear model.

The vapor pressure of water at various temperatures follows:(a) Draw a scatter diagram of these data. What type of relationship seems appropriate in relating y to x?(b) Fit a simple linear regression model to these data. (c) Test for significance of regression using 0.05. What conclusions can you draw? (d) Plot the residuals from the simple linear regression model versus . What do you conclude about model adequacy? yi Customer x y Customer x y 1 679 0.79 26 1434 0.31 2 292 0.44 27 837 4.20 3 1012 0.56 28 1748 4.88 4 493 0.79 29 1381 3.48 5 582 2.70 30 1428 7.58 6 1156 3.64 31 1255 2.63 7 997 4.73 32 1777 4.99 8 2189 9.50 33 370 0.59 9 1097 5.34 34 2316 8.19 10 2078 6.85 35 1130 4.79 continued (e) The ClausisClapeyron relationship states that ln where is the vapor pressure of water. Repeat parts (a)(d). using an appropriate transformation.

An electric utility is interested in developing a model relating peak hour demand ( y in kilowatts) to total monthly energy usage during the month (x, in kilowatt hours). Data for 50 residential customers are shown in the following table.Customer x y Customer x y 11 1818 5.84 36 463 0.51 12 1700 5.21 37 770 1.74 13 747 3.25 38 724 4.10 14 2030 4.43 39 808 3.94 15 1643 3.16 40 790 0.96 16 414 0.50 41 783 3.29 17 354 0.17 42 406 0.44 18 1276 1.88 43 1242 3.24 19 745 0.77 44 658 2.14 20 795 3.70 45 1746 5.71 21 540 0.56 46 895 4.12 22 874 1.56 47 1114 1.90 23 1543 5.28 48 413 0.51 24 1029 0.64 49 1787 8.33 25 710 4.00 50 3560 14.94 (a) Draw a scatter diagram of y versus x. (b) Fit the simple linear regression model. (c) Test for significance of regression using 0.05. (d) Plot the residuals versus and comment on the underlying regression assumptions. Specifically, does it seem that the equality of variance assumption is satisfied? (e) Find a simple linear regression model using as the response. Does this transformation on y stabilize the inequality of variance problem noted in part (d) above?

A study was conducted attempting to relate home ownership to family income. Twenty households were selected and family income was estimated, along with information concerning home ownership (y 1 indicates yes and y 0 indicates no). The data are shown below. Home Ownership Household Income Status 1 38,000 0 2 51,200 1 3 39,600 0 4 43,400 1 5 47,700 0 6 53,000 0 7 41,500 1 8 40,800 0 9 45,400 1 10 52,400 1 11 38,700 1 12 40,100 0 13 49,500 1 14 38,000 0 15 42,000 1 16 54,000 1 17 51,700 1 18 39,400 0 19 40,900 0 20 52,800 1(a) Fit a logistic regression model to the response variable y. Use a simple linear regression model as the structure for the linear predictor. (b) Is the logistic regression model in part (a) adequate? (c) Provide an interpretation of the parameter 1 in this model.

The compressive strength of an alloy fastener used in aircraft construction is being studied. Ten loads were selected over the range 2500 4300 psi and a number of fasteners were tested at those loads. The numbers of fasteners failing at each load were recorded. The complete test data follow.Load, x (psi) Sample Size, n Number Failing, r 2500 50 10 2700 70 17 2900 100 30 3100 60 21 3300 40 18 3500 85 43 3700 90 54 3900 50 33 4100 80 60 4300 65 51 (a) Fit a logistic regression model to the data. Use a simple linear regression model as the structure for the linear predictor. (b) Is the logistic regression model in part (a) adequate?

The market research department of a soft drink manufacturer is investigating the effectiveness of a price discount coupon on the purchase of a two-liter beverage product. A sample of 5500 customers was given coupons for varying price discounts between 5 and 25 cents. The response variable was the number of coupons in each price discount category redeemed after one month. The data are shown below. (a) Fit a logistic regression model to the data. Use a simple linear regression model as the structure for the linear predictor. (b) Is the logistic regression model in part (a) adequate? (c) Draw a graph of the data and the fitted logistic regression model.(d) Expand the linear predictor to include a quadratic term. Is there any evidence that this quadratic term is required in the model? (e) Draw a graph of this new model on the same plot that you prepared in part (c). Does the expanded model visually provide a better fit to the data than the original model from part (a)?

A study was performed to investigate new automobile purchases. A sample of 20 families was selected. Each family was surveyed to determine the age of their oldest vehicle and their total family income. A follow-up survey was conducted six months later to determine if they had actually purchased a new vehicle during that time period ( y = 1 indicates yes and y = 0 indicates no). The data from this study are shown in the following table. Income, x1 Age, x2 y Income, x1 Age, x2 y 45,000 2 0 37,000 5 1 40,000 4 0 31,000 7 1 60,000 3 1 40,000 4 1 50,000 2 1 75,000 2 0 55,000 2 0 43,000 9 1 50,000 5 1 49,000 2 0 35,000 7 1 37,500 4 1 65,000 2 1 71,000 1 0 53,000 2 0 34,000 5 0 48,000 1 0 27,000 6 0 (a) Fit a logistic regression model to the data. (b) Is the logistic regression model in part (a) adequate? (c) Interpret the model coefficients 1 and 2. (d) What is the estimated probability that a family with an income of $45,000 and a car that is five years old will purchase a new vehicle in the next six months? (e) Expand the linear predictor to include an interaction term. Is there any evidence that this term is required in the model?

Show that, for the simple linear regression model, the following statements are true: (a) (b) (c)

An article in the IEEE Transactions on Instrumentation and Measurement [Direct, Fast, and Accurate Measurement of VT and K of MOS Transistor Using VT-Sift Circuit (1991, Vol. 40, pp. 951955)] described the use of a simple linear regression model to express drain current y (in milliamperes) as a function of ground-to-source voltage x (in volts). The data are as follows:(a) Draw a scatter diagram of these data. Does a straight-line relationship seem plausible? (b) Fit a simple linear regression model to these data. (c) Test for significance of regression using 0.05. What is the P-value for this test? (d) Find a 95% confidence interval estimate on the slope. (e) Test the hypothesis H0: 0 0 versus H1: 0 0 using 0.05. What conclusions can you draw?

The strength of paper used in the manufacture of cardboard boxes ( y) is related to the percentage of hardwood concentration in the original pulp (x). Under controlled conditions, a pilot plant manufactures 16 samples, each from a different batch of pulp, and measures the tensile strength. The data are shown in the table that follows:(a) Fit a simple linear regression model to the data. (b) Test for significance of regression using 0.05. (c) Construct a 90% confidence interval on the slope 1. (d) Construct a 90% confidence interval on the intercept 0. (e) Construct a 95% confidence interval on the mean strength at x 2.5. (f) Analyze the residuals and comment on model adequacy.

Consider the following data. Suppose that the relationship between Y and x is hypothesized to be Y (0  1x  ) 1 . Fit an appropriate model to the data. Does the assumed model form seem reasonable? x 10 15 18 12 9 8 11 6 y 0.1 0.13 0.09 0.15 0.20 0.21 0.18 0.24 y

The following data, adapted from Montgomery, Peck, and Vining (2006), present the number of certified mental defectives per 10,000 of estimated population in the United Kingdom ( y) and the number of radio receiver licenses issued (x) by the BBC (in millions) for the years 1924 through 1937. Fit a regression model relating y and x. Comment on the model. Specifically, does the existence of a strong correlation imply a cause-and-effect relationship?

Consider the weight and blood pressure data in Exercise 11-72. Fit a no-intercept model to the data, and compare it to the model obtained in Exercise 11-70. Which model is superior?

An article in Air and Waste [Update on Ozone Trends in Californias South Coast Air Basin (Vol. 43, 1993)] studied the ozone levels on the South Coast air basin of California for the years 19761991. The author believes that the number of days that the ozone level exceeds 0.20 parts per million depends on the seasonal meteorological index (the seasonal average 850 millibar temperature). The data follow: (a) Construct a scatter diagram of the data. (b) Fit a simple linear regression model to the data. Test for significance of regression. (c) Find a 95% CI on the slope (d) Analyze the residuals and comment on model adequacy

An article in the Journal of Applied Polymer Science (Vol. 56, pp. 471476, 1995) studied the effect of the mole. ratio of sebacic acid on the intrinsic viscosity of copolyesters. The data follow:(a) Construct a scatter diagram of the data. (b) Fit a simple linear repression model. (c) Test for significance of regression. Calculate R2 for the model. (d) Analyze the residuals and comment on model adequacy.

Two different methods can be used for measuring the temperature of the solution in a Hall cell used in aluminum smelting, a thermocouple implanted in the cell and an indirect measurement produced from an IR device. The indirect method is preferable because the thermocouples are eventually destroyed by the solution. Consider the following 10 measurements:(a) Construct a scatter diagram for these data, letting x thermocouple measurement and y IR measurement. (b) Fit a simple linear regression model. (c) Test for significance a regression and calculate R2 . What conclusions can you draw? (d) Is there evidence to support a claim that both devices produce equivalent temperature measurements? Formulate and test an appropriate hypothesis to support this claim. (e) Analyze the residuals and comment on model adequacy.

The grams of solids removed from a material ( y) is thought to be related to the drying time. Ten observations obtained from an experimental study follow: (a) Construct a scatter diagram for these data. (b) Fit a simple linear regression model. (c) Test for significance of regression. (d) Based on these data, what is your estimate of the mean grams of solids removed at 4.25 hours? Find a 95% confidence interval on the mean. (e) Analyze the residuals and comment on model adequacy.

Cesium atoms cooled by laser light could be used to build inexpensive atomic clocks. In a study in IEEE Transactions on Instrumentation and Measurement (2001, Vol. 50, pp. 12241228), the number of atoms cooled by lasers of various powers were counted.(a) Graph the data and fit a regression line to predict the number of atoms from laser power. Comment on the adequacy of a linear model. (b) Is there a significant regression at ? What is the P-value? (c) Estimate the correlation coefficient. (d) Test the hypothesis that against the alternative with . What is the P-value? (e) Compute a 95% confidence interval for the slope coefficient.

The following data related diamond carats to purchase prices. It appeared in Singapores Business Times, February 18, 2000.  0  0.05  0  0.05 (a) Graph the data. What is the relation between carat and price? Is there an outlier? (b) What would you say to the person who purchased the diamond that was an outlier? (c) Fit two regression models, one with all the data and the other with unusual data omitted. Estimate the slope coeffi- cient with a 95% confidence interval in both cases. Comment on any difference. 1

The following table shows the population and the average count of wood storks sighted per sample period for South Carolina from 1991 to 2004. Fit a regression line with population as the response and the count of wood storks as the predictor. Such an analysis might be used to evaluate the relationship between storks and babies. Is regression significant at ? What do you conclude about the role of regression analysis to establish a cause-and-effect relationship?

Suppose that we have n pairs of observations (xi , yi ) such that the sample correlation coefficient r is unity (approximately). Now let zi y2 i and consider the sample correlation coefficient for the n-pairs of data (xi , zi ). Will this sample correlation coefficient be approximately unity? Explain why or why not.

Consider the simple linear regression model Y 0  1x  , with E() 0, V() 2 , and the errors  uncorrelated. (a) Show that cov (b) Show that cov . 11-9

Consider the simple linear regression model Y 0  1x  , with E() 0, V() 2 , and the errors  uncorrelated. (a) Show that E( ) E(MSE) 2 . (b) Show that E(MSR) 2  1 2 Sx x. 11-100.

Show that E(MSR) 2  1 2 Sx x. 11-100. Suppose that we have assumed the straightline regression model but the response is affected by a second variable x2 such that the true regression function is Is the estimator of the slope in the simple linear regression model unbiased? 1

Suppose that we are fitting a line and we wish to make the variance of the regression coefficient as small as possible. Where should the observations xi , i 1, 2, p , n, be taken so as to minimize V( )? Discuss the practical implications of this allocation of the xi .

Weighted Least Squares. Suppose that we are fitting the line Y 0  1x  , but the variance 1 1 E1Y 2 0  1x1  2x2 Y 0  1x1    2 1Y, 12 0 1 0, 12 x2 Sx x. of Y depends on the level of x; that is, where the wi are constants, often called weights. Show that for an objective function in which each squared residual is multiplied by the reciprocal of the variance of the corresponding observation, the resulting weighted least squares normal equations are Find the solution to these normal equations. The solutions are weighted least squares estimators of 0 and 1. 11-103. Consider

Consider a situation where both Y and X are random variables. Let sx and sy be the sample standard deviations of the observed xs and ys, respectively. Show that an alternative expression for the fitted simple linear regression model is

Suppose that we are interested in fitting a simple linear regression model Y 0  1x  , where the intercept, 0, is known. (a) Find the least squares estimator of 1. (b) What is the variance of the estimator of the slope in part (a)? (c) Find an expression for a 100(1 )% confidence interval for the slope 1. Is this interval longer than the corresponding interval for the case where both the intercept and slope are unknown? Justify your answer. y y