Does a large value of R2 always imply that at least one of the independent variables

If 0 and 1 are the least-squares estimates for the intercept and slope in a simple linear regression model, show that the least-squares equation y = 0 + 1x always goes through the point (x, y). [Hint: Substitute x for x in the least-squares equation and use the fact that 0 = y 1x.]

Applet Exercise How can you improve your understanding of what the method of least-squares actually does? Access the applet Fitting a Line Using Least Squares (at www.thomsonedu. com/statistics/wackerly). The data that appear on the first graph is from Example 11.1. a What are the slope and intercept of the blue horizontal line? (See the equation above the graph.) What is the sum of the squares of the vertical deviations between the points on the horizontal line and the observed values of the ys? Does the horizontal line fit the data well? Click the button Display/Hide Error Squares. Notice that the areas of the yellow boxes are equal to the squares of the associated deviations. How does SSE compare to the sum of the areas of the yellow boxes? b Click the button Display/Hide Error Squares so that the yellow boxes disappear. Place the cursor on right end of the blue line. Click and hold the mouse button and drag the line so that the slope of the blue line becomes negative. What do you notice about the lengths of the vertical red lines? Did SSE increase of decrease? Does the line with negative slope appear to fit the data well? c Drag the line so that the slope is near 0.8. What happens as you move the slope closer to 0.7? Did SSE increase or decrease? When the blue line is moved, it is actually pivoting around a fixed point. What are the coordinates of that pivot point? Are the coordinates of the pivot point consistent with the result you derive in Exercise 11.1? d Drag the blue line until you obtain a line that visually fits the data well. What are the slope and intercept of the line that you visually fit to the data? What is the value of SSE for the line that you visually fit to the data? Click the button Find Best Model to obtain the least-squares line. How does the value of SSE compare to the SSE associated with the line that you visually fit to the data? How do the slope and intercept of the line that you visually fit to the data compare to slope and intercept of the least-squares line?

Fit a straight line to the five data points in the accompanying table. Give the estimates of 0 and 1. Plot the points and sketch the fitted line as a check on the calculations. y 3.0 2.0 1.0 1.0 0.5 x 2.0 1.0 0.0 1.0 2.0

Auditors are often required to compare the audited (or current) value of an inventory item with the book (or listed) value. If a company is keeping its inventory and books up to date, there should be a strong linear relationship between the audited and book values. A company sampled ten inventory items and obtained the audited and book values given in the accompanying table. Fit the model Y = 0 + 1x + to these data. Item Audit Value (yi) Book Value (xi) 1 9 10 2 14 12 37 9 4 29 27 5 45 47 6 109 112 7 40 36 8 238 241 9 60 59 10 170 167 a What is your estimate for the expected change in audited value for a one-unit change in book value? b If the book value is x = 100, what would you use to estimate the audited value?

What did housing prices look like in the good old days? The median sale prices for new single-family houses are given in the accompanying table for the years 1972 through 1979.1 Letting Y denote the median sales price and x the year (using integers 1, 2,..., 8), fit the model Y = 0 + 1x + . What can you conclude from the results? Year Median Sales Price (1000) 1972 (1) $27.6 1973 (2) $32.5 1974 (3) $35.9 1975 (4) $39.3 1976 (5) $44.2 1977 (6) $48.8 1978 (7) $55.7 1979 (8) $62.9

Applet Exercise Refer to Exercises 11.2 and 11.5. The data from Exercise 11.5 appear in the graph under the heading Another Example in the applet Fitting a Line Using Least Squares. Again, the horizontal blue line that initially appears on the graph is a line with 0 slope. a What is the intercept of the line with 0 slope? What is the value of SSE for the line with 0 slope? b Do you think that a line with negative slope will fit the data well? If the line is dragged to produce a negative slope, does SSE increase or decrease? c Drag the line to obtain a line that visually fits the data well. What is the equation of the line that you obtained? What is the value of SSE? What happens to SSE if the slope (and intercept) of the line is changed from the one that you visually fit? d Is the line that you visually fit the least-squares line? Click on the button Find Best Model to obtain the line with smallest SSE. How do the slope and intercept of the least-squares line compare to the slope and intercept of the line that you visually fit in part (c)? How do the SSEs compare? e Refer to part (a). What is the y-coordinate of the point around which the blue line pivots? f Click on the button Display/Hide Error Squares. What do you observe about the size of the yellow squares that appear on the graph? What is the sum of the areas of the yellow squares?

Applet Exercise Move down to the portion of the applet labeled Curvilinear Relationship associated with the applet Fitting a Line Using Least Squares. a Does it seem like a straight line will provide a good fit to the data in the graph? Does it seem that there is likely to be some functional relationship between E(Y ) and x? b Is there any straight line that fits the data better than the one with 0 slope? c If you fit a line to a data set and obtain that the best fitting line has 0 slope, does that mean that there is no functional relationship between E(Y ) and the independent variable? Why?

Laboratory experiments designed to measure LC50 (lethal concentration killing 50% of the test species) values for the effect of certain toxicants on fish are run by two different methods. One method has water continuously flowing through laboratory tanks, and the other method has static water conditions. For purposes of establishing criteria for toxicants, the Environmental Protection Agency (EPA) wants to adjust all results to the flow-through condition. Thus, a model is needed to relate the two types of observations. Observations on toxicants examined under both static and flow-through conditions yielded the data in the accompanying table (measurements in parts per million, ppm). Fit the model Y = 0 + 1x + . Toxicant LC50 Flow-Through (y) LC50 Static (x) 1 23.00 39.00 2 22.30 37.50 3 9.40 22.20 4 9.70 17.50 5 .15 .64 6 .28 .45 7 .75 2.62 8 .51 2.36 9 28.00 32.00 10 .39 .77 a What interpretation can you give to the results? b Estimate the flow-through value for a toxicant with an LC50 static value of x = 12 ppm.

Information about eight four-cylinder automobiles judged to be among the most fuel efficient in 2006 is given in the following table. Engine sizes are in total cylinder volume, measured in liters (L). Car Cylinder Volume (x) Horsepower (y) Honda Civic 1.8 51 Toyota Prius 1.5 51 VW Golf 2.0 115 VW Beetle 2.5 150 Toyota Corolla 1.8 126 VW Jetta 2.5 150 Mini Cooper 1.6 118 Toyota Yaris 1.5 106 a Plot the data points on graph paper. b Find the least-squares line for the data. c Graph the least-squares line to see how well it fits the data. d Use the least-squares line to estimate the mean horsepower rating for a fuel-efficient automobile with cylinder volume 1.9 L

Some data obtained by C. E. Marcellari2 on the height x and diameter y of shells appear in the following table. If we consider the model E(Y ) = 1x, then the slope 1 is the ratio of the mean diameter to the height. Use the following data and the result of Exercise 11.10 to obtain the least-squares estimate of the mean diameter to height ratio. Specimen Diameter (y) Height (x) OSU 36651 185 78 OSU 36652 194 65 OSU 36653 173 77 OSU 36654 200 76 OSU 36655 179 72 OSU 36656 213 76 OSU 36657 134 75 OSU 36658 191 77 OSU 36659 177 69 OSU 36660 199 65 2. Source: Carlos E. Marcellari, Revision of S

Processors usually preserve cucumbers by fermenting them in a low-salt brine (6% to 9% sodium chloride) and then storing them in a high-salt brine until they are used by processors to produce various types of pickles. The high-salt brine is needed to retard softening of the pickles and to prevent freezing when they are stored outside in northern climates. Data showing the reduction in firmness of pickles stored over time in a low-salt brine (2% to 3%) are given in the accompanying table.3 Weeks (x) in Storage at 72F 0 4 14 32 52 Firmness (y) in pounds 19.8 16.5 12.8 8.1 7.5 a Fit a least-squares line to the data. b As a check on your calculations, plot the five data points and graph the line. Does the line appear to provide a good fit to the data points? c Use the least-squares line to estimate the mean firmness of pickles stored for 20 weeks.

J. H. Matis and T. E. Wehrly5 report the following table of data on the proportion of green sunfish that survive a fixed level of thermal pollution for varying lengths of time. Proportion of Survivors (y) Scaled Time (x) 1.00 .10 .95 .15 .95 .20 .90 .25 .85 .30 .70 .35 .65 .40 .60 .45 .55 .50 .40 .55 a Fit the linear model Y = 0 + 1x + . Give your interpretation. b Plot the points and graph the result of part (a). Does the line fit through the points?

a Derive the following identity: SSE = n i=1 (yi yi) 2 = n i=1 (yi 0 1xi) 2 = n i=1 (yi y) 2 1 n i=1 (xi x)(yi y) = Syy 1 Sxy . Notice that this provides an easier computational method of finding SSE. b Use the computational formula for SSE derived in part (a) to prove that SSE Syy . [Hint: 1 = Sxy/Sxx .]

An experiment was conducted to observe the effect of an increase in temperature on the potency of an antibiotic. Three 1-ounce portions of the antibiotic were stored for equal lengths of time at each of the following Fahrenheit temperatures: 30, 50, 70, and 90. The potency readings observed at the end of the experimental period were as shown in the following table. Potency Readings (y) 38, 43, 29 32, 26, 33 19, 27, 23 14, 19, 21 Temperature (x) 30 50 70 90 a Find the least-squares line appropriate for this data. b Plot the points and graph the line as a check on your calculations. c Calculate S2.

a Calculate SSE and S2 for Exercise 11.5. b It is sometimes convenient, for computational purposes, to have x-values spaced symmetrically and equally about zero. The x-values can be rescaled (or coded) in any convenient manner, with no loss of information in the statistical analysis. Refer to Exercise 11.5. Code the x-values (originally given on a scale of 1 to 8) by using the formula x = x 4.5 .5 . Then fit the model Y = 0 + 1 x + . Calculate SSE. (Notice that the x -values are integers symmetrically spaced about zero.) Compare the SSE with the value obtained in part (a).

a Calculate SSE and S2 for Exercise 11.8. b Refer to Exercise 11.8. Code the x-values in a convenient manner and fit a simple linear model to the LC50 measurements presented there. Compute SSE and compare your answer to the result of part (a).

A study was conducted to determine the effects of sleep deprivation on subjects ability to solve simple problems. The amount of sleep deprivation varied over 8, 12, 16, 20, and 24 hours without sleep. A total of ten subjects participated in the study, two at each sleep-deprivation level. After his or her specified sleep-deprivation period, each subject was administered a set of simple addition problems, and the number of errors was recorded. The results shown in the following table were obtained. Number of Errors (y) 8, 6 6, 10 8, 14 14, 12 16, 12 Number of Hours without Sleep (x) 8 12 16 20 24 a Find the least-squares line appropriate to these data. b Plot the points and graph the least-squares line as a check on your calculations. c Calculate S2

Under the assumptions of Exercise 11.20, find Cov(0, 1). Use this answer to show that 0 and 1 are independent if n i=1 xi = 0. [Hint: Cov(0, 1) = Cov(Y 1x, 1). Use Theorem 5.12 and the results of this section.]

Under the assumptions of Exercise 11.20, find the MLE of 2.

Refer to Exercise 11.3. a Do the data present sufficient evidence to indicate that the slope 1 differs from zero? (Test at the 5% significance level.) b What can be said about the attained significance level associated with the test implemented in part (a) using a table in the appendix? c Applet Exercise What can be said about the attained significance level associated with the test implemented in part (a) using the appropriate applet? d Find a 95% confidence interval for 1

Refer to Exercise 11.13. Do the data present sufficient evidence to indicate that the size x of the anchovy catch contributes information for the prediction of the price y of the fish meal? a Give bounds on the attained significance level. b Applet Exercise What is the exact p-value? c Based on your answers to parts (a) and/or (b), what would you conclude at the = .10 level of significance?

Do the data in Exercise 11.19 present sufficient evidence to indicate that the number of errors is linearly related to the number of hours without sleep? a Give bounds on the attained significance level. b Applet Exercise Determine the exact p-value. c Based on your answers to parts (a) and/or (b), what would you conclude at the = .05 level of significance? d Would you expect the relationship between y and x to be linear if x were varied over a wider range, say, from x = 4 to x = 48? e Give a 95% confidence interval for the slope. Provide a practical interpretation for this interval estimate.

Most sophomore physics students are required to conduct an experiment verifying Hookes law. Hookes law states that when a force is applied to a body that is long in comparison to its cross-sectional area, the change y in its length is proportional to the force x; that is, y = 1x, where 1 is a constant of proportionality. The results of a physics students laboratory experiment are shown in the following table. Six lengths of steel wire, .34 millimeter (mm) in diameter and 2 meters (m) long, were used to obtain the six force-length change measurements. Force Change in Length x (kg) (y) (mm) 29.4 4.25 39.2 5.25 49.0 6.50 58.8 7.85 68.6 8.75 78.4 10.00 a Fit the model, Y = 0 + 1x + , to the data, using the method of least squares. b Find a 95% confidence interval for the slope of the line. c According to Hookes law, the line should pass through the point (0, 0); that is, 0 should equal 0. Test the hypothesis that E(Y ) = 0 when x = 0. Give bounds for the attained significance level. d Applet Exercise What is the exact p-value? e What would you conclude at the = .05 level?

Use the properties of the least-squares estimators given in Section 11.4 to complete the following.a Show that under the null hypothesis H0 : i = i0 T = i i0 S cii possesses a t distribution with n 2 df, where i = 1, 2. b Derive the confidence intervals for i given in this section.

Suppose that Y1, Y2,..., Yn are independent, normally distributed random variables with E(Yi) = 0 + 1xi and V(Yi) = 2, for i = 1, 2,..., n. Show that the likelihood ratio test of H0 : 1 = 0 versus Ha : 1 = 0 is equivalent to the t test given in this section.

Let Y1, Y2,..., Yn be as given in Exercise 11.28. Suppose that we have an additional set of independent random variables W1, W2,..., Wm, where Wi is normally distributed with E(Wi) = 0 + 1ci and V(Wi) = 2, for i = 1, 2,..., m. Construct a test of H0 : 1 = 1 against the Ha : 1 = 1. 6

Using a chemical procedure called differential pulse polarography, a chemist measured the peak current generated (in microamperes, A) when solutions containing different amounts of nickel (measured in parts per billion, ppb) are added to different portions of the same buffer.8 Is there sufficient evidence to indicate that peak current increases as nickel concentrations increase? Use = .05. x = Ni (ppb) y = Peak Current (A) 19.1 .095 38.2 .174 57.3 .256 76.2 .348 95 .429 114 .500 131 .580 150 .651 170 .722

Refer to Exercises 11.5 and 11.17. a Is there sufficient evidence to indicate that the median sales price for new single-family houses increased over the period from 1972 through 1979 at the .01 level of significance? b Estimate the expected yearly increase in median sale price by constructing a 99% confidence interval.

Refer to Exercise 11.8 and 11.18. Is there evidence of a linear relationship between flow-through and static LC50s? Test at the .05 significance level.

Refer to Exercise 11.33. Is there evidence of a linear relationship between flow-through and static LC50s? a Give bounds for the attained significance level. b Applet Exercise What is the exact p-value?

For the simple linear regression model Y = 0 + 1x + with E() = 0 and V() = 2, use the expression for V(a00 + a11) derived in this section to show that V(0 + 1x ) =
1 n + (x x)2 Sxx  2 . For what value of x does the confidence interval for E(Y ) achieve its minimum length?

Refer to Exercise 11.13 and 11.24. Find the 90% confidence interval for the mean price per ton of fish meal if the anchovy catch is 5 million metric tons

Using the model fit to the data of Exercise 11.8, construct a 95% confidence interval for the mean value of flow-through LC50 measurements for a toxicant that has a static LC50 of 12 parts per million. (Also see Exercise 11.18.)

Refer to Exercise 11.3. Find a 90% confidence interval for E(Y ) when x = 0. Then find 90% confidence intervals for E(Y ) when x = 2 and x = +2. Compare the lengths of these intervals. Plot these confidence limits on the graph you constructed for Exercise 11.3.

Refer to Exercise 11.16. Find a 95% confidence interval for the mean potency of a 1-ounce portion of antibiotic stored at 65F

Refer to Exercise 11.4. Suppose that the sample given there came from a large but finite population of inventory items. We wish to estimate the population mean of the audited values, using the fact that book values are known for every item on inventory. If the population contains N items and E(Yi) = i = 0 + 1xi, then the population mean is given by Y = 1 N N i=1 i = 0 + 1 1 N N i=1 xi = 0 + 1x . a Using the least-squares estimators of 0 and 1, show that Y can be estimated by Y = y + 1(x x). (Notice that y is adjusted up or down, depending on whether x is larger or smaller than x .) b Using the data of Exercise 11.4 and the fact that x = 74.0, estimate Y , the mean of the audited values, and place a 2-standard-deviation bound on the error of estimation. (Regard the xi-values as constants when computing the variance of Y .)

Suppose that the model Y = 0 + 1x + is fit to the n data points (y1, x1), . . . , (yn , xn ). At what value of x will the length of the prediction interval for Y be minimized?

Refer to Exercises 11.5 and 11.17. Use the data and model given there to construct a 95% prediction interval for the median sale price in 1980.

Refer to Exercise 11.43. Find a 95% prediction interval for the median sale price for the year 1981. Repeat for 1982. Would you feel comfortable in using this model and the data of Exercise 11.5 to predict the median sale price for the year 1988?

Refer to Exercises 11.8 and 11.18. Find a 95% prediction interval for a flow-through LC50 if the static LC50 is observed to be 12 parts per million. Compare the length of this interval to that of the interval found in Exercise 11.37.

Refer to Exercise 11.16. Find a 95% prediction interval for the potency of a 1-ounce portion of antibiotic stored at 65F. Compare this interval to that calculated in Exercise 11.39.

Refer to Exercise 11.14. Find a 95% prediction interval for the proportion of survivors at time x = .60.

The accompanying table gives the peak power load for a power plant and the daily high temperature for a random sample of 10 days. Test the hypothesis that the population correlation coefficient between peak power load and high temperature is zero versus the alternative that it is positive. Use = .05. Bound or determine the attained significance level. Day High Temperature (F) Peak Load 1 95 214 2 82 152 3 90 156 4 81 129 5 99 254 6 100 266 7 93 210 8 95 204 9 93 213 10 87 150

Applet Exercise Refer to Example 11.1 and Exercise 11.2. Access the applet Fitting a Line Using Least Squares. The data that appear on the first graph is from Example 11.1. a Drag the blue line to obtain an equation that visually fits the data well. What do you notice about the values of SSE and r 2 as the fit of the line improves? Why does r 2 increase as SSE decreases? b Click the button Find Best Model to obtain the least-squares line. What is the value of r 2? What is the value of the correlation coefficient?

In Exercise 11.8 both the flow-through and static LC50 values could be considered random variables. Using the data of Exercise 11.8, test to see whether the correlation between static and flow-through values significantly differs from zero. Use = .01. Bound or determine the associated p-value.

Is the plant density of a species related to the altitude at which data are collected? Let Y denote the species density and X denote the altitude. A fit of a simple linear regression model using 14 observations yielded y = 21.6 7.79x and r 2 = .61. a What is the value of the correlation coefficient r? b What proportion of the variation in densities is explained by the linear model using altitude as the independent variable? c Is there sufficient evidence at the = .05 to indicate that plant densities decrease with an increase in altitude?

The correlation coefficient for the heights and weights of ten offensive backfield football players was determined to be r = .8261. a What percentage of the variation in weights was explained by the heights of the players? b What percentage of the variation in heights was explained by the weights of the players? c Is there sufficient evidence at the = .01 level to claim that heights and weights are positively correlated? d Applet Exercise What is the attained significance level associated with the test performed in part (c)?

Suppose that we seek an intuitive estimator for = Cov(X, Y ) X Y . a The method-of-moments estimator of Cov(X, Y ) = E[(X X )(Y Y )] is Cov
(X, Y ) = 1 n n i=1 (Xi X)(Yi Y ). Show that the method-of-moments estimators for the standard deviations of X and Y are X = 0 1 n n i=1 (Xi X)2 and Y = 0 1 n n i=1 (Yi Y )2. b Substitute the estimators for their respective parameters in the definition of and obtain the method-of-moments estimator for . Compare your estimator to r, the maximumlikelihood estimator for presented in this section.

Consider the simple linear regression model based on normal theory. If we are interested in testing H0 : 1 = 0 versus various alternatives, the statistic T = 1 0 S/ Sxx possesses a t distribution with n 2 df if the null hypothesis is true. Show that the equation for T can also be written as T = r n 2 1 r 2

Refer to Exercise 11.55. Isr = .8 big enough to claim > 0 at the = .05 significance level? a Assume n = 5 and implement the test. b Assume n = 12 and implement the test. c Applet Exercise Determine the p-values for the tests implemented in parts (a) and (b). d Did you reach the same conclusions in parts (a) and (b)? Why or why not? e Why is the p-value associated with the test in part (b) so much smaller that the p-value associated with the test performed in part (a)?

Refer to Exercises 11.55 and 11.56. a What term in the T statistic determines whether the value of t is positive or negative? b What quantities determine the size of |t|?

Refer to Exercise 11.55. If n = 4, what is the smallest value ofr that will allow you to conclude that > 0 at the = .05 level of significance?

Refer to Exercises 11.55 and 11.58. If n = 20, what is the largest value r that will allow you to conclude that < 0 at the = .05 level of significance?

Refer to Example 11.10. Find a 90% prediction interval for the strength of concrete when the water/cement ratio is 1.5.

Refer to Example 11.11. Calculate the correlation coefficientr between the variables ln W and ln l. What proportion of the variation in y = ln w is explained by x = ln l?

It is well known that large bodies of water have a mitigating effect on the temperature of the surrounding land masses. On a cold night in central Florida, temperatures were recorded at equal distances along a transect running downwind from a large lake. The resulting data are given in the accompanying table.Site (x) Temperature F, (y) 1 37.00 2 36.25 3 35.41 4 34.92 5 34.52 6 34.45 7 34.40 8 34.00 9 33.62 10 33.90 Notice that the temperatures drop rapidly and then level off as we move away from the lake. The suggested model for these data is E(Y ) = 0e1 x . a Linearize the model and estimate the parameters by the method of least squares. b Find a 90% confidence interval for 0. Give an interpretation of the result.

Refer to Exercise 11.14. One model proposed for these data on the proportion of survivors of thermal pollution is E(Y ) = exp(0x1 ). Linearize this model and estimate the parameters by using the method of least squares and the data of Exercise 11.14. (Omit the observation with y = 1.00.)

In the biological and physical sciences, a common model for proportional growth over time is E(Y ) = 1 et , where Y denotes a proportion and t denotes time. Y might represent the proportion of eggs that hatch, the proportion of an organism filled with diseased cells, the proportion of patients reacting to a drug, or the proportion of a liquid that has passed through a porous medium. With n observations of the form (yi, ti), outline how you would estimate and then form a confidence interval for .

Refer to Exercise 11.3. Fit the model suggested there by use of matrices.

Use the matrix approach to fit a straight line to the data in the accompanying table, plot the points, and then sketch the fitted line as a check on the calculations. The data points are the same as for Exercises 11.3 and 11.66 except that they are translated 1 unit in the positive direction along the x-axis. What effect does symmetric spacing of the x-values about x = 0 have on the form of the (X X) matrix and the resulting calculations? y x 3 1 2 0 1 1 1 2 .5 3

Fit the quadratic model Y = 0 +1x +2x 2 + to the data points in the following table. Plot the points and sketch the fitted parabola as a check on the calculations. y x 1 3 0 2 0 1 1 0 1 1 0 2 0 3

The manufacturer of Lexus automobiles has steadily increased sales since the 1989 launch of that brand in the United States. However, the rate of increase changed in 1996 when Lexus introduced a line of trucks. The sales of Lexus vehicles from 1996 to 2003 are shown in the accompanying table.10 x y 1996 18.5 1997 22.6 1998 27.2 1999 31.2 2000 33.0 2001 44.9 2002 49.4 2003 35.0 a Letting Y denote sales and x denote the coded year (7 for 1996, 5 for 1997, through 7 for 2003), fit the model Y = 0 + 1x + . b For the same data, fit the model Y = 0 + 1x + 2x 2 + .

Consider the general linear model Y = 0 + 1x1 + 2x2 ++ k xk + , where E() = 0 and V() = 2. Notice that i = a , where the vector a is defined by aj = $ 1, if j = i, 0, if j = i. Use this to verify that E(i) = i and V(i) = cii2, where cii is the element in row i and column i of (X X)1

Refer to Exercise 11.69. a Is there evidence of a quadratic effect in the relationship between Y and x? (Test H0 : 2 = 0.) Use = .10. b Find a 90% confidence interval for 2.

The experimenter who collected the data in Exercise 11.68 claims that the minimum value of E(Y ) occurs at x = 1. Test this claim at the 5% significance level. [Hint: E(Y ) = 0 + 1x + 2x 2 has its minimum at the point x0, which satisfies the equation 1 + 22x0 = 0.]

An experiment was conducted to investigate the effect of four factorstemperature T1, pressure P, catalyst C, and temperature T2on the yield Y of a chemical. a The values (or levels) of the four factors used in the experiment are shown in the accompanying table. If each of the four factors is coded to produce the four variables x1, x2, x3, and x4, respectively, give the transformation relating each coded variable to its corresponding original. T1 x1 P x2 C x3 T2 x4 50 1 10 1 1 1 100 1 70 1 20 1 2 1 200 1 b Fit the linear model Y = 0 + 1x1 + 2x2 + 3x3 + 4x4 + to the following table of data. x4 +1 1 x3 x3 1 1 1 1 1 x2 1 22.2 24.5 24.4 25.9 x1 1 19.4 24.1 25.2 28.4 1 22.1 19.6 23.5 16.5 +1 x2 1 14.2 12.7 19.3 16.0 c Do the data present sufficient evidence to indicate that T1 contributes information for the estimation of Y ? Does P? DoesC? Does T2? (Test the hypotheses, respectively, that 1 = 0, 2 = 0, 3 = 0, and 4 = 0.) Give bounds for the p-value associated with each test. What would you conclude if you used = .01 in each case?

Refer to Exercise 11.74. Find a 90% confidence interval for the expected yield, given that T1 = 50, P = 20, C = 1, and T2 = 200.

The results that follow were obtained from an analysis of data obtained in a study to assess the relationship between percent increase in yield (Y ) and base saturation (x1, pounds/acre), phosphate saturation (x2, BEC%), and soil pH (x3). Fifteen responses were analyzed in the study. The least-squares equation and other useful information follow. y = 38.83 0.0092x1 0.92x2 + 11.56x3, Syy = 10965.46, SSE = 1107.01, 104 (X X) 1 = 151401.8 2.6 100.5 28082.9 2.6 1.0 0.0 0.4 100.5 0.0 8.1 5.2 28082.9 0.4 5.2 6038.2 . a Is there sufficient evidence that, with all independent variables in the model, 2 < 0? Test at the = .05 level of significance. b Give a 95% confidence interval for the mean percent increase in yield if x1 = 914, x2 = 65 and x3 = 6

Refer to Exercise 11.76. Give a 95% prediction interval for the percent increase in yield in a field with base saturation = 914 pounds/acre, phosphate saturation = 65%, and soil pH = 6.

Refer to Exercise 11.69. Find a 98% prediction interval for Lexus sales in 2004. Use the quadratic model.

Refer to Exercises 11.74 and 11.75. Find a 90% prediction interval for Y if T1 = 50, P = 20, C = 1, and T2 = 200.

In Exercise 11.80, you used an F test to test the same hypothesis that was tested in Exercise 11.31 via a t test. Consider the general simple linear regression case and the F and t statistics that can be used to implement the test of H0 : 1 = 0 versus Ha : 1 = 0. Show that in general F = t 2. Compare the value of F obtained in Exercise 11.80 to the corresponding value of t obtained in Exercise 11.31.

Refer to Exercise 11.76 where we obtained the following information when fitting a multiple regression model to 15 responses; y = 38.83 0.0092x1 0.92x2 + 11.56x3, Syy = 10965.46, SSE = 1107.01. a Is there sufficient evidence to conclude that at least one of the independent variables contributes significant information for the prediction of Y ? b Calculate the value of the multiple coefficient of determination. Interpret the value of R2.

Refer to Exercises 11.76 and 11.82. Does including the variables phosphate saturation x2 and pH x3 contribute to a significantly better fit of the model to the data? The reduced linear regression model, Y = 0 + 1x1 + was fit and we observed SSER = 5470.07. a Implement the appropriate test of hypothesis at the = .05 level of significance. b What is the smallest value of SSER that would have allowed you to conclude that at least one of the variables (phosphate saturation and/or pH) contributed to a better fit of the model to the data?

We have fit a model with k independent variables, and wish to test the null hypothesis H0 : 1 = 2 == k = 0. a Show that the appropriate F-distributed test statistic can be expressed as F = n (k + 1) k R2 1 R2 . b If k = 1 how does the value of F from part (a) compare to the expression for the T statistic derived in Exercise 11.55?

A real estate agents computer data listed the selling price Y (in thousands of dollars), the living area x1 (in hundreds of square feet), the number of floors x2, number of bedrooms x3, and number of bathrooms x4 for newly listed condominiums. The multiple regression model E(Y ) = 0 + 1x1 + 2x2 + 3x3 + 4x4 was fit to the data obtained by randomly selecting 15 condos currently on the market. a If R2 = .942, is there sufficient evidence that at least one of the independent variables contributes significant information for the prediction of selling price? b If Syy = 16382.2, what is SSE?

Refer to Exercise 11.85. A realtor suspects that square footage x1 might be the most important predictor variable and that the other variables can be eliminated from the model without much loss in prediction information. The simple linear regression model for selling price versus square footage was fit to the 15 data points that were used in Exercise 11.85, and the realtor observed that SSE = 1553. Can the additional independent variables used to fit the model in Exercise 11.85 be dropped from the model without losing predictive information? Test at the = .05 significance level.

Does a large value of R2 always imply that at least one of the independent variables should be retained in the regression model? Does a small value of R2 always indicate that none of the independent variables are useful for prediction of the response? a Suppose that a model with k = 4 independent variables is fit using n = 7 data points and that R2 = .9. How many numerator and denominator degrees of freedom are associated with the F statistic for testing H0 : 1 = 2 = 3 = 4 = 0? Use the result in Exercise 11.84(a) to compute the value of the appropriate F statistic. Can H0 be rejected at the = .10 significance level? b Refer to part (a). What do you observe about the relative sizes of n and k? What impact does this have on the value of F? c A model with k = 3 independent variables is fit to n = 44 data points resulting in R2 = .15. How many numerator and denominator degrees of freedom are associated with the F statistic for testing H0 : 1 = 2 = 3 = 0? Use the result in Exercise 11.84(a) to compute the value of the appropriate F statistic. Can H0 be rejected at the = .10 significance level? d Refer to part (c). What do you observe about the relative sizes of n and k? What impact does this have on the value of F?

Does a large value of R2 always imply that at least one of the independent variables should be retained in the regression model? Does a small value of R2 always indicate that none of the independent variables are useful for prediction of the response? a Suppose that a model with k = 4 independent variables is fit using n = 7 data points and that R2 = .9. How many numerator and denominator degrees of freedom are associated with the F statistic for testing H0 : 1 = 2 = 3 = 4 = 0? Use the result in Exercise 11.84(a) to compute the value of the appropriate F statistic. Can H0 be rejected at the = .10 significance level? b Refer to part (a). What do you observe about the relative sizes of n and k? What impact does this have on the value of F? c A model with k = 3 independent variables is fit to n = 44 data points resulting in R2 = .15. How many numerator and denominator degrees of freedom are associated with the F statistic for testing H0 : 1 = 2 = 3 = 0? Use the result in Exercise 11.84(a) to compute the value of the appropriate F statistic. Can H0 be rejected at the = .10 significance level? d Refer to part (c). What do you observe about the relative sizes of n and k? What impact does this have on the value of F? Are the following statements true or false? a If Model I is fit, the estimate for 2 is based on 16 df. b If Model II is fit, we can perform a t test to determine whether x2 contributes to a better fit of the model to the data. c If Models I and II are both fit, then SSEI SSEII. d If Models I and II are fit, then 2 I 2 II. e Model II is a reduction of model I. f Models I and III can be compared using the complete/reduced model technique presented in Section 11.14.

Refer to the three models given in Exercise 11.88. Let R2 I , R2 II, and R2 III denote the coefficients of determination for models I, II, and III. Are the following statements true or false? a R2 I R2 II. b R2 I R2 III. c R2 II R2 III

Refer to Exercise 11.74. Test the hypothesis at the 5% level of significance that neither T1 nor T2 affects the yield.

Utility companies, which must plan the operation and expansion of electricity generation, are vitally interested in predicting customer demand over both short and long periods of time. A short-term study was conducted to investigate the effect of each months mean daily temperature x1 and of cost per kilowatt-hour, x2 on the mean daily consumption (in kWh) per household. The company officials expected the demand for electricity to rise in cold weather (due to heating), fall when the weather was moderate, and rise again when the temperature rose and there was a need for air conditioning. They expected demand to decrease as the cost per kilowatt-hour increased, reflecting greater attention to conservation. Data were available for 2 years, a period during which the cost per kilowatt-hour x2 increased due to the increasing costs of fuel. The company officials fitted the model Y = 0 + 1x1 + 2x 2 1 + 3x2 + 4x1x2 + 5x 2 1 x2 + to the data in the following table and obtained y = 325.60611.383x1 +.113x 2 1 21.699x2 + .873x1x2 .009x 2 1 x2 with SSE = 152.177. Mean Daily Consumption Price per kWh ( x2) (kWh) per Household 8 Mean daily F temperature (x1) 31 34 39 42 47 56 Mean daily consumption (y) 55 49 46 47 40 43 10 Mean daily F temperature (x1) 32 36 39 42 48 56 Mean daily consumption (y) 50 44 42 42 38 40 8 Mean daily F temperature (x1) 62 66 68 71 75 78 Mean daily consumption (y) 41 46 44 51 62 73 10 Mean daily F temperature (x1) 62 66 68 72 75 79 Mean daily consumption (y) 39 44 40 44 50 55 When the model Y = 0 1x1 +2x 2 1 + was fit, the prediction equation was y = 130.009 3.302x1 + .033x 2 1 with SSE = 465.134. Test whether the terms involving x2(x2, x1x2, x 2 1 x2) contribute to a significantly better fit of the model to the data. Give bounds for the attained significance level.

Refer to Example 11.19. Using the reduced model, construct a 95% confidence interval for the expected abrasion resistance of rubber when x1 = 1 and x2 = 1

Refer to Example 11.19. Construct individual tests of the three hypotheses H0 : 3 = 0, H0 : 4 = 0, and H0 : 5 = 0. Use a 1% level of significance on each test. (If multiple tests are to be conducted on the same set of data, it is wise to use a very small level on each test.)

At temperatures approaching absolute zero (273C), helium exhibits traits that defy many laws of conventional physics. An experiment has been conducted with helium in solid form at various temperatures near absolute zero. The solid helium is placed in a dilution refrigerator along with a solid impure substance, and the fraction (in weight) of the impurity passing through the solid helium is recorded. (The phenomenon of solids passing directly through solids is known as quantum tunneling.) The data are given in the following table. Proportion of Impurity Passing C Temperature (x) Through Helium (y) 262.0 .315 265.0 .202 256.0 .204 267.0 .620 270.0 .715 272.0 .935 272.4 .957 272.7 .906 272.8 .985 272.9 .987 a Fit a least-squares line to the data. b Test the null hypothesis H0: 1 = 0 against the alternative hypothesis Ha : 1 < 0, at the = .01 level of significance. c Find a 95% prediction interval for the percentage of the solid impurity passing through solid helium at 273C. (This value of x is outside the experimental region where use of the model for prediction may be dangerous.)

A study was conducted to determine whether a linear relationship exists between the breaking strength y of wooden beams and the specific gravity x of the wood. Ten randomly selected beams of the same cross-sectional dimensions were stressed until they broke. The breaking strengths and the density of the wood are shown in the accompanying table for each of the ten beams. Beam Specific Gravity (x) Strength (y) 1 .499 11.14 2 .558 12.74 3 .604 13.13 4 .441 11.51 5 .550 12.38 6 .528 12.60 7 .418 11.13 8 .480 11.70 9 .406 11.02 10 .467 11.41 a Fit the model Y = 0 + 1x + . b Test H0: 1 = 0 against the alternative hypothesis, Ha: 1 = 0. c Estimate the mean strength for beams with specific gravity .590, using a 90% confidence interval.

A response Y is a function of three independent variables x1, x2, and x3 that are related as follows: Y = 0 + 1x1 + 2x2 + 3x3 + . a Fit this model to the n = 7 data points shown in the accompanying table. y x1 x2 x3 1 3 5 1 0 201 0 1 3 1 1 0 4 0 2 1 3 1 320 1 3351 b Predict Y when x1 = 1, x2 = 3, x3 = 1. Compare with the observed response in the original data. Why are these two not equal? c Do the data present sufficient evidence to indicate that x3 contributes information for the prediction of Y ? (Test the hypothesis H0: 3 = 0, using = .05.) d Find a 95% confidence interval for the expected value of Y , given x1 = 1, x2 = 3, and x3 = 1. e Find a 95% prediction interval for Y , given x1 = 1, x2 = 3, and x3 = 1.

If values of independent variables are equally spaced, what is the advantage of coding to new variables that represent symmetric spacing about the origin?

Suppose that you wish to fit a straight line to a set of n data points, where n is an even integer, and that you can select the n values of x in the interval 9 x 9. How should you select the values of x so as to minimize V(1)?

The data in the accompanying table come from the comparison of the growth rates for bacteria types A and B. The growth Y recorded at five equally spaced (and coded) points of time is shown in the table. Time Bacteria Type 2 10 1 2 A 8.0 9.0 9.1 10.2 10.4 B 10.0 10.3 12.2 12.6 13.9 a Fit the linear model Y = 0 + 1x1 + 2x2 + 3x1x2 + to the n = 10 data points. Let x1 = 1 if the point refers to bacteria type B and let x1 = 0 if the point refers to type A. Let x2 = coded time. b Plot the data points and graph the two growth lines. Notice that 3 is the difference between the slopes of the two lines and represents timebacteria interaction. c Predict the growth of type A at time x2 = 0 and compare the answer with the graph. Repeat the process for type B. d Do the data present sufficient evidence to indicate a difference in the rates of growth for the two types of bacteria? e Find a 90% confidence interval for the expected growth for type B at time x2 = 1. f Find a 90% prediction interval for the growth Y of type B at time x2 = 1.

The following model was proposed for testing whether there was evidence of salary discrimination against women in a state university system: Y = 0 + 1x1 + 2x2 + 3x1x2 + 4x 2 2 + , where Y = annual salary (in thousands of dollars), x1 = $ 1, if female, 0, if male, x2 = amount of experience (in years). When this model was fit to data obtained from the records of 200 faculty members, SSE = 783.90. The reduced model Y = 0 + 1x2 + 2x 2 2 + was also fit and produced a value of SSE = 795.23. Do the data provide sufficient evidence to support the claim that the mean salary depends on the gender of the faculty members? Use = .05.

Show that the least-squares prediction equation y = 0 + 1x1 ++ k xk passes through the point (x 1, x 2,..., xk , y).

An experiment was conducted to determine the effect of pressure and temperature on the yield of a chemical. Two levels of pressure (in pounds per square inch, psi) and three of temperature were used: Pressure (psi) Temperature (F) 50 100 80 200 300 One run of the experiment at each temperaturepressure combination gave the data listed in the following table. Yield Pressure (psi) Temperature (F) 21 50 100 23 50 200 26 50 300 22 80 100 23 80 200 28 80 300 a Fit the model Y = 0+1x1+2x2+3x 2 2 +, where x1 = pressure and x2 = temperature. b Test to see whether 3 differs significantly from zero, with = .05. c Test the hypothesis that temperature does not affect the yield, with = .05.

Let (X, Y ) have a bivariate normal distribution. A test of H0: = 0 against Ha: = 0 can be derived as follows. a Let Syy = n i=1(yi y)2 and Sxx = n i=1(xi x)2. Show that 1 = r 0 Syy Sxx . b Conditional on Xi = xi , for i = 1, 2,..., n, show that under H0 : = 0 1 (n 2)Sxx (Syy (1 r 2) has a t distribution with (n 2) df. c Conditional on Xi = xi , for i = 1, 2,..., n, conclude that T = r n 2 1 r 2 has a t distribution with (n 2) df, under H0 : = 0. Hence, conclude that T has the same distribution unconditionally.

Labor and material costs are two basic components in the cost of construction. Changes in the component costs of course lead to changes in total construction costs. The accompanying table tracks changes in construction cost and cost of all construction materials for 8 consecutive months. Index of All Construction Construction Month Cost (y) Materials (x) January 193.2 180.0 February 193.1 181.7 March 193.6 184.1 April 195.1 185.3 May 195.6 185.7 June 198.1 185.9 July 200.9 187.7 August 202.7 189.6 Do the data provide sufficient evidence to indicate a nonzero correlation between the monthly construction costs and indexes of all construction materials? Give the attained significance level.

The data in the following table give the miles per gallon obtained by a test automobile when using gasolines of varying octane levels. Miles per Gallon (y) Octane (x) 13.0 89 13.2 93 13.0 87 13.6 90 13.3 89 13.8 95 14.1 100 14.0 98 a Calculate the value of r. b Do the data provide sufficient evidence to indicate that octane level and miles per gallon are dependent? Give the attained significance level, and indicate your conclusion if you wish to implement an = .05 level test.

Applet Exercise Access the applet Removing Points from Regression. Sometimes removing a point from those used to fit a regression model produces a fitted model substantially different that the one obtained using all of the data (such a point is called a high-influence point). a The top graph gives a data set and fitted regression line useful for predicting a students weight given his or her height. Click on any data points to remove them and refit the regression model. Can you find a high influence data point in this data set? b Scroll down to the second graph that relates quantitative SAT score to high school rank. Does the slope of the fitted regression line surprise you? Can you find a high-influence data point? Does removing that data point produce a regression line that better meets your expectation regarding the relationship between quantitative SAT scores and class rank? c Scroll down to the remainder of the data sets and explore what happens when different data points are removed.