View solution: Regression and Predictions. Exercises 1328 use the same data sets as

Notation and Terminology If we use the paired height/pulse data for females from Data Set 1 in Appendix B, we get this regression equation: y ^ = 73.9 + 0.0223 x , where x represents height (cm) and the pulse rate is in beats per minute. What does the symbol y ^ represent? In this case, what does the predictor variable represent? What does the response variable represent?

. BestFit Line In what sense is the regression line the straight line that best fits the points in a scatterplot?

Correlation and Slope Formula 103 shows that the slope of a regression line can be found by evaluating r s y / s x . What do we know about the graph of the regression line if r is a positive value? What do we know about the graph of the regression line if r is a negative value?

Notation What is the difference between the regression equation y ^ = b 0 + b 1 x and the regression equation y = 0 + 1 x ?

Making Predictions. In Exercises 58, let the predictor variable x be the first variable given. Use the given data to find the regression equation and the best predicted value of the response variable. Be sure to follow the prediction procedure summarized in Figure 105 Old Faithful For 40 eruptions of the Old Faithful geyser in Yellowstone National Park, duration times (sec) were recorded along with the time intervals (min) after the eruptions. The linear correlation coefficient is r = 0.687 and the regression equation is y ^ = 47.4 + 0.180 x , where x represents duration time. The mean of the 40 duration times is 245.0 sec and the mean of the 40 interval times is 91.4 min. What is the best predicted interval time following an eruption with a duration time of 120 min?

Making Predictions. In Exercises 58, let the predictor variable x be the first variable given. Use the given data to find the regression equation and the best predicted value of the response variable. Be sure to follow the prediction procedure summarized in Figure 106 Old Faithful For 40 eruptions of the Old Faithful geyser in Yellowstone National Park, duration times (sec) were recorded along with the heights (ft) of the eruptions. The linear correlation coefficient is r = 0.0915 and the regression equation is y ^ = 119 + 0.0331 x , where x represents duration time. The mean of the 40 duration times is 245.0 sec and the mean of the 40 heights is 127.2 ft. What is the best predicted height of an eruption with a duration time of 120 min?

Making Predictions. In Exercises 58, let the predictor variable x be the first variable given. Use the given data to find the regression equation and the best predicted value of the response variable. Be sure to follow the prediction procedure summarized in Figure 107 Heights of Fathers and Sons The heights (in inches) of a sample of 10 father/son pairs of subjects were measured. The linear correlation coefficient is r = 0.149 and the regression equation is y ^ = 66.8 + 0.016 x , where x represents the father's height. The mean of the 10 father heights is 69.7 in. and the mean of the 10 son heights is 68.0 in. What is the best predicted height of a son who has a father with a height of 72.0 in.?

Making Predictions. In Exercises 58, let the predictor variable x be the first variable given. Use the given data to find the regression equation and the best predicted value of the response variable. Be sure to follow the prediction procedure summarized in Figure 108 Cereal Killers The amounts of sugar (grams of sugar per gram of cereal) and calories (per gram of cereal) were recorded for a sample of 16 different cereals. The linear correlation coefficient is r = 0.765 and the regression equation is y ^ = 3.46 + 1.01 x , where x represents the amount of sugar. The mean of the 16 amounts of sugar is 0.295 grams and the mean of the 16 calorie counts is 3.76. What is the best predicted calorie count for a cereal with a measured sugar amount of 0.40 g?

Finding the Equation of the Regression Line. In Exercises 9 and 10, use the given data to find the equation of the regression line. Examine the scatterplot and identify a characteristic of the data that is ignored by the regression line.x 10 8 13 9 11 14 6 4 12 7 5 y 9.14 8.14 8.74 8.77 9.26 8.10 6.13 3.10 9.13 7.26 4.74

Finding the Equation of the Regression Line. In Exercises 9 and 10, use the given data to find the equation of the regression line. Examine the scatterplot and identify a characteristic of the data that is ignored by the regression line. x 10 8 13 9 11 14 6 4 12 7 5 y 7.46 6.77 12.74 7.11 7.81 8.84 6.08 5.39 8.15 6.42 5.73

Effects of an Outlier Refer to the Minitabgenerated scatterplot given in Exercise 11 of Section 102. a. Using the pairs of values for all 10 points, find the equation of the regression line. b. After removing the point with coordinates (10, 10), use the pairs of values for the remaining 9 points and find the equation of the regression line. c. Compare the results from parts (a) and (b).

Effects of Clusters Refer to the Minitabgenerated scatterplot given in Exercise 12 of Section 102. a. Using the pairs of values for all 8 points, find the equation of the regression line.b. Using only the pairs of values for the 4 points in the lower left corner, find the equation of the regression line. c. Using only the pairs of values for the 4 points in the upper right corner, find the equation of the regression line. d. Compare the results from parts (a), (b), and (c).

Regression and Predictions. Exercises 1328 use the same data sets as Exercises 1328 in Section 102. In each case, find the regression equation, letting the first variable be the predictor (x) variable. Find the indicated predicted value by following the prediction procedure summarized in Figure 105.Lemons and Car Crashes Find the best predicted crash fatality rate for a year in which there are 500 metric tons of lemon imports. Lemon Imports 230 265 358 480 530 Crash Fatality Rate 15.9 15.7 15.4 15.3 14.9

Regression and Predictions. Exercises 1328 use the same data sets as Exercises 1328 in Section 102. In each case, find the regression equation, letting the first variable be the predictor (x) variable. Find the indicated predicted value by following the prediction procedure summarized in Figure 105.PSAT and SAT Scores One subject not included in the given table had a PSAT score of 229. Find the best predicted SAT score for this student. Is the result close to the reported value of 2400? Given that the data are from volunteered responses, are the results valid? PSAT 183 207 167 206 197 142 193 176 SAT 2200 2040 1890 2380 2290 2070 2370 1980

Regression and Predictions. Exercises 1328 use the same data sets as Exercises 1328 in Section 102. In each case, find the regression equation, letting the first variable be the predictor (x) variable. Find the indicated predicted value by following the prediction procedure summarized in Figure 105.Campus Crime Find the best predicted number of burglaries for Ohio State, which had an enrollment of 51,800 students. Is the predicted value close to 329, which was the actual number of burglaries? Enrollment 32 31 53 28 27 36 42 30 34 46 Burglaries 103 103 86 57 32 131 157 20 27 161

Regression and Predictions. Exercises 1328 use the same data sets as Exercises 1328 in Section 102. In each case, find the regression equation, letting the first variable be the predictor (x) variable. Find the indicated predicted value by following the prediction procedure summarized in Figure 105.Altitude and Temperature At 6327 ft (or 6.327 thousand feet), the author recorded the temperature. Find the best predicted temperature at that altitude. How does the result compare to the actual recorded value of 48 F ? Altitude 3 10 14 22 28 31 33 Temperature 57 37 24 5 30 41 54

Regression and Predictions. Exercises 1328 use the same data sets as Exercises 1328 in Section 102. In each case, find the regression equation, letting the first variable be the predictor (x) variable. Find the indicated predicted value by following the prediction procedure summarized in Figure 105.Town Courts The court for the town of Beekman had income of $83,941 (or $83.941 thousand). Find the best predicted salary for the justice. Is the result close to the actual salary of $26,088? Court Income 65 404 1567 1131 272 252 111 154 32 Justice Salary 30 44 92 56 46 61 25 26 18

Regression and Predictions. Exercises 1328 use the same data sets as Exercises 1328 in Section 102. In each case, find the regression equation, letting the first variable be the predictor (x) variable. Find the indicated predicted value by following the prediction procedure summarized in Figure 105.Auction Bids An item not included in the table is a pair of New York Knicks tickets. The auctioneer started the bidding at $300. Find the best predicted winning bid. Is the result close to the actual winning bid of $250? Opening Bid 1500 500 500 400 300 Winning Bid 650 175 125 275 125

Regression and Predictions. Exercises 1328 use the same data sets as Exercises 1328 in Section 102. In each case, find the regression equation, letting the first variable be the predictor (x) variable. Find the indicated predicted value by following the prediction procedure summarized in Figure 105.Galaxy Distances The cluster Hydra has a measured redshift of 0.0126. Find the best predicted distance to that cluster. Is the result close to the actual distance of 0.18 billion lightyears? Redshift 0.0233 0.0539 0.0718 0.0395 0.0438 0.0103 Distance 0.32 0.75 1.00 0.55 0.61 0.14

Regression and Predictions. Exercises 1328 use the same data sets as Exercises 1328 in Section 102. In each case, find the regression equation, letting the first variable be the predictor (x) variable. Find the indicated predicted value by following the prediction procedure summarized in Figure 105.Diamond Prices Not included in the table below is a diamond with a weight of 1.50 carats. Find the best predicted price for this diamond. Is the result close to the actual price of $16,097? What is wrong with predicting the price of a 1.50carat diamond? Weight 0.3 0.4 0.5 0.5 1.0 0.7 Price 510 1151 1343 1410 5669 2277

Regression and Predictions. Exercises 1328 use the same data sets as Exercises 1328 in Section 102. In each case, find the regression equation, letting the first variable be the predictor (x) variable. Find the indicated predicted value by following the prediction procedure summarized in Figure 105.Measuring Seals from Photos Find the best predicted weight of a seal if the overhead width measured from a photograph is 2 cm. Can the prediction be correct? What is wrong with predicting the weight in this case? Overhead Width 7.2 7.4 9.8 9.4 8.8 8.4 Weight 116 154 245 202 200 191

Regression and Predictions. Exercises 1328 use the same data sets as Exercises 1328 in Section 102. In each case, find the regression equation, letting the first variable be the predictor (x) variable. Find the indicated predicted value by following the prediction procedure summarized in Figure 105.Car Repair Costs Find the best predicted repair costs for a fullrear crash for a Volkswagon Passat, given that its repair cost from a fullfront crash is $4594. How does the result compare to the $982 actual repair cost from a fullrear crash? Front 936 978 2252 1032 3911 4312 3469 Rear 1480 1202 802 3191 1122 739 2767

Regression and Predictions. Exercises 1328 use the same data sets as Exercises 1328 in Section 102. In each case, find the regression equation, letting the first variable be the predictor (x) variable. Find the indicated predicted value by following the prediction procedure summarized in Figure 105. Blood Pressure Measurements Find the best predicted systolic blood pressure in the left arm given that the systolic blood pressure in the right arm is 100 mm Hg. Right Arm 102 101 94 79 79 Left Arm 175 169 182 146 144

Regression and Predictions. Exercises 1328 use the same data sets as Exercises 1328 in Section 102. In each case, find the regression equation, letting the first variable be the predictor (x) variable. Find the indicated predicted value by following the prediction procedure summarized in Figure 105.Crickets and Temperature Find the best predicted temperature at a time when a cricket chirps 3000 times in 1 minute. What is wrong with this predicted value? Chirps in 1 min 882 1188 1104 864 1200 1032 960 900 Temperature ( F ) 69.7 93.3 84.3 76.3 88.6 82.6 71.6 79.6

Regression and Predictions. Exercises 1328 use the same data sets as Exercises 1328 in Section 102. In each case, find the regression equation, letting the first variable be the predictor (x) variable. Find the indicated predicted value by following the prediction procedure summarized in Figure 105.Gas Prices One gas station not included in the table below had a listed price of $2.78 for regular gas. Find the best predicted price of premium gas at this station. Is the result close to the actual price of $2.93 for premium gas? Regular 2.77 2.77 2.79 2.81 2.78 2.86 2.75 2.77 MidGrade 3.00 2.77 2.89 2.93 2.93 2.96 2.86 2.91 Premium 3.07 3.09 3.00 3.06 3.03 3.06 3.02 3.03

Regression and Predictions. Exercises 1328 use the same data sets as Exercises 1328 in Section 102. In each case, find the regression equation, letting the first variable be the predictor (x) variable. Find the indicated predicted value by following the prediction procedure summarized in Figure 105.Gas Prices Using the data from the preceding exercise, find the best predicted price for midgrade gas for a station that posted $2.78 as the price of regular gas. Is the result close to the actual price of $2.84 for midgrade gas?

Regression and Predictions. Exercises 1328 use the same data sets as Exercises 1328 in Section 102. In each case, find the regression equation, letting the first variable be the predictor (x) variable. Find the indicated predicted value by following the prediction procedure summarized in Figure 105. Sports Find the best predicted circumference of a marble with a diameter of 1.50 cm. How does the result compare to the actual circumference of 4.7 cm? Baseball Basketball Golf Soccer Tennis PingPong Volleyball Softball Diameter 7.4 23.9 4.3 21.8 7.0 4.0 20.9 9.7 Circumference 23.2 75.1 13.5 68.5 22.0 12.6 65.7 30.5 Volume 212.2 7148.1 41.6 5424.6 179.6 33.5 4780.1 477.9

Regression and Predictions. Exercises 1328 use the same data sets as Exercises 1328 in Section 102. In each case, find the regression equation, letting the first variable be the predictor (x) variable. Find the indicated predicted value by following the prediction procedure summarized in Figure 105.Sports Using the data from the preceding exercise, find the best predicted volume of a marble with a diameter of 1.50 cm. How does the result compare to the actual volume of 1.8 cm 3 ?

Large Data Sets. Exercises 2932 use the same Appendix B data sets as Exercises 2932 in Section 102. In each case, find the regression equation, letting the first variable be the predictor (x) variable. Find the indicated predicted values following the prediction procedure summarized in Figure 105 IQ and Brain Volume Refer to Data Set 6 in Appendix B and use the paired data consisting of IQ score and brain volume ( cm 3 ) . Find the best predicted IQ score for someone with a brain volume of 1000 cm 3

Large Data Sets. Exercises 2932 use the same Appendix B data sets as Exercises 2932 in Section 102. In each case, find the regression equation, letting the first variable be the predictor (x) variable. Find the indicated predicted values following the prediction procedure summarized in Figure 106 Flight Delays Refer to Data Set 15 in Appendix B and use the departure delay times and the arrival delay times. Find the best predicted arrival delay time for a flight with no departure delay.

Large Data Sets. Exercises 2932 use the same Appendix B data sets as Exercises 2932 in Section 102. In each case, find the regression equation, letting the first variable be the predictor (x) variable. Find the indicated predicted values following the prediction procedure summarized in Figure 107 Word Counts of Men and Women Refer to Data Set 17 in Appendix B and use the word counts measured from men and women in couple relationships listed in the first two columns of Data Set 17. Find the best predicted word count for a woman who is in a couple relationship with a man having a word count of 10,000.

Large Data Sets. Exercises 2932 use the same Appendix B data sets as Exercises 2932 in Section 102. In each case, find the regression equation, letting the first variable be the predictor (x) variable. Find the indicated predicted values following the prediction procedure summarized in Figure 108 Earthquakes Refer to Data Set 16 in Appendix B and use the magnitudes and depths from the earthquakes. Find the best predicted depth of an earthquake with a magnitude of 1.50.

Equivalent Hypothesis Tests Explain why a test of the null hypothesis H 0 : = 0 is equivalent to a test of the null hypothesis H 0 : 1 = 0 , where r is the linear correlation coefficient for a population of paired data, and 1 is the slope of the regression line for that same population

LeastSquares Property According to the leastsquares property, the regression line minimizes the sum of the squares of the residuals. Refer to the data in Table 101. a. Find the sum of squares of the residuals. b. Show that the regression equation y ^ = 120 + 2.00 x results in a larger sum of squares of residuals.