The golden ratio is = (1 + 5)/2. John Putz, a mathematician who was interested in music

The midterm and final exam scores of 10 students in a statistics course are tabulated as shown. (a) Calculate the least squares regression line for these data. (b) Plot the points and the least squares regression line on the same graph. (c) Find the value of \(\widehat{\sigma^2}\).

Using an Instron 4204, rectangular strips of Plexiglas® were stretched to failure in a tensile test. The following data give the change in length, in millimeters \((\mathrm{mm})\), before breaking \((x)\) and the cross–sectional area in square millimeters \(\left(m m^{2}\right)(y)\): (5.28, 52.36) (5.40, 52.58) (4.65, 51.07) (4.76, 52.28) (5.55, 53.02) (5.73, 52.10) (5.84, 52.61) (4.97, 52.21) (5.50, 52.39) (6.24, 53.77) (a) Find the equation of the least squares regression line. (b) Plot the points and the line on the same graph. (c) Interpret your output. Equation Transcription: Text Transcription: (mm) (x) (mm^2) (y)

The final grade in a calculus course was predicted on the basis of the student's high school grade point average in mathematics, Scholastic Aptitude Test (SAT) score in mathematics, and score on a mathematics entrance examination. The predicted grades \(x\) and the earned grades \(y\) for 10 students are given (2.0 represents a \(C\), \(2.3\_ a \(\mathrm{C}+, 2.7 \mathrm{a} \mathrm{B}-\), etc.) (a) Calculate the least squares regression line for these data. (b) Plot the points and the least squares regression line on the same graph. (c) Find the value of \(\widehat{\sigma^{2}}\). Equation Transcription: a Text Transcription: x y C 2.3 a C+,2.7aB? Widechat sigma^2

Astudent who considered himself to be a “car guy” was interested in how the horsepower and weight of a car affected the time that it takes the car to go from \(0\) to \(60 \mathrm{mph}\). The following table gives, for each of 14 cars, the horsepower, the time in seconds to go from 0 to \(60 \mathrm{mph}\), and the weight in pounds: (a) Calculate the least squares regression line for “\(0-60\)” versus horsepower. (b) Plot the points and the least squares regression line on the same graph. (c) Calculate the least squares regression line for “\(0–60\)” versus weight. (d) Plot the points and the least squares regression line on the same graph. (e) Which of the two variables, horsepower or weight, has the most effect on the “\(0–60\)” time? Equation Transcription: Text Transcription: 0 60 mph 0-60

The “golden ratio” is \(\phi=(1+\sqrt 5)\). John Putz, a mathematician who was interested in music, analyzed Mozart’s sonata movements, which are divided into two distinct sections, both of which are repeated in performance (see References). The length of the “Exposition” in measures is represented by a and the length of the “Development and Recapitulation” is represented by b. Putz’s conjecture was that Mozart divided his movements close to the golden ratio. That is, Putz was interested in studying whether a scatter plot of a + b against b not only would be linear, but also would actually fall along the line \(y = \phi x\). Here are the data in tabular form, in which the first column identifies the piece and movement by the Köchel cataloging system: (a) Make a scatter plot of the points a + b against the points b. Is this plot linear? (b) Find the equation of the least squares regression line. Superimpose it on the scatter plot. (c) On the scatter plot, superimpose the line \(y = \phi x\). Compare this line with the least squares regression line (graphically if you wish). (d) Find the sample mean of the points (a + b)/b. Is the mean close to \(\phi\)?

Show that the residuals, Yi Y 6i (i = 1, 2, ... , n), from the least squares fit of the simple linear regression model sum to zero.

In some situations where the regression model is useful, it is known that the mean of Y when X = 0 is equal to 0, i.e., where i for i = 1, 2, ... , n are independent and N(0, ). (a) Obtain the maximum likelihood estimators, and , of under this model. (b) Find the distributions of . (You may use, without proof, the fact that and are independent, together with Theorem 9.3-1.)

The midterm and final exam scores of 10 students in a statistics course are tabulated as shown. (a) Calculate the least squares regression line for these data. (b) Plot the points and the least squares regression line on the same graph. (c) Find the value of 72. Midterm Final Midterm Final 70 87 67 73 74 79 70 83 80 88 64 79 84 98 74 91 80 96 82 94

The final grade in a calculus course was predicted on the basis of the students high school grade point average in mathematics, Scholastic Aptitude Test (SAT) score in mathematics, and score on a mathematics entrance examination. The predicted grades x and the earned grades y for 10 students are given (2.0 represents a C, 2.3 a C+, 2.7 a B, etc.). (a) Calculate the least squares regression line for these data. (b) Plot the points and the least squares regression line on the same graph. (c) Find the value of 72. xyxy 2.0 1.3 2.7 3.0 3.3 3.3 4.0 4.0 3.7 3.3 3.7 3.0 2.0 2.0 3.0 2.7 2.3 1.7 2.3 3.0

A student who considered himself to be a “car guy” was interested in how the horsepower and weight of a car affected the time that it takes the car to go from 0 to 60 mph. The following table gives, for each of 14 cars, the horsepower, the time in seconds to go from 0 to 60 mph, and the weight in pounds: (a) Calculate the least squares regression line for “0–60” versus horsepower. (b) Plot the points and the least squares regression line on the same graph. (c) Calculate the least squares regression line for “0–60” versus weight. (d) Plot the points and the least squares regression line on the same graph. (e) Which of the two variables, horsepower or weight, has the most effect on the “0–60” time?

Let x and y equal the ACT scores in social science and natural science, respectively, for a student who is applying for admission to a small liberal arts college. A sample of n = 15 such students yielded the following data: xy xy xy 32 28 30 27 26 32 23 25 17 23 16 22 23 24 20 30 21 28 23 32 17 18 24 31 26 31 18 18 30 26 (a) Calculate the least squares regression line for these data. (b) Plot the points and the least squares regression line on the same graph. (c) Find point estimates for , , and 2.

The Federal Trade Commission measured the number of milligrams of tar and carbon monoxide (CO) per cigarette for all domestic cigarettes. Let x and y equal the measurements of tar and CO, respectively, for 100- millimeter filtered and mentholated cigarettes. A sample of 12 brands yielded the following data: Brand x y Brand x y Capri 9 6 Now 3 4 Carlton 4 6 Salem 17 18 Kent 14 14 Triumph 6 8 Kool Milds 12 12 True 7 8 Marlboro Lights 10 12 Vantage 8 13 Merit Ultras 5 7 Virginia Slims 15 13 (a) Calculate the least squares regression line for these data. (b) Plot the points and the least squares regression line on the same graph. (c) Find point estimates for , , and 2.

The data in the following table, part of a set of data collected by Ledolter and Hogg (see References), provide the number of miles per gallon (mpg) for city and highway driving of 2007 midsize-model cars, as well as the curb weight of the cars: mpg mpg Curb Type City Hwy Weight Ford Fusion V6 SE 20 28 3230 Chevrolet Sebring Sedan Base 24 32 3287 Toyota Camry Solara SE 24 34 3240 Honda Accord Sedan 20 29 3344 Audi A6 3.2 21 29 3825 BMW 5-series 525i Sedan 20 29 3450 Chrysler PT Cruiser Base 22 29 3076 Mercedes E-Class E350 Sedan 19 26 3740 Volkswagen Passat Sedan 2.0T 23 32 3305 Nissan Altima 2.5 26 35 3055 Kia Optima LX 24 34 3142 (a) Find the least squares regression line for highway mpg (y) and city mpg (x). (b) Plot the points and the least squares regression line on the same graph. (c) Repeat parts (a) and (b) for the regression of highway mpg (y) on curb weight (x).

The data in the following table, part of a set of data collected by Ledolter and Hogg (see References), provide the number of miles per gallon (mpg) for city and highway driving of 2007 midsize-model cars, as well as the curb weight of the cars: mpg mpg Curb Type City Hwy Weight Ford Fusion V6 SE 20 28 3230 Chevrolet Sebring Sedan Base 24 32 3287 Toyota Camry Solara SE 24 34 3240 Honda Accord Sedan 20 29 3344 Audi A6 3.2 21 29 3825 BMW 5-series 525i Sedan 20 29 3450 Chrysler PT Cruiser Base 22 29 3076 Mercedes E-Class E350 Sedan 19 26 3740 Volkswagen Passat Sedan 2.0T 23 32 3305 Nissan Altima 2.5 26 35 3055 Kia Optima LX 24 34 3142 (a) Find the least squares regression line for highway mpg (y) and city mpg (x). (b) Plot the points and the least squares regression line on the same graph. (c) Repeat parts (a) and (b) for the regression of highway mpg (y) on curb weight (x).(mm), before breaking (x) and the crosssectional area in square millimeters (mm2) (y): (5.28, 52.36) (5.40, 52.58) (4.65, 51.07) (4.76, 52.28) (5.55, 53.02) (5.73, 52.10) (5.84, 52.61) (4.97, 52.21) (5.50, 52.39) (6.24, 53.77) (a) Find the equation of the least squares regression line. (b) Plot the points and the line on the same graph. (c) Interpret your output.

The golden ratio is = (1 + 5)/2. John Putz, a mathematician who was interested in music, analyzed Mozarts sonata movements, which are divided into two distinct sections, both of which are repeated in performance (see References). The length of the Exposition in measures is represented by a and the length of the Development and Recapitulation is represented by b. Putzs conjecture was that Mozart divided his movements close to the golden ratio. That is, Putz was interested in studying whether a scatter plot of a + b against b not only would be linear, but also would actually fall along the line y = x. Here are the data in tabular form, in which the first column identifies the piece and movement by the Kchel cataloging system: (a) Make a scatter plot of the points a + b against the points b. Is this plot linear? (b) Find the equation of the least squares regression line. Superimpose it on the scatter plot. Kchel a ba + b Kchel a ba + b 279, I 38 62 100 279, II 28 46 74 279, III 56 102 158 280, I 56 88 144 280, II 24 36 60 280, III 77 113 190 281, I 40 69 109 281, II 46 60 106 282, I 15 18 33 282, III 39 63 102 283, I 53 67 120 283, II 14 23 37 283, III 102 171 273 284, I 51 76 127 309, I 58 97 155 311, I 39 73 112 310, I 49 84 133 330, I 58 92 150 330, III 68 103 171 332, I 93 136 229 332, III 90 155 245 333, I 63 102 165 333, II 31 50 81 457, I 74 93 167 533, I 102 137 239 533, II 46 76 122 545, I 28 45 73 547a, I 78 118 196 570, I 79 130 209 (c) On the scatter plot, superimpose the line y = x. Compare this line with the least squares regression line (graphically if you wish). (d) Find the sample mean of the points (a + b)/b. Is the mean close to ?