Question
2. We use the form = a + bx for the least-squares line. In some computer printouts, the least-squares equation is not given directly. Instead,
2. We use the form
= a + bx
for the least-squares line. In some computer printouts, the least-squares equation is not given directly. Instead, the value of the constant a is given, and the coefficient b of the explanatory or predictor variable is displayed. Sometimes a is referred to as the constant, and sometimes as the intercept. Data from a report showed the following relationship between elevation (in thousands of feet) and average number of frost-free days per year in a state.A Minitab printout provides the following information.
| Predictor | Coef | SE Coef | T | P |
| Constant | 315.81 | 28.31 | 11.24 | 0.002 |
| Elevation | -28.275 | 3.511 | -8.79 | 0.003 |
| S = 11.8603 | R-Sq = 94.2% |
Notice that "Elevation" is listed under "Predictor." This means that elevation is the explanatory variable x. Its coefficient is the slope b. "Constant" refers to a in the equation
= a + bx.
(a) Use the printout to write the least-squares equation.
| = | + x |
(b) For each 1000-foot increase in elevation, how many fewer frost-free days are predicted? (Round your answer to three decimal places.) (c) The printout gives the value of the coefficient of determination r2. What is the value of r? Be sure to give the correct sign for r based on the sign of b. (Round your answer to four decimal places.) (d) What percentage of the variation in y can be explained by the corresponding variation in x and the least-squares line? % What percentage is unexplained? %
3. The following Minitab display gives information regarding the relationship between the body weight of a child (in kilograms) and the metabolic rate of the child (in 100 kcal/ 24 hr).
| Predictor | Coef | SE Coef | T | P |
| Constant | 0.8462 | 0.4148 | 2.06 | 0.84 |
| Weight | 0.38594 | 0.02978 | 13.52 | 0.000 |
| S = 0.517508 | R-Sq =98.0% |
(a) Write out the least-squares equation.
| = | + x |
(b) For each 1 kilogram increase in weight, how much does the metabolic rate of a child increase? (Use 5 decimal places.) (c) What is the value of the correlation coefficientr? (Use 3 decimal places.)
4. An economist is studying the job market in Denver area neighborhoods. Letxrepresent the total number of jobs in a given neighborhood, and letyrepresent the number of entry-level jobs in the same neighborhood. A sample of six Denver neighborhoods gave the following information (units in hundreds of jobs).
| x | 15 | 34 | 52 | 28 | 50 | 25 |
|---|---|---|---|---|---|---|
| y | 1 | 2 | 7 | 5 | 9 | 3 |
Complete parts (a) through (e), givenx=204,y=27,x2=7994,y2=169,xy=1112,andr0.8654.(a)Draw a scatter diagram displaying the data.
A scatter diagram has a horizontal axis labeled "x(total number of jobs (in hundreds))" with values from 15 to 52 and a vertical axis labeled "y(number of entry-level jobs (in hundreds))" with values from 1 to 9. The scatter diagram has 6 points. A pattern goes up and right from(15, 1)to(52, 9).
A scatter diagram has a horizontal axis labeled "x(total number of jobs (in hundreds))" with values from 15 to 52 and a vertical axis labeled "y(number of entry-level jobs (in hundreds))" with values from 1 to 9. The scatter diagram has 6 points. A pattern goes down and right from(15, 9)to(52, 1).
A scatter diagram has a horizontal axis labeled "x(total number of jobs (in hundreds))" with values from 1 to 9 and a vertical axis labeled "y(number of entry-level jobs (in hundreds))" with values from 15 to 52. The scatter diagram has 6 points. A pattern goes down and right from(1, 52)to(9, 15).
A scatter diagram has a horizontal axis labeled "x(total number of jobs (in hundreds))" with values from 1 to 9 and a vertical axis labeled "y(number of entry-level jobs (in hundreds))" with values from 15 to 52. The scatter diagram has 6 points. A pattern goes up and right from(1, 15)to(9, 52).
(b)Verify the given sumsx,y,x2,y2,xy, and the value of the sample correlation coefficientr. (Round your value forrto four decimal places.)x=y=x2=y2=xy=r=(c)Findx, andy. Then find the equation of the least-squares line=a+bx. (Round your answers to four decimal places.)x=y== + x(d)Graph the least-squares line. Be sure to plot the point (x,y) as a point on the line.
A graph containing a trend line and a point has a horizontal axis labeled "x(total number of jobs(in hundreds))" with values from 1 to 9 and a vertical axis labeled "y(number of entrylevel jobs(in hundreds))" with values from 15 to 52. The trend line enters the window in the first quadrant, goes down and right, passes through the approximate point(1.7, 46.8),passes through the approximate point(7.3, 21.2),and exits the window in the first quadrant. The point(4.5, 34)is plotted on the trend line.
A graph containing a trend line and a point has a horizontal axis labeled "x(total number of jobs(in hundreds))" with values from 15 to 52 and a vertical axis labeled "y(number of entrylevel jobs(in hundreds))" with values from 1 to 9. The trend line enters the window in the first quadrant, goes up and right, passes through the approximate point(20, 1.9),passes through the approximate point(48, 7.1),and exits the window in the first quadrant. The point(34, 4.5)is plotted on the trend line.
A graph containing a trend line and a point has a horizontal axis labeled "x(total number of jobs(in hundreds))" with values from 1 to 9 and a vertical axis labeled "y(number of entrylevel jobs(in hundreds))" with values from 15 to 52. The trend line enters the window in the first quadrant, goes up and right, passes through the approximate point(1.9, 23.5),passes through the approximate point(7.1, 44.5),and exits the window in the first quadrant. The point(4.5, 34)is plotted on the trend line.
A graph containing a trend line and a point has a horizontal axis labeled "x(total number of jobs(in hundreds))" with values from 15 to 52 and a vertical axis labeled "y(number of entrylevel jobs(in hundreds))" with values from 1 to 9. The trend line enters the window in the first quadrant, goes down and right, passes through the approximate point(20, 7.3),passes through the approximate point(48, 1.7),passes through the approximate point(51.8, 0.9),and exits the window in the first quadrant. The point(34, 4.5)is plotted on the trend line.
(e)Find the value of the coefficient of determinationr2. What percentage of the variation inycan beexplainedby the corresponding variation inxand the least-squares line? What percentage isunexplained? (Round your answer forr2to four decimal places. Round your answers for the percentages to two decimal place.)
r2=
explained %
unexplained %
(f)For a neighborhood withx=31hundred jobs, how many are predicted to be entry level jobs? (Round your answer to two decimal places.)
hundred jobs
5. You are the foreman of the Bar-S cattle ranch in Colorado. A neighboring ranch has calves for sale, and you are going to buy some calves to add to the Bar-S herd. How much should a healthy calf weigh? Letxbe the age of the calf (in weeks), and letybe the weight of the calf (in kilograms).
| x | 1 | 5 | 10 | 16 | 26 | 36 |
| y | 39 | 45 | 73 | 100 | 150 | 200 |
Complete parts (a) through (e), givenx=94,y=607,x2=2354,y2=81,375,xy=13,694,andr0.9975.(a) Draw a scatter diagram displaying the data.
(b) Verify the given sumsx,y,x2,y2,xy, and the value of the sample correlation coefficientr. (Round your value forrto four decimal places.)
| x= | |
| y= | |
| x2= | |
| y2= | |
| xy= | |
| r= |
(c) Findx, andy. Then find the equation of the least-squares line=a+bx. (Round your answers to four decimal places.)
| x | = | |
| y | = | |
| = | + x |
(d) Graph the least-squares line. Be sure to plot the point (x,y) as a point on the line.
(e) Find the value of the coefficient of determinationr2. What percentage of the variation inycan beexplainedby the corresponding variation inxand the least-squares line? What percentage isunexplained? (Round your answer forr2to four decimal places. Round your answers for the percentages to two decimal place.)
| r2= | |
| explained | % |
| unexplained | % |
(f) The calves you want to buy are12weeks old. What does the least-squares line predict for a healthy weight? (Round your answer to two decimal places.) kg
6. Do heavier cars really use more gasoline? Suppose a car is chosen at random. Letxbe the weight of the car (in hundreds of pounds), and letybe the miles per gallon (mpg).
| x | 29 | 42 | 33 | 47 | 23 | 40 | 34 | 52 |
|---|---|---|---|---|---|---|---|---|
| y | 28 | 19 | 26 | 13 | 29 | 17 | 21 | 14 |
Complete parts (a) through (e), givenx=300,y=167,x2=11,892,y2=3757,xy=5868,and
r0.9460.
(a)Draw a scatter diagram displaying the data.
A scatter diagram has a horizontal axis labeled "x(weight of the car (in hundreds of pounds))" with values from 13 to 29 and a vertical axis labeled "y(miles per gallon)" with values from 23 to 52. The scatter diagram has 8 points. A pattern goes down and right from(13, 52)to(29, 23).
A scatter diagram has a horizontal axis labeled "x(weight of the car (in hundreds of pounds))" with values from 23 to 52 and a vertical axis labeled "y(miles per gallon)" with values from 13 to 29. The scatter diagram has 8 points. A pattern goes up and right from(23, 13)to(52, 29).
A scatter diagram has a horizontal axis labeled "x(weight of the car (in hundreds of pounds))" with values from 13 to 29 and a vertical axis labeled "y(miles per gallon)" with values from 23 to 52. The scatter diagram has 8 points. A pattern goes up and right from(13, 23)to(29, 52).
A scatter diagram has a horizontal axis labeled "x(weight of the car (in hundreds of pounds))" with values from 23 to 52 and a vertical axis labeled "y(miles per gallon)" with values from 13 to 29. The scatter diagram has 8 points. A pattern goes down and right from(23, 29)to(52, 13).
(b)Verify the given sumsx,y,x2,y2,xy, and the value of the sample correlation coefficientr. (Round your value forrto four decimal places.)x=y=x2=y2=xy=r=(c)Findx, andy. Then find the equation of the least-squares line=a+bx. (Round your answers to four decimal places.)x=y== + x(d)Graph the least-squares line. Be sure to plot the point (x,y) as a point on the line.
A graph containing a trend line and a point has a horizontal axis labeled "x(weight of the car (in hundreds of pounds))" with values from 13 to 29 and a vertical axis labeled "y(miles per gallon)" with values from 23 to 52. The trend line enters the window in the first quadrant, goes down and right, passes through the approximate point(13.8, 47.8),passes through the approximate point(27.9, 27.2),and exits the window in the first quadrant. The approximate point(20.9, 37.5)is plotted on the trend line.
A graph containing a trend line and a point has a horizontal axis labeled "x(weight of the car (in hundreds of pounds))" with values from 13 to 29 and a vertical axis labeled "y(miles per gallon)" with values from 23 to 52. The trend line enters the window in the first quadrant, goes up and right, passes through the approximate point(13.6, 26.6),passes through the approximate point(28.1, 48.4),and exits the window in the first quadrant. The approximate point(20.9, 37.5)is plotted on the trend line.
A graph containing a trend line and a point has a horizontal axis labeled "x(weight of the car (in hundreds of pounds))" with values from 23 to 52 and a vertical axis labeled "y(miles per gallon)" with values from 13 to 29. The trend line enters the window in the first quadrant, goes up and right, passes through the approximate point(26, 13.6),passes through the approximate point(49, 28.1),and exits the window in the first quadrant. The approximate point(37.5, 20.9)is plotted on the trend line.
A graph containing a trend line and a point has a horizontal axis labeled "x(weight of the car (in hundreds of pounds))" with values from 23 to 52 and a vertical axis labeled "y(miles per gallon)" with values from 13 to 29. The trend line enters the window in the first quadrant, goes down and right, passes through the approximate point(26, 27.9),passes through the approximate point(49, 13.8),and exits the window in the first quadrant. The approximate point(37.5, 20.9)is plotted on the trend line.
(e)Find the value of the coefficient of determinationr2. What percentage of the variation inycan beexplainedby the corresponding variation inxand the least-squares line? What percentage isunexplained? (Round your answer forr2to four decimal places. Round your answers for the percentages to two decimal place.)
r2=
explained %
unexplained %
(f)Suppose a car weighsx=41(hundred pounds). What does the least-squares line forecast fory= miles per gallon? (Round your answer to two decimal places.)
mpg
7. Letxbe the percentage of 16- to 19-year-olds not in school and not high school graduates. Letybe the reported violent crimes per 1000 residents. Six small cities in Arkansas (Blytheville, El Dorado, Hot Springs, Jonesboro, Rogers, and Russellville) reported the following information aboutxandy.
| x | 23.7 | 19.5 | 18.0 | 14.9 | 19.0 | 17.5 |
| y | 13.2 | 4.1 | 9.4 | 1.3 | 0.8 | 3.6 |
Complete parts (a) through (e), givenx=112.6,y=32.4,x2=2155.2,y2=294.7,xy=659.56,andr0.7259.(a) Draw a scatter diagram displaying the data.
(b) Verify the given sumsx,y,x2,y2,xy, and the value of the sample correlation coefficientr. (Round your value forrto four decimal places.)
| x= | |
| y= | |
| x2= | |
| y2= | |
| xy= | |
| r= |
(c) Findx, andy. Then find the equation of the least-squares line=a+bx. (Round your answer to four decimal places.)
| x | = | |
| y | = | |
| = | + x |
(d) Graph the least-squares line. Be sure to plot the point (x,y) as a point on the line.
(e) Find the value of the coefficient of determinationr2. What percentage of the variation inycan beexplainedby the corresponding variation inxand the least-squares line? What percentage isunexplained? (Round your answer forr2to four decimal places. Round your answers for the percentages to two decimal place.)
| r2= | |
| explained | % |
| unexplained | % |
(f) If the percentage of 16- to 19-year-olds not in school and not graduates reaches22.5%in a similar city, what is the predicted rate of violent crimes per 1000 residents? (Round your answer to two decimal places.) violent crimes per 1000 residents
8.
Consider the following data. Do heavier cars really use more gasoline? Suppose a car is chosen at random. Letxbe the weight of the car (in hundreds of pounds), and letybe the miles per gallon (mpg). The following information is based on data taken fromConsumer Reports(Vol. 62, No. 4).
| x | 25 | 47 | 32 | 47 | 23 | 40 | 34 | 52 |
| y | 33 | 17 | 25 | 13 | 29 | 17 | 21 | 14 |
Given the least-squares line=45.01610.6371x: (a) Make a residual plot for the least-squares model. Be sure to plot the horizontal line aty= 0.
(b) Use the residual plot to comment about the appropriateness of the least squares model for these data.The residuals do not seem to be scattered randomly around the horizontal line at 0. There appear to be outliers.The residuals seem to be scattered randomly around the horizontal line at 0. There do not appear to be any outliers. The residuals do not seem to be scattered randomly around the horizontal line at 0. There do not appear to be any outliers.The residuals seem to be scattered randomly around the horizontal line at 0. They appear to be outliers.
9.
(a) Suppose you are given the following (x,y) data pairs.
| x | 1 | 3 | 4 |
| y | 2 | 1 | 8 |
Find the least-squares equation for these data (rounded to four digits after the decimal). = + x (b) Now suppose you are given these (x,y) data pairs.
| x | 2 | 1 | 8 |
| y | 1 | 3 | 4 |
Find the least-squares equation for these data (rounded to four digits after the decimal). = + x (c) In the data for parts (a) and (b), did we simply exchange thexandyvalues of each data pair?YesNo
(d) Solve your answer from part (a) forx(rounded to four digits after the decimal). x= + y Do you get the least-squares equation of part (b) with the symbolsxandyexchanged?YesNo
(e) In general, suppose we have the least-squares equationy=a+bxfor a set of data pairs (x,y). If we solve this equation forx, will wenecessarilyget the least-squares equation for the set of data pairs(y,x),(withxandyexchanged)? Explain using parts (a) through (d).Switchingxandyvalues sometimes produces the same least-squares equation and sometimes it is different.In general, switchingxandyvalues produces a different least-squares equation. In general, switchingxandyvalues produces the same least-squares equation.
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Sampling
Authors: Steven K Thompson
3rd Edition
111816296X, 9781118162965
