Question
1. How much should a healthy Shetland pony weigh? Let x be the age of the pony (in months), and let y be the average
1. How much should a healthy Shetland pony weigh? Letxbe the age of the pony (in months), and letybe the average weight of the pony (in kilograms).
| x | 3 | 6 | 12 | 16 | 24 |
| y | 60 | 95 | 140 | 166 | 173 |
(a) Make a scatter diagram of the data and visualize the line you think best fits the data.
A scatter diagram has a horizontal axis labeled "x(age (in months))" with values from 60 to 173 and a vertical axis labeled "y(average weight (in kg))" with values from 3 to 24. The scatter diagram has 5 points. A pattern goes up and right from(60, 3)to(173, 24).
A scatter diagram has a horizontal axis labeled "x(age (in months))" with values from 3 to 24 and a vertical axis labeled "y(average weight (in kg))" with values from 60 to 173. The scatter diagram has 5 points. A pattern goes down and right from(3, 173)to(24, 60).
A scatter diagram has a horizontal axis labeled "x(age (in months))" with values from 60 to 173 and a vertical axis labeled "y(average weight (in kg))" with values from 3 to 24. The scatter diagram has 5 points. A pattern goes down and right from(60, 24)to(173, 3).
A scatter diagram has a horizontal axis labeled "x(age (in months))" with values from 3 to 24 and a vertical axis labeled "y(average weight (in kg))" with values from 60 to 173. The scatter diagram has 5 points. A pattern goes up and right becoming less steep from(3, 60)to(24, 173).
(b) Would you say the correlation is low, moderate, or strong? low
moderate
strong
Would you say the correlation is positive or negative? positive
negative
(c) Use a calculator to verify that
x=61,
x2=1021,
y=634,
y2=89,710,
and
xy=9,238.
Computer. (Round your answer to four decimal places.) Asxincreases from 3 to24months, does the value ofrimply thatyshould tend to increase or decrease? Explain your answer. Given our value ofr,yshould tend to remain constant asxincreases.Given our value ofr, we can not draw any conclusions for the behavior ofyasxincreases. Given our value ofr,yshould tend to decrease asxincreases.Given our value ofr,yshould tend to increase asxincreases.
2.
Letxbe the average number of employees in a group health insurance plan, and letybe the average administrative cost as a percentage of claims.
| x | 3 | 7 | 15 | 33 | 73 |
| y | 40 | 35 | 30 | 27 | 19 |
(a)
Make a scatter diagram of the data and visualize the line you think best fits the data.
A scatter diagram has a horizontal axis labeled "x(average number of employees)" with values from 3 to 73 and a vertical axis labeled "y(average cost as percentage)" with values from 19 to 40. The scatter diagram has 5 points. A pattern goes up and right becoming less steep from(3, 19)to(73, 40).
A scatter diagram has a horizontal axis labeled "x(average number of employees)" with values from 3 to 73 and a vertical axis labeled "y(average cost as percentage)" with values from 19 to 40. The scatter diagram has 5 points. A pattern goes down and right from(3, 40)to(73, 19).
A scatter diagram has a horizontal axis labeled "x(average number of employees)" with values from 3 to 73 and a vertical axis labeled "y(average cost as percentage)" with values from 19 to 40. The scatter diagram has 5 points. A pattern goes down and right from(13, 40)to(73, 19).
A scatter diagram has a horizontal axis labeled "x(average number of employees)" with values from 3 to 73 and a vertical axis labeled "y(average cost as percentage)" with values from 19 to 40. The scatter diagram has 5 points. A pattern goes up and right from(13, 19)to(73, 40).
(b)
Would you say the correlation is low, moderate, or strong? positive or negative?
moderate and negativemoderate and positive low and negativelow and positivestrong and negativestrong and positive
(c)
Use a calculator to verify thatx=131,x2=6701,y=151,y2=4815,andxy=3093.Computer. (Round your answer to four decimal places.)
r=
Asxincreases, does the value ofrimply thatyshould tend to increase or decrease? Explain.
Given our value ofr,yshould tend to remain constant asxincreases.Given our value ofr,yshould tend to decrease asxincreases. Given our value ofr,yshould tend to increase asxincreases.Given our value ofr, we cannot draw any conclusions for the behavior ofyasxincreases.
3.
Can a low barometer reading be used to predict maximum wind speed of an approaching tropical cyclone? For a random sample of tropical cyclones, letxbe the lowest pressure (in millibars) as a cyclone approaches, and letybe the maximum wind speed (in miles per hour) of the cyclone.
| x | 1004 | 975 | 992 | 935 | 983 | 922 |
| y | 40 | 100 | 65 | 145 | 82 | 155 |
(a) Make a scatter diagram of the data and visualize the line you think best fits the data.
A scatter diagram has a horizontal axis labeled "x(lowest pressure (in millibars))" with values from 922 to 1,004 and a vertical axis labeled "y(maximum wind speed (in mph))" with values from 40 to 155. The scatter diagram has 6 points. A pattern goes up and right from(922, 40)to(1,004, 155).
A scatter diagram has a horizontal axis labeled "x(lowest pressure (in millibars))" with values from 40 to 155 and a vertical axis labeled "y(maximum wind speed (in mph))" with values from 922 to 1,004. The scatter diagram has 6 points. A pattern goes down and right becoming more steep from(40, 1,004)to(155, 922).
A scatter diagram has a horizontal axis labeled "x(lowest pressure (in millibars))" with values from 922 to 1,004 and a vertical axis labeled "y(maximum wind speed (in mph))" with values from 40 to 155. The scatter diagram has 6 points. A pattern goes down and right becoming more steep from(922, 155)to(1,004, 40).
A scatter diagram has a horizontal axis labeled "x(lowest pressure (in millibars))" with values from 40 to 155 and a vertical axis labeled "y(maximum wind speed (in mph))" with values from 922 to 1,004. The scatter diagram has 6 points. A pattern goes up and right from(40, 922)to(155, 1,004).
(b) Would you say the correlation is low, moderate, or strong? lowmoderate strong
Would you say the correlation is positive or negative? positivenegative
(c) Use a calculator to verify that
x=5811,
x2=5,633,303,
y=587,
y2=67,599
and
xy=561,231.
Computer. (Round your answer to four decimal places.) Asxincreases, does the value ofrimply thatyshould tend to increase or decrease? Explain your answer. Given our value ofr,yshould tend to decrease asxincreases.Given our value ofr,yshould tend to remain constant asxincreases. Given our value ofr, we can not draw any conclusions for the behavior ofyasxincreases.Given our value ofr,yshould tend to increase asxincreases.
4.
Is the magnitude of an earthquake related to the depth below the surface at which the quake occurs? Letxbe the magnitude of an earthquake (on the Richter scale), and letybe the depth (in kilometers) of the quake below the surface at the epicenter.
| x | 3.1 | 4.0 | 3.3 | 4.5 | 2.6 | 3.2 | 3.4 |
| y | 4.6 | 9.9 | 11.2 | 10.0 | 7.9 | 3.9 | 5.5 |
(a)
Make a scatter diagram of the data. Then visualize the line you think best fits the data.
A scatter diagram has a horizontal axis labeled "x(magnitude of an earthquake (on the Richter scale))" with values from 2.6 to 4.5 and a vertical axis labeled "y(depth (in kilometers))" with values from 3.9 to 11.2. The scatter diagram has 7 points. The points are scattered from 2.6 to 4.5 on the horizontal axis centered at 3.44 and from 3.9 to 11.2 on the vertical axis centered at 7.57.
A scatter diagram has a horizontal axis labeled "x(magnitude of an earthquake (on the Richter scale))" with values from 2.6 to 4.5 and a vertical axis labeled "y(depth (in kilometers))" with values from 3.9 to 11.2. The scatter diagram has 7 points. A pattern goes up and right from(2.6, 3.9)to(4.5, 11.2).
A scatter diagram has a horizontal axis labeled "x(magnitude of an earthquake (on the Richter scale))" with values from 3.9 to 11.2 and a vertical axis labeled "y(depth (in kilometers))" with values from 2.6 to 4.5. The scatter diagram has 7 points. The points are scattered from 3.9 to 11.2 on the horizontal axis centered at 7.57 and from 2.6 to 4.5 on the vertical axis centered at 3.44.
A scatter diagram has a horizontal axis labeled "x(magnitude of an earthquake (on the Richter scale))" with values from 3.9 to 11.2 and a vertical axis labeled "y(depth (in kilometers))" with values from 2.6 to 4.5. The scatter diagram has 7 points. A pattern goes up and right from(3.9, 2.6)to(11.2, 4.5).
(b)
Use a calculator to verify thatx=24.1,x2=85.31,y=53.0,y2=452.48andxy=187.54.
Computer. (Round your answer to four decimal places.)
Asxincreases, does the value ofrimply thatyshould tend to increase or decrease? Explain your answer.
Given our value ofr, we can not draw any conclusions for the behavior ofyasxincreases.Given our value ofr,yshould tend to remain constant asxincreases. Given our value ofr,yshould tend to increase asxincreases.Given our value ofr,yshould tend to decrease asxincreases.
5.
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 constantais given, and the coefficientbof the explanatory or predictor variable is displayed. Sometimesais 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 | -28.005 | 3.511 | -8.79 | 0.003 |
| S = 11.8603 | R-Sq =96.0% |
Notice that "Elevation" is listed under "Predictor." This means that elevation is the explanatory variablex. Its coefficient is the slopeb. "Constant" refers toain 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 determinationr2. What is the value ofr? Be sure to give the correct sign forrbased on the sign ofb. (Round your answer to four decimal places.) (d) What percentage of the variation inycan be explained by the corresponding variation inxand the least-squares line? % What percentage is unexplained? %
6.
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 | 17 | 30 | 52 | 28 | 50 | 25 |
|---|---|---|---|---|---|---|
| y | 1 | 4 | 5 | 5 | 9 | 3 |
Complete parts (a) through (e), givenx=202,y=27,x2=7802,y2=157,xy=1062,andr0.8115.
(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 1 to 9 and a vertical axis labeled "y(number of entry-level jobs (in hundreds))" with values from 17 to 52. The scatter diagram has 6 points. A pattern goes down and right from(1, 52)to(9, 17).
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 17 to 52. The scatter diagram has 6 points. A pattern goes up and right from(1, 17)to(9, 52).
A scatter diagram has a horizontal axis labeled "x(total number of jobs (in hundreds))" with values from 17 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 becoming less steep from(17, 9)to(52, 1).
A scatter diagram has a horizontal axis labeled "x(total number of jobs (in hundreds))" with values from 17 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(17, 1)to(52, 9).
(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 17 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, 6.8),passes through the approximate point(48, 2.1),and exits the window in the first quadrant. The approximate point(33.7, 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 17 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, 2.4),passes through the approximate point(48, 6.7),and exits the window in the first quadrant. The approximate point(33.7, 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 17 to 52. The trend line enters the window in the first quadrant, goes down and right, passes through the approximate point(2.1, 45.2),passes through the approximate point(6.8, 22.7),and exits the window in the first quadrant. The approximate point(4.5, 33.7)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 17 to 52. The trend line enters the window in the first quadrant, goes up and right, passes through the approximate point(2.4, 24.7),passes through the approximate point(6.7, 43.1),and exits the window in the first quadrant. The approximate point(4.5, 33.7)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=33hundred jobs, how many are predicted to be entry level jobs? (Round your answer to two decimal places.)
hundred jobs
7.
It is thought that basketball teams that make too many fouls in a game tend to lose the game even if they otherwise play well. Letxbe the number of fouls more than (i.e., over and above) the opposing team. Letybe the percentage of times the team with the larger number of fouls wins the game.
| x | 0 | 4 | 5 | 6 |
| y | 48 | 42 | 33 | 26 |
Complete parts (a) through (e), givenx=15,y=149,x2=77,y2=5833,xy=489,and
r0.9106.
(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) If a team hadx=3fouls over and above the opposing team, what does the least-squares equation forecast fory? (Round your answer to two decimal places.) %
8.
Does prison really deter violent crime? Letxrepresent percent change in the rate of violent crime andyrepresent percent change in the rate of imprisonment in the general U.S. population. For 7 recent years, the following data have been obtained.
| x | 5.9 | 5.8 | 4.2 | 5.2 | 6.2 | 6.5 | 11.1 |
| y | 1.0 | 3.8 | 7.6 | 4.0 | 3.6 | 0.1 | 4.4 |
Complete parts (a) through (e), givenx=44.9,
y=17.3,
x2=317.03,y2=121.53,
xy=107.83,
andr0.0656.(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) Considering the values ofrandr2, does it make sense to use the least-squares line for prediction? Explain your answer. The correlation between the variables is so low that it does not make sense to use the least-squares line for prediction.The correlation between the variables is so high that it makes sense to use the least-squares line for prediction. The correlation between the variables is so high that it does not make sense to use the least-squares line for prediction.The correlation between the variables is so low that it makes sense to use the least-squares line for prediction.
9.
Letxbe a random variable that represents the percentage of successful free throws a professional basketball player makes in a season. Letybe a random variable that represents the percentage of successful field goals a professional basketball player makes in a season. A random sample ofn= 6 professional basketball players gave the following information.
| x | 67 | 64 | 75 | 86 | 73 | 73 |
| y | 44 | 39 | 48 | 51 | 44 | 51 |
(a) Verify thatx=438,y=277,x2=32264,y2=12899,xy=20365, andr0.803.
| x | |
| y | |
| x2 | |
| y2 | |
| xy | |
| r |
(b) Use a 5% level of significance to test the claim that> 0. (Round your answers to two decimal places.)
| t | |
| criticalt |
ConclusionReject the null hypothesis, there is sufficient evidence that> 0.Reject the null hypothesis, there is insufficient evidence that> 0. Fail to reject the null hypothesis, there is insufficient evidence that> 0.Fail to reject the null hypothesis, there is sufficient evidence that> 0.
(c) Verify thatSe3.1357,a9.918,b0.4966, andx73.000.
| Se | |
| a | |
| b | |
| x |
(d) Find the predicted percentageof successful field goals for a player withx=65%successful free throws. (Round your answer to two decimal places.) % (e) Find a 90% confidence interval forywhenx=65. (Round your answers to one decimal place.)
| lower limit | % |
| upper limit | % |
(f) Use a 5% level of significance to test the claim that> 0. (Round your answers to two decimal places.)
| t | |
| criticalt |
ConclusionReject the null hypothesis, there is sufficient evidence that> 0.Reject the null hypothesis, there is insufficient evidence that> 0. Fail to reject the null hypothesis, there is insufficient evidence that> 0.Fail to reject the null hypothesis, there is sufficient evidence that> 0.
(g) Find a 90% confidence interval for. (Round your answers to three decimal places.)
| lower limit | |
| upper limit |
Interpret its meaning. For every percentage increase in successful free throws, the percentage of successful field goals increases by an amount that falls within the confidence interval.For every percentage increase in successful free throws, the percentage of successful field goals decreases by an amount that falls outside the confidence interval. For every percentage increase in successful free throws, the percentage of successful field goals decreases by an amount that falls within the confidence interval.For every percentage increase in successful free throws, the percentage of successful field goals increases by an amount that falls outside the confidence interval.
10.
Letxbe a random variable that represents the batting average of a professional baseball player. Letybe a random variable that represents the percentage of strikeouts of a professional baseball player. A random sample ofn= 6 professional baseball players gave the following information.
| x | 0.324 | 0.292 | 0.340 | 0.248 | 0.367 | 0.269 |
| y | 3.1 | 7.9 | 4.0 | 8.6 | 3.1 | 11.1 |
(a) Verify thatx=1.84,y=37.8,x2=0.574394,y2=294.8,xy=10.9276, andr-0.877.
| x | |
| y | |
| x2 | |
| y2 | |
| xy | |
| r |
(b) Use a10%level of significance to test the claim that0. (Use 2 decimal places.)
| t | |
| criticalt |
ConclusionReject the null hypothesis, there is sufficient evidence thatdiffers from 0.Reject the null hypothesis, there is insufficient evidence thatdiffers from 0. Fail to reject the null hypothesis, there is insufficient evidence thatdiffers from 0.Fail to reject the null hypothesis, there is sufficient evidence thatdiffers from 0.
(c) Verify thatSe1.8078,a26.419, andb-65.605.
| Se | |
| a | |
| b |
(d) Find the predicted percentageof strikeouts for a player with anx=0.3batting average. (Use 2 decimal places.) % (e) Find a95%confidence interval forywhenx=0.3. (Use 2 decimal places.)
| lower limit | % |
| upper limit | % |
(f) Use a10%level of significance to test the claim that0. (Use 2 decimal places.)
| t | |
| criticalt |
ConclusionReject the null hypothesis, there is sufficient evidence thatdiffers from 0.Reject the null hypothesis, there is insufficient evidence thatdiffers from 0. Fail to reject the null hypothesis, there is insufficient evidence thatdiffers from 0.Fail to reject the null hypothesis, there is sufficient evidence thatdiffers from 0.
(g) Find a95%confidence interval forand interpret its meaning. (Use 2 decimal places.)
| lower limit | |
| upper limit |
Interpretation For every unit increase in batting average, the percentage strikeouts increases by an amount that falls within the confidence interval.For every unit increase in batting average, the percentage strikeouts decreases by an amount that falls outside the confidence interval. For every unit increase in batting average, the percentage strikeouts increases by an amount that falls outside the confidence interval.For every unit increase in batting average, the percentage strikeouts decreases by an amount that falls within the confidence interval.
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Sampling
Authors: Steven K Thompson
3rd Edition
111816296X, 9781118162965
