Deck 9: Re-Expressing Data: Get It Straight

Question 1

One of the important factors determining a car's fuel efficiency is its weight.This relationship is examined for 11 cars,and the association is shown in the scatterplot below.
Describe the association between these variables shown in the scatterplot.

A)Fairly Exponential,positive,strong relational
B)Fairly Quadratic,positive,weak relationship
C)Fairly linear,negative,strong relationship
D)Fairly linear,negative,weak relationship
E)Fairly linear,positive,strong relationship

Answer 1

Fairly linear,negative,strong relationship

Question 2

The model
= 12 + 20(dia)can be used to predict the breaking strength of a rope (in kilograms)from its diameter (in metres).According to this model,how much force should a rope one-half metre in diameter be able to withstand?

A)256 kg
B)16 kg
C)484 kg
D)4.7 kg
E)22 kg

Answer 2

484 kg

Question 3

For the model
= 2.93(
),predict y when x = 2.Round to two decimal places.

A)396.97
B)49.17
C)16.53
D)135.48
E)39.85

Answer 3

135.48

Question 4

One of the important factors determining a car's fuel efficiency is its weight.This relationship is examined for 11 cars,and the association is shown in the scatterplot below.
If a linear model is considered,the regression analysis is as follows:
Dependent variable: MPG
R-squared = 84.7%
VARIABLE COEFFICIENT
Intercept 47.1181
Weight -7.34614
The residuals plot is:
Based upon the residuals plot,do you think that this linear model is appropriate?

A)Yes,residuals show no pattern.
B)Yes,residuals show a linear pattern.
C)No,residuals show a curved pattern.
D)No,residuals show no pattern.
E)Yes,residuals show a curved pattern.

Question 5

For the model
= 0.1
+ 2.1x + 8,predict y when x = 3.Round to two decimal places.

A)33.6
B)17
C)-33.6
D)15.2
E)-15.2

Question 6

One of the important factors determining a car's fuel efficiency is its weight.This relationship is examined for 11 cars,and the association is shown in the scatterplot below.
If a linear model is considered,the regression analysis is as follows:
Dependent variable: MPG
R-squared = 84.7%
VARIABLE COEFFICIENT
Intercept 47.1181
Weight -7.34614
What does the slope say about this relationship?

A)Gas mileage increases an average of 7.346 mpg for each thousand pounds of weight.
B)Gas mileage decreases an average of 7.346 mpg for each thousand pounds of weight.
C)Gas mileage increases an average of 4.712 mpg for each thousand pounds of weight.
D)Gas mileage decreases an average of 4.712 mpg for each thousand pounds of weight.
E)Gas mileage decreases an average of .7346 mpg for each thousand pounds of weight.

Question 7

The model
= 3.30 + 0.235(speed)can be used to predict the stopping distance (in centimetres)for a car traveling at a specific speed (in km/h).According to this model,about how much distance will a car going 65 km/h need to stop?

A)345.0 cm
B)729.0 cm
C)4.3 cm
D)18.6 cm
E)27.0 cm

Question 8

For the model
= 2.72 + 2.3x,predict y when x = 5.Round to two decimal places.

A)14.22
B)13.49
C)14.79
D)-14.22
E)-14.79

Question 9

It's easy to measure the circumference of a tree's trunk,but not so easy to measure its height.Foresters developed a model for ponderosa pine trees that they use to predict the tree's height (h,in metres)from the circumference of its trunk (c,in centimetres): ln
= -1.2 + 1.4(ln c).A lumberjack finds a tree with a circumference of 30 cm.How tall does this model estimate the tree to be?

A)35 m
B)83 m
C)11 m
D)5 m
E)19 m

Question 10

For the model
= 1.77 + 1.6x,predict y when x = 2.Round to two decimal places.

A)4.97
B)0.20
C)-0.70
D)-1.43
E)0.17

Question 11

For the model
= 4.16 + 7.9
,predict y when x = 4.Round to two decimal places.

A)130.56
B)19.96
C)35.76
D)0.00
E)-11.64

Question 12

One of the important factors determining a car's fuel efficiency is its weight.This relationship is examined for 11 cars,and the association is shown in the scatterplot below.
If a linear model is considered,the regression analysis is as follows:
Dependent variable: MPG
R-squared = 84.7%
VARIABLE COEFFICIENT
Intercept 47.1181
Weight -7.34614
The residuals plot is:
Based upon the information above,what re-expression of Y would be appropriate?

A)log y gallons per mile
B)100 ∙

Gallons per 100 miles
C)-100 ∙

Gallons per 100 miles
D)100 ∙

Gallons per 100 miles
E)100 ∙

Gallons per 100 miles

Question 13

For the model ln
= 3.21 + 8.1x,predict y when x = 5.Round to two decimal places.

A)1.64
B)43.71
C)9,616,384,107,947,345,920.00
D)

Is undefined for x = 5.
E)3.78

Question 14

The relationship between two quantities x and y is examined.The relationship appears to be fairly linear.A linear model is considered,and the regression analysis is as follows:
Dependent variable: y
R-squared = 87.9%
VARIABLE COEFFICIENT
Intercept 37.74
X -9.97
What does the slope say about the relationship between x and y?

A)For each increase in x of 1,the corresponding average decrease in y is 9.97.
B)For each increase in x of 1,the corresponding average increase in y is 37.74.
C)For each increase in x of 1,the corresponding average decrease in y is 37.74.
D)For each increase in x of 1,the corresponding average increase in y is 9.97.
E)For each increase in x of 1,y decreases by and average of 87.9%.

Question 15

For the model
= 4.3 + 0.1x,predict y when x = 5.Round to two decimal places.

A)23.04
B)22.23
C)24.27
D)4.8
E)2.19

Question 16

For the model log
= 2.1 + 0.5x,predict y when x = 2.Round to two decimal places.

A)158,489.32
B)1,267.63
C)1,258.93
D)1,274.13
E)0.49

Question 17

For the model
= 2.8
,predict y when x = 4.Round to two decimal places.

A)3.76
B)11.2
C)4.88
D)4.48
E)7.03

Question 18

The relationship between two quantities x and y is examined,and the association is shown in the scatterplot below.
Describe the association between these variables shown in the scatterplot.

A)Fairly linear,negative,weak relationship
B)Fairly linear,negative,strong relationship
C)Fairly linear,positive,weak relationship
D)Fairly linear,positive,strong relationship
E)Fairly exponential,strong relationship

Question 19

For the model
= 1.4 + 0.08 log x,predict y when x = 2.Round to two decimal places.

A)6.72
B)9.4
C)1.42
D)1.56
E)-0.94

Question 20

The relationship between two quantities x and y is examined,and the association is shown in the scatterplot below.
Describe the association between these variables shown in the scatterplot.

A)Fairly linear,weak relationship
B)Fairly exponential,strong relationship
C)Fairly linear,strong relationship
D)Fairly exponential,weak relationship
E)Fairly quadratic,weak relationship

Question 21

The consumer price index (CPI)is a measure of the relative cost of goods in the a given country for a particular year.The table below shows the CPI for a country for the stated years beginning in 1940.
Create an appropriate model to re-express the CPI.What re-expression of the CPI does this model involve?

A)-

B)

C)log (CPI)
D)

E)

Question 22

The relationship between two quantities X and Y is examined,and the association is shown in the scatterplot below.
Should you re-express these data before trying to fit a model? Explain.

A)Data should be re-expressed.Scatterplot shows upward curvature and increasing scatter as X increases.
B)Data should be re-expressed.Scatterplot shows linear relationship and uniform scatter.
C)Data does not need to be re-expressed.Scatterplot shows upward curvature and increasing scatter as X increases.
D)Data should be re-expressed.Scatterplot shows linear relationship.
E)Data does not need to be re-expressed.Scatterplot shows linear relationship.

Question 23

Dioxins are a class of long-lived and highly toxic pollutants.The topsoil concentration in parts per million (ppm)are shown in the table below as a function of distance from the dump.
Create an appropriate model to re-express the concentration (ppm).What re-expression of the concentration (ppm)does this model involve?

A)

B)

C)

D)

E)log (ppm)

Question 24

The consumer price index (CPI)is a measure of the relative cost of goods in a country for a particular year.The table below shows the CPI for various years beginning in 1940.
a)Re-express the CPI by taking the log of the data,then plot the re-expressed data as a scatterplot.
b)Determine the regression equation and correlation coefficient for the re-expressed data.
c)Use your model from part (b)to predict the CPI in 2010.

Question 25

The relationship between two quantities X and Y is examined,and the association is shown in the scatterplot below.
If a linear model is considered,the regression analysis is as follows:
Dependent variable: Y
R-squared = 84.7%
VARIABLE COEFFICIENT
Intercept 1.2305
X .4443
What does the slope say about this relationship?

A)For every increase in X of 1,the corresponding average increase in Y is .4443
B)For every increase in X of .5,the corresponding average decrease in Y is .4443
C)For every increase in X of .5,the corresponding average increase in Y is .4443
D)For every increase in X of 1,the corresponding average increase in Y is 1.2305
E)For every increase in X of 1,the corresponding average decrease in Y is .4443

Question 26

The consumer price index (CPI)is a measure of the relative cost of goods in the a given country for a particular year.The table below shows the CPI for a country for the stated years beginning in 1940.
Re-express the CPI.Then determine the regression equation for the re-expressed data.Use your model to predict the CPI in 2,017.

A)289.3
B)277
C)296.5
D)261.2
E)241.8

Question 27

A university radio station has been airing a public service announcement (PSA)that gives a telephone number students can call to get information on upcoming campus events.In order to determine the optimal number of times per day to run the PSA,the station has been counting the number of calls generated each day as a function of the number of times the PSA was aired.The results of the experiment are summarized in the table below.
a)Re-express the data by squaring the number of calls per day,then plot the re-expressed data as a scatterplot.
b)Determine the regression equation and correlation coefficient for the re-expressed data.
c)Interpret the slope of your model.Comment on the meaning and usefulness of this slope.

Question 28

Dioxins are a class of long-lived and highly toxic pollutants.The topsoil concentration of dioxins in parts per million (ppm)are shown in the table below as a function of the distance from the dump.
a)Re-express the dioxin concentration by taking the reciprocal square root of the data,then plot the re-expressed data as a scatterplot.
b)Determine the regression equation and correlation coefficient for the re-expressed data.
c)Use your model from part (b)to predict the dioxin concentration at a distance of 500 metres.

Question 29

Dioxins are a class of long-lived and highly toxic pollutants.The topsoil concentration in parts per million (ppm)are shown in the table below as a function of distance from the dump.
Create an appropriate model to re-express the concentration (ppm).Then determine the regression equation and correlation coefficient for the re-expressed data.Round all figures to 4 decimal places.

A)Re-expression:

;y = .0012x + 0.0218,r = 0.9985
B)Re-expression:

;y = .0012x + .0500,r = 1
C)Re-expression:

;y = .0012x + 0.0233,r = 0.9970
D)Re-expression:

;y = .0012x + 0.0251,r = 0.9928
E)Re-expression:

;y = .0012x + 0.0277,r = 0.9873

Question 30

The relationship between two quantities X and Y is examined,and the association is shown in the scatterplot below.
What re-expression of Y should be tried as a starting point?

A)log (Y)against X
B)

Against X
C)

Against X
D)

Against X
E)

Against X

Question 31

A forester would like to know how big a maple tree might be at age 50 years.She gathers data from some trees that have been cut down,and plots the diameters (in inches)of the trees against their ages (in years).First she makes a linear model.The scatterplot and residuals plot are shown.If she uses this model to try to predict the diameter of a 50-year old maple tree,would you expect that estimate to be fairly accurate,too low,or too high? Explain.

Question 32

A company manufactures of polypropylene rope in six different sizes.To assess the strength of the ropes they test two samples of each size to see how much force (in kilograms)the ropes will hold without breaking.The table shows the results of the tests.We want to create a model for predicting the breaking strength from the diameter of the rope.Find a model that uses re-expressed data to straighten the scatterplot.

Question 33

Doctors studying how the human body assimilates medication inject some patients with penicillin,and then monitor the concentration of the drug (in units/cc)in the patients' blood for seven hours.The data are shown in the scatterplot.First they tried to fit a linear model.Now the researchers try a new model,using the re-expression log(Concentration).Examine the regression analysis and the residuals plot below.Explain why you think this model is better than the original linear model.

Question 34

A forester would like to know how big a maple tree might be at age 50 years.She gathers data from some trees that have been cut down,and plots the diameters (in inches)of the trees against their ages (in years).First she makes a linear model.The scatterplot and residuals plot are shown.Do you think the linear model is appropriate? Explain.

Question 35

A forester would like to know how big a maple tree might be at age 50 years.She gathers data from some trees that have been cut down,and plots the diameters (in inches)of the trees against their ages (in years).First she makes a linear model.The scatterplot and residuals plot are shown.Describe the association shown in the scatterplot.

Question 36

A psychology graduate student is studying the effect of alcohol intoxication on a person's ability to perform manual tasks.In a very simple experiment,a subject's blood alcohol level is brought to a certain level and then the subject is presented with 10 rope knots.The subject then attempts to untie as many knots as he or she can in 30 seconds.The results of the experiments are summarized in the table below.
a)Re-express the number of knots untied by taking the inverse of each count,then plot the re-expressed data as a scatterplot.
b)Determine the regression equation and correlation coefficient for the re-expressed data.
c)Interpret the y-intercept of your model.Does it have meaning?

Question 37

Dioxins are a class of long-lived and highly toxic pollutants.The topsoil concentration in parts per million (ppm)are shown in the table below as a function of distance from the dump.
Create an appropriate model to re-express the Concentration (ppm).Then use your model to predict the dioxin concentration at a distance of 400 metres.Round all figures to 4 decimal places.

A)3.56 ppm
B)3.9477 ppm
C)3.9196 ppm
D)3.8796 ppm
E)3.9714 ppm

Question 38

The consumer price index (CPI)is a measure of the relative cost of goods in the a given country for a particular year.The table below shows the CPI for a country for the stated years beginning in 1940.
Re-express the CPI.Then determine the regression equation and correlation coefficient for the re-expressed data.

A)Re-expression: log(CPI);y = 0.0184x - 34.5699,r = 0.9874
B)Re-expression: - log(CPI);y = -0.0184x - 34.5699,r = 0.9780
C)Re-expression:

;y = 0.0295x - 32.2299,r = 0.9874
D)Re-expression: -

;y = 0.0755x - 36.9099,r = 0.9856
E)Re-expression:

;y = 0.0184x - 34.5699,r = 1.0379

Question 39

A forester would like to know how big a maple tree might be at age 50 years.She gathers data from some trees that have been cut down,and plots the diameters (in inches)of the trees against their ages (in years).First she makes a linear model.Now she re-expresses the data,using the logarithm of age to try to predict the diameter of the tree.Here are the regression analysis and the residuals plot.Explain why you think this is a better model.

Question 40

Doctors studying how the human body assimilates medication inject some patients with penicillin,and then monitor the concentration of the drug (in units/cc)in the patients' blood for seven hours.The data are shown in the scatterplot.First they tried to fit a linear model.The regression analysis and residuals plot are shown.Is that estimate likely to be accurate,too low,or too high? Explain.

Question 41

Doctors studying how the human body assimilates medication inject a patient with penicillin,and then monitor the concentration of the drug in the patient's blood for several hours.The data are shown in the table.
Use your model to estimate what the concentration of penicillin will be after 8 hours.

Question 42

A rock is dropped from a tall building and its distance (in feet)below the point of release is recorded as accurately as possible at various times after the moment of release.The results are shown in the table.Use technology to create a curved model for the data.

Question 43

Use technology to create a curved model for the data.

Question 44

A company manufactures of polypropylene rope in six different sizes.To assess the strength of the ropes they test two samples of each size to see how much force (in kilograms)the ropes will hold without breaking.The table shows the results of the tests.We want to create a model for predicting the breaking strength from the diameter of the rope.Use technology to create a curved model for the data.

Question 45

QuarkNet,a project funded by the National Science Foundation and the U.S.Department of Energy,poses the following problem on its website:
"Last year,deep within the Soudan mine,QuarkNet teachers began a long-term experiment to measure the amount of carbon-14 remaining in an initial 100-gram sample at 2000-year intervals.The experiment will be complete in the year 32001.Fortunately,a method for sending information backwards in time will be discovered in the year 29998,so,although the experiment is far from over,the results are in."
Here is a portion of the data:
A scatterplot of these data looks like:
Straighten the scatterplot by re-expressing these data and create an appropriate model for predicting the mass from the year.

Question 46

QuarkNet,a project funded by the National Science Foundation and the U.S.Department of Energy,poses the following problem on its website:
"Last year,deep within the Soudan mine,QuarkNet teachers began a long-term experiment to measure the amount of carbon-14 remaining in an initial 100-gram sample at 2000-year intervals.The experiment will be complete in the year 32001.Fortunately,a method for sending information backwards in time will be discovered in the year 29998,so,although the experiment is far from over,the results are in."
Here is a portion of the data:
A scatterplot of these data looks like:
Use your model to estimate what the mass will be after 7500 years.

Question 47

A forester would like to know how big a maple tree might be at age 50 years.She gathers data from some trees that have been cut down,and plots the diameters (in inches)of the trees against their ages (in years).First she makes a linear model.Now she re-expresses the data,using the logarithm of age to try to predict the diameter of the tree.Here are the regression analysis and the residuals plot.Using this model,predict the diameter of a maple tree at age 50 years.

Question 48

Use technology to create a curved model for the data.

Question 49

Doctors studying how the human body assimilates medication inject a patient with penicillin,and then monitor the concentration of the drug in the patient's blood for several hours.The data are shown in the table.
Straighten the scatterplot by re-expressing these data and create an appropriate model for predicting the concentration of penicillin.

Question 50

QuarkNet,a project funded by the National Science Foundation and the U.S.Department of Energy,poses the following problem on its website:
"Last year,deep within the Soudan mine,QuarkNet teachers began a long-term experiment to measure the amount of carbon-14 remaining in an initial 100-gram sample at 2000-year intervals.The experiment will be complete in the year 32001.Fortunately,a method for sending information backwards in time will be discovered in the year 29998,so,although the experiment is far from over,the results are in."
Here is a portion of the data:
A scatterplot of these data looks like:
Can you use your model to predict when 50 g of the sample will be left? Explain.

Question 51

A company manufactures of polypropylene rope in six different sizes.To assess the strength of the ropes they test two samples of each size to see how much force (in kilograms)the ropes will hold without breaking.The table shows the results of the tests.We want to create a model for predicting the breaking strength from the diameter of the rope.The company is thinking of introducing a new 25 mm rope.How strong should it be? (Write a sentence in context based on one of your models. )