Linear Regression and Correlation Homework

Solve these problems, then check your answers against the given solutions.

Exercises

Exercise 12.1

For each situation below, state the independent variable and the dependent variable.

  1. A study is done to determine if elderly drivers are involved in more motor vehicle fatalities than all other drivers. The number of fatalities per 100,000 drivers is compared to the age of drivers.
  2. A study is done to determine if the weekly grocery bill changes based on the number of family members.
  3. Insurance companies base life insurance premiums partially on the age of the applicant.
  4. Utility bills vary according to power consumption.
  5. A study is done to determine if a higher education reduces the crime rate in a population.
Exercise 12.3
The average number of people in a family that received welfare for various years is given below.

Year
 Welfare family size
1969
 4.0
1973 3.6
1975
 3.2
1979
 3.0
1983
 3.0
1988
 3.0
1991
 2.9

Table 12.3

  1. Using "year " as the independent variable and "welfare family size" as the dependent variable, make a scatter plot of the data.
  2. Calculate the least squares line. Put the equation in the form of: \hat y = a +bx
  3. Find the correlation coefficient. Is it significant?
  4. Pick two years between 1969 and 1991 and find the estimated welfare family sizes.
  5. Use the two points in (d) to plot the least squares line on your graph from (b).
  6. Based on the above data, is there a linear relationship between the year and the average number of people in a welfare family?
  7. Using the least squares line, estimate the welfare family sizes for 1960 and 1995. Does the least squares line give an accurate estimate for those years? Explain why or why not.
  8. Are there any outliers in the above data?
  9. What is the estimated average welfare family size for 1986? Does the least squares line give an accurate estimate for that year? Explain why or why not.
  10. What is the slope of the least squares (best-fit) line? Interpret the slope.

Exercise 12.5
The height (sidewalk to roof) of notable tall buildings in America is compared to the number of stories of the building (beginning at street level).

Height (in feet)
 Stories
1050 57
428 28
362 26
529 40
790 60
401 22
380 38
1454 110
1127 100
700 46

Table 12.4

  1. Using "stories" as the independent variable and "height" as the dependent variable, make a scatter plot of the data.
  2. Does it appear from inspection that there is a relationship between the variables?
  3. Calculate the least squares line. Put the equation in the form of: \hat y = a + bx
  4. Find the correlation coefficient. Is it significant?
  5. Find the estimated heights for 32 stories and for 94 stories.
  6. Use the two points in (e) to plot the least squares line on your graph from (b).
  7. Based on the above data, is there a linear relationship between the number of stories in tall buildings and the height of the buildings?
  8. Are there any outliers in the above data? If so, which point(s)?
  9. What is the estimated height of a building with 6 stories? Does the least squares line give an accurate estimate of height? Explain why or why not.
  10. Based on the least squares line, adding an extra story adds about how many feet to a building?
  11. What is the slope of the least squares (best-fit) line? Interpret the slope.

Exercise 12.7
The percent of female wage and salary workers who are paid hourly rates is given below for the years 1979 - 1992.

Year
 Percent of workers paid hourly rates
1979
 61.2
1980 60.7
1981
 61.3
1982
 61.3
1983
 61.8
1984
 61.7
1985
 61.8
1986
 62.0
1987
 62.7
1990
 62.8
1992
 62.9

Table 12.6

  1.  Using "year" as the independent variable and "percent" as the dependent variable, make a scatter plot of the data.
  2. Does it appear from inspection that there is a relationship between the variables? Why or why not?
  3. Calculate the least squares line. Put the equation in the form of: \hat y = a + bx
  4. Find the correlation coefficient. Is it significant?
  5. Find the estimated percents for 1991 and 1988.
  6. Use the two points in (e) to plot the least squares line on your graph from (b).
  7. Based on the above data, is there a linear relationship between the year and the percent of female wage and salary earners who are paid hourly rates?
  8. Are there any outliers in the above data? What is the estimated percent for the year 2050? Does the least squares line give an accurate estimate for that year? Explain why or why not?
  9. What is the slope of the least squares (best-fit) line? Interpret the slope.

Exercise 12.9
  1. Using "size" as the independent variable and "cost" as the dependent variable, make a scatter plot.
  2. Does it appear from inspection that there is a relationship between the variables? Why or why not?
  3. Calculate the least squares line. Put the equation in the form of: \hat y= a + bx
  4. Find the correlation coefficient. Is it significant?
  5. If the laundry detergent were sold in a 40 ounce size, find the estimated cost.
  6. If the laundry detergent were sold in a 90 ounce size, find the estimated cost.
  7. Use the two points in (e) and (f) to plot the least squares line on your graph from (a).
  8. Does it appear that a line is the best way to fit the data? Why or why not?
  9. Are there any outliers in the above data?
  10. Is the least squares line valid for predicting what a 300 ounce size of the laundry detergent would cost? Why or why not?
  11. What is the slope of the least squares (best-fit) line? Interpret the slope.
Exercise 12.11
According to flyer by a Prudential Insurance Company representative, the costs of approximate probate fees and taxes for selected net taxable estates are as follows:

Net Taxable Estate ($) Approximate Probate Fees and Taxes ($)
600,000 30,000
750,000 92,500
1,000,000 203,000
1,500,000 438,000
2,000,000 688,000
2,500,000 1,037,000
3,000,000 1,350,000

Table 12.9

  1. Decide which variable should be the independent variable and which should be the dependent variable.
  2. Make a scatter plot of the data.
  3. Does it appear from inspection that there is a relationship between the variables? Why or why not?
  4. Calculate the least squares line. Put the equation in the form of: \hat y = a + bx
  5. Find the correlation coefficient. Is it significant?
  6. Find the estimated total cost for a net taxable estate of $1,000,000. Find the cost for $2,500,000.
  7. Use the two points in (f) to plot the least squares line on your graph from (b).
  8. Does it appear that a line is the best way to fit the data? Why or why not?
  9. Are there any outliers in the above data?
  10. Based on the above, what would be the probate fees and taxes for an estate that does not have any assets?
  11. What is the slope of the least squares (best-fit) line? Interpret the slope.
Exercise 12.13
Below are the average heights for American boys.

Age (years)
 Height (cm)
birth 50.8
2 83.8
3 91.4
5 106.6
7 119.3
10 137.1
14 157.5

Table 12.11
  1. Decide which variable should be the independent variable and which should be the dependent variable.
  2. Make a scatter plot of the data.
  3. Does it appear from inspection that there is a relationship between the variables? Why or why not?
  4. Calculate the least squares line. Put the equation in the form of: \hat y= a + bx
  5. Find the correlation coefficient. Is it significant?
  6. Find the estimated average height for a one year-old. Find the estimated average height for an eleven year-old.
  7. Use the two points in (f) to plot the least squares line on your graph from (b).
  8. Does it appear that a line is the best way to fit the data? Why or why not?
  9. Are there any outliers in the above data?
  10. Use the least squares line to estimate the average height for a sixty-two year-old man. Do you think that your answer is reasonable? Why or why not?
  11. What is the slope of the least squares (best-fit) line? Interpret the slope.
Exercise 12.15
(Use the following state information).

State # Letters in name
 Year entered the Union
 Rank for entering the Union
 Area (square miles)
Alabama 7 1819 22 52,423
Colorado 1876 38 104,100
Hawaii 1959 50 10,932
Iowa 1846 29 56,276
Maryland 1788 7 12,407
Missouri 1821 24 69,709
New Jersey 1787 3 8,722
Ohio 1803 17 44,828
South Carolina 13 1788 8 32,008
Utah
1896 45 84,904
Wisconsin 1848 30 65,499

Table 12.13

We are interested in whether or not the number of letters in a state name depends upon the year the state entered the Union.

  1. Decide which variable should be the independent variable and which should be the dependent variable.
  2. Make a scatter plot of the data.
  3. Does it appear from inspection that there is a relationship between the variables? Why or why not?
  4. Calculate the least squares line. Put the equation in the form of: \hat y= a + bx
  5. Find the correlation coefficient. What does it imply about the significance of the relationship?
  6. Find the estimated number of letters (to the nearest integer) a state would have if it entered the Union in 1900. Find the estimated number of letters a state would have if it entered the Union in 1940.
  7. Use the two points in (f) to plot the least squares line on your graph from (b).
  8. Does it appear that a line is the best way to fit the data? Why or why not?
  9. Use the least squares line to estimate the number of letters a new state that enters the Union this year would have. Can the least squares line be used to predict it? Why or why not?

Exercise 12.21

Try these multiple choice question

A correlation coefficient of -0.95 means there is a _________between the two variables.

  1. Strong positive correlation
  2. Weak negative correlation
  3. Strong negative correlation
  4. No Correlation

Exercise 12.26
We are interested in exploring the relationship between the weight of a vehicle and its fuel effi- ciency (gasoline mileage). The data in the table show the weights, in pounds, and fuel efficiency, measured in miles per gallon, for a sample of 12 vehicles.

Weight
Fuel Efficiency
2715 24
2570 28
2610 29
2750 38
3000 25
3410 22
3640 20
3700 26
3880 21
3900 18
4060 18
4710 15

Table 12.17

  1. Graph a scatterplot of the data.
  2. Find the correlation coefficient and determine if it is significant.
  3. Find the equation of the best fit line.
  4. Write the sentence that interprets the meaning of the slope of the line in the context of the data.
  5. What percent of the variation in fuel efficiency is explained by the variation in the weight of the vehicles, using the regression line? (State your answer in a complete sentence in the context of the data.)
  6. Accurately graph the best fit line on your scatterplot.
  7. For the vehicle that weights 3000 pounds, find the residual (y-yhat). Does the value predicted by the line underestimate or overestimate the observed data value?
  8. Identify any outliers, using either the graphical or numerical procedure demonstrated in the textbook.
  9. The outlier is a hybrid car that runs on gasoline and electric technology, but all other vehicles in the sample have engines that use gasoline only. Explain why it would be appropriate to remove the outlier from the data in this situation. Remove the outlier from the sample data. Find the new correlation coefficient, coefficient of determination, and best fit line.
  10. Compare the correlation coefficients and coefficients of determination before and after removing the outlier, and explain in complete sentences what these numbers indicate about how the model has changed.

Source: Barbara Illowsky and Susan Dean, https://archive.org/details/CollaborativeStatisticsHomeworkBook/page/nundefined/mode/1up
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