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Linear Regression and Correlation: Homework
Summary: This module provides a homework for Linear Regression and Correlation as a part of Collaborative Statistics collection (col10522) by Barbara Illowsky and Susan Dean.
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Download handout (.pdf)Exercise 1
For each situation below, state the independent variable and the dependent variable.
- a. 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.
- b. A study is done to determine if the weekly grocery bill changes based on the number of family members.
- c. Insurance companies base life insurance premiums partially on the age of the applicant.
- d. Utility bills vary according to power consumption.
- e. A study is done to determine if a higher education reduces the crime rate in a population.
Solution 1
- a. Independent: Age; Dependent: Fatalities
- d. Independent: Power Consumption; Dependent: Utility
Exercise 2
In 1990 the number of driver deaths per 100,000 for the different age groups was as follows (Source: The National Highway Traffic Safety Administration's National Center for Statistics and Analysis):
| Age | Number of Driver Deaths per 100,000 |
|---|---|
| 15-24 | 28 |
| 25-39 | 15 |
| 40-69 | 10 |
| 70-79 | 15 |
| 80+ | 25 |
- a. For each age group, pick the midpoint of the interval for the x value. (For the 80+ group, use 85.)
- b. Using “ages” as the independent variable and “Number of driver deaths per 100,000” as the dependent variable, make a scatter plot of the data.
- c. Calculate the least squares (best–fit) line. Put the equation in the form of:
- d. Find the correlation coefficient. Is it significant?
- e. Pick two ages and find the estimated fatality rates.
- f. Use the two points in (e) to plot the least squares line on your graph from (b).
- g. Based on the above data, is there a linear relationship between age of a driver and driver fatality rate?
Exercise 3
The average number of people in a family that received welfare for various years is given below. (Source: House Ways and Means Committee, Health and Human Services Department)
| 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 |
- a. Using “year” as the independent variable and “welfare family size” as the dependent variable, make a scatter plot of the data.
- b. Calculate the least squares line. Put the equation in the form of:
- c. Find the correlation coefficient. Is it significant?
- d. Pick two years between 1969 and 1991 and find the estimated welfare family sizes.
- e. Use the two points in (d) to plot the least squares line on your graph from (b).
- f. Based on the above data, is there a linear relationship between the year and the average number of people in a welfare family?
- g. 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.
- h. Are there any outliers in the above data?
- i. 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.
Solution 3
- b.
- c. -0.8533, Yes
- g. No
- h. No.
- i. 2.97, Yes
Exercise 4
Use the AIDS data from the practice for this section, but this time use the columns “year #” and “# new AIDS deaths in U.S.” Answer all of the questions from the practice again, using the new columns.
Exercise 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). (Source: Microsoft Bookshelf)
| Height (in feet) | Stories |
|---|---|
| 1050 | 57 |
| 428 | 28 |
| 362 | 26 |
| 529 | 40 |
| 790 | 60 |
| 401 | 22 |
| 380 | 38 |
| 1454 | 110 |
| 1127 | 100 |
| 700 | 46 |
- a. Using “stories” as the independent variable and “height” as the dependent variable, make a scatter plot of the data.
- b. Does it appear from inspection that there is a relationship between the variables?
- c. Calculate the least squares line. Put the equation in the form of:
- d. Find the correlation coefficient. Is it significant?
- e. Find the estimated heights for 32 stories and for 94 stories.
- f. Use the two points in (e) to plot the least squares line on your graph from (b).
- g. Based on the above data, is there a linear relationship between the number of stories in tall buildings and the height of the buildings?
- h. Are there any outliers in the above data? If so, which point(s)?
- i. 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.
- j. Based on the least squares line, adding an extra story adds about how many feet to a building?
Solution 5
- b. Yes
- c.
- d. 0.9436; yes
- e. 478.70 feet; 1207.73 feet
- g. Yes
- h. Yes;
- i. 172.98; No
- j. 11.7585 feet
Exercise 6
Below is the life expectancy for an individual born in the United States in certain years. (Source: National Center for Health Statistics)
| Year of Birth | Life Expectancy |
|---|---|
| 1930 | 59.7 |
| 1940 | 62.9 |
| 1950 | 70.2 |
| 1965 | 69.7 |
| 1973 | 71.4 |
| 1982 | 74.5 |
| 1987 | 75 |
| 1992 | 75.7 |
- a. Decide which variable should be the independent variable and which should be the dependent variable.
- b. Draw a scatter plot of the ordered pairs.
- c. Calculate the least squares line. Put the equation in the form of:
- d. Find the correlation coefficient. Is it significant?
- e. Find the estimated life expectancy for an individual born in 1950 and for one born in 1982.
- f. Why aren’t the answers to part (e) the values on the above chart that correspond to those years?
- g. Use the two points in (e) to plot the least squares line on your graph from (b).
- h. Based on the above data, is there a linear relationship between the year of birth and life expectancy?
- i. Are there any outliers in the above data?
- j. Using the least squares line, find the estimated life expectancy for an individual born in 1850. Does the least squares line give an accurate estimate for that year? Explain why or why not.
Exercise 7
The percent of female wage and salary workers who are paid hourly rates is given below for the years 1979 - 1992. (Source: Bureau of Labor Statistics, U.S. Dept. of Labor)
| 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 |
- a. Using “year” as the independent variable and “percent” as the dependent variable, make a scatter plot of the data.
- b. Does it appear from inspection that there is a relationship between the variables? Why or why not?
- c. Calculate the least squares line. Put the equation in the form of:
- d. Find the correlation coefficient. Is it significant?
- e. Find the estimated percents for 1991 and 1988.
- f. Use the two points in (e) to plot the least squares line on your graph from (b).
- g. 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?
- h. Are there any outliers in the above data?
- i. 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?
Solution 7
- b. Yes
- c.
- d. 0.9448; Yes
- e. 62.9206; 62.4237
- h. No
- i. 72.639; No
Exercise 8
The maximum discount value of the Entertainment® card for the “Fine Dining” section, Edition 10, for various pages is given below.
| Page number | Maximum value ($) |
|---|---|
| 4 | 16 |
| 14 | 19 |
| 25 | 15 |
| 32 | 17 |
| 43 | 19 |
| 57 | 15 |
| 72 | 16 |
| 85 | 15 |
| 90 | 17 |
- a. Decide which variable should be the independent variable and which should be the dependent variable.
- b. Draw a scatter plot of the ordered pairs.
- c. Calculate the least squares line. Put the equation in the form of:
- d. Find the correlation coefficient. Is it significant?
- e. Find the estimated maximum values for the restaurants on page 10 and on page 70.
- f. Use the two points in (e) to plot the least squares line on your graph from (b).
- g. Does it appear that the restaurants giving the maximum value are placed in the beginning of the “Fine Dining” section? How did you arrive at your answer?
- h. Suppose that there were 200 pages of restaurants. What do you estimate to be the maximum value for a restaurant listed on page 200?
- i. Is the least squares line valid for page 200? Why or why not?
The next two questions refer to the following data: The cost of a leading liquid laundry detergent in different sizes is given below.
| Size (ounces) | Cost ($) | Cost per ounce |
|---|---|---|
| 16 | 3.99 | |
| 32 | 4.99 | |
| 64 | 5.99 | |
| 200 | 10.99 |
Exercise 9
- a. Using “size” as the independent variable and “cost” as the dependent variable, make a scatter plot.
- b. Does it appear from inspection that there is a relationship between the variables? Why or why not?
- c. Calculate the least squares line. Put the equation in the form of:
- d. Find the correlation coefficient. Is it significant?
- e. If the laundry detergent were sold in a 40 ounce size, find the estimated cost.
- f. If the laundry detergent were sold in a 90 ounce size, find the estimated cost.
- g. Use the two points in (e) and (f) to plot the least squares line on your graph from (a).
- h. Does it appear that a line is the best way to fit the data? Why or why not?
- i. Are there any outliers in the above data?
- j. Is the least squares line valid for predicting what a 300 ounce size of the laundry detergent would cost? Why or why not?
Solution 9
- b. Yes
- c.
- d. 0.9986; Yes
- e. $5.08
- f. $6.93
- i. No
- j. Not valid
Exercise 10
- a. Complete the above table for the cost per ounce of the different sizes.
- b. Using “Size” as the independent variable and “Cost per ounce” as the dependent variable, make a scatter plot of the data.
- c. Does it appear from inspection that there is a relationship between the variables? Why or why not?
- d. Calculate the least squares line. Put the equation in the form of:
- e. Find the correlation coefficient. Is it significant?
- f. If the laundry detergent were sold in a 40 ounce size, find the estimated cost per ounce.
- g. If the laundry detergent were sold in a 90 ounce size, find the estimated cost per ounce.
- h. Use the two points in (f) and (g) to plot the least squares line on your graph from (b).
- i. Does it appear that a line is the best way to fit the data? Why or why not?
- j. Are there any outliers in the above data?
- k. Is the least squares line valid for predicting what a 300 ounce size of the laundry detergent would cost per ounce? Why or why not?
Exercise 11
According to flyer by a Prudential Insurance Company representative, the costs of approximate probate fees and taxes for selected net taxable estates are as follows:
| Net Taxable Estate ($) | 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 |
- a. Decide which variable should be the independent variable and which should be the dependent variable.
- b. Make a scatter plot of the data.
- c. Does it appear from inspection that there is a relationship between the variables? Why or why not?
- d. Calculate the least squares line. Put the equation in the form of:
- e. Find the correlation coefficient. Is it significant?
- f. Find the estimated total cost for a net taxable estate of $1,000,000. Find the cost for $2,500,000.
- g. Use the two points in (f) to plot the least squares line on your graph from (b).
- h. Does it appear that a line is the best way to fit the data? Why or why not?
- i. Are there any outliers in the above data?
- j. Based on the above, what would be the probate fees and taxes for an estate that does not have any assets?
Solution 11
- c. Yes
- d.
- e. 0.9964; Yes
- f. $208,872.49; $1,028,318.20
- h. Yes
- i. No
Exercise 12
The following are advertised sale prices of color televisions at Anderson’s.
| Size (inches) | Sale Price ($) |
|---|---|
| 9 | 147 |
| 20 | 197 |
| 27 | 297 |
| 31 | 447 |
| 35 | 1177 |
| 40 | 2177 |
| 60 | 2497 |
- a. Decide which variable should be the independent variable and which should be the dependent variable.
- b. Make a scatter plot of the data.
- c. Does it appear from inspection that there is a relationship between the variables? Why or why not?
- d. Calculate the least squares line. Put the equation in the form of:
- e. Find the correlation coefficient. Is it significant?
- f. Find the estimated sale price for a 32 inch television. Find the cost for a 50 inch television.
- g. Use the two points in (f) to plot the least squares line on your graph from (b).
- h. Does it appear that a line is the best way to fit the data? Why or why not?
- i. Are there any outliers in the above data?
Exercise 13
Below are the average heights for American boys. (Source: Physician’s Handbook, 1990)
| 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 |
- a. Decide which variable should be the independent variable and which should be the dependent variable.
- b. Make a scatter plot of the data.
- c. Does it appear from inspection that there is a relationship between the variables? Why or why not?
- d. Calculate the least squares line. Put the equation in the form of:
- e. Find the correlation coefficient. Is it significant?
- f. Find the estimated average height for a one year–old. Find the estimated average height for an eleven year–old.
- g. Use the two points in (f) to plot the least squares line on your graph from (b).
- h. Does it appear that a line is the best way to fit the data? Why or why not?
- i. Are there any outliers in the above data?
- j. 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?
Solution 13
- c. Yes
- d.
- e. 0.9761; yes
- f. 72.2 cm; 143.13 cm
- h. Yes
- i. No
- j. 505.0 cm; No
Exercise 14
The following chart gives the gold medal times for every other Summer Olympics for the women’s 100 meter freestyle (swimming).
| Year | Time (seconds) |
|---|---|
| 1912 | 82.2 |
| 1924 | 72.4 |
| 1932 | 66.8 |
| 1952 | 66.8 |
| 1960 | 61.2 |
| 1968 | 60.0 |
| 1976 | 55.65 |
| 1984 | 55.92 |
| 1992 | 54.64 |
- a. Decide which variable should be the independent variable and which should be the dependent variable.
- b. Make a scatter plot of the data.
- c. Does it appear from inspection that there is a relationship between the variables? Why or why not?
- d. Calculate the least squares line. Put the equation in the form of:
- e. Find the correlation coefficient. Is the decrease in times significant?
- f. Find the estimated gold medal time for 1932. Find the estimated time for 1984.
- g. Why are the answers from (f) different from the chart values?
- h. Use the two points in (f) to plot the least squares line on your graph from (b).
- i. Does it appear that a line is the best way to fit the data? Why or why not?
- j. Use the least squares line to estimate the gold medal time for the next Summer Olympics. Do you think that your answer is reasonable? Why or why not?
The next three questions 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 |
Exercise 15
We are interested in whether or not the number of letters in a state name depends upon the year the state entered the Union.
- a. Decide which variable should be the independent variable and which should be the dependent variable.
- b. Make a scatter plot of the data.
- c. Does it appear from inspection that there is a relationship between the variables? Why or why not?
- d. Calculate the least squares line. Put the equation in the form of:
- e. Find the correlation coefficient. What does it imply about the significance of the relationship?
- f. 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.
- g. Use the two points in (f) to plot the least squares line on your graph from (b).
- h. Does it appear that a line is the best way to fit the data? Why or why not?
- i. 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?
Solution 15
- c. No
- d.
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