73.
The average number of people in a family who attended college for various years is given in Table 12.31.
| Year | No. of Family Members Attending College |
|---|---|
| 1969 | 4.0 |
| 1973 | 3.6 |
| 1975 | 3.2 |
| 1979 | 3.0 |
| 1983 | 3.0 |
| 1988 | 3.0 |
| 1991 | 2.9 |
Table
12.31
- Using year as the independent variable and number of family members attending college as the dependent variable, draw a scatter plot of the data.
- Calculate the least-squares line. Put the equation in the form ŷ = a + bx.
- Does the y-intercept, a, have any meaning here?
- Find the correlation coefficient. Is it significant?
- Pick two years between 1969 and 1991 and find the estimated number of family members attending college.
- Based on the data in Table 12.31, is there a linear relationship between the year and the average number of family members attending college?
- Using the least-squares line, estimate the number of family members attending college for 1960 and 1995. Does the least-squares line give an accurate estimate for those years? Explain why or why not.
- Are there any outliers in the data?
- What is the estimated average number of family members attending college for 1986? Does the least-squares line give an accurate estimate for that year? Explain why or why not.
- What is the slope of the least-squares (best-fit) line? Interpret the slope.
74.
The percent of female wage and salary workers who are paid hourly rates is given in Table 12.32 for the years 1979 to 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
12.32
- Using year as the independent variable and percent of workers paid hourly rates as the dependent variable, draw a scatter plot of the data.
- Does it appear from inspection that there is a relationship between the variables? Why or why not?
- Does the y-intercept, a, have any meaning here?
- Calculate the least-squares line. Put the equation in the form ŷ = a + bx.
- Find the correlation coefficient. Is it significant?
- Find the estimated percentages for 1991 and 1988.
- Based on the data, is there a linear relationship between the year and the percentage of female wage and salary earners who are paid hourly rates?
- Are there any outliers in the data?
- What is the estimated percentage for the year 2050? Does the least-squares line give an accurate estimate for that year? Explain why or why not.
- What is the slope of the least-squares (best-fit) line? Interpret the slope.
Use the following information to answer the next two exercises. The cost of a leading liquid laundry detergent in different sizes is given in Table 12.33.
| Size (ounces) | Cost ($) | Cost per Ounce |
|---|---|---|
| 16 | 3.99 | |
| 32 | 4.99 | |
| 64 | 5.99 | |
| 200 | 10.99 |
Table
12.33
75.
- Using size as the independent variable and cost as the dependent variable, draw a scatter plot.
- Does it appear from inspection that there is a relationship between the variables? Why or why not?
- Calculate the least-squares line. Put the equation in the form ŷ = a + bx.
- Find the correlation coefficient. Is it significant?
- If the laundry detergent were sold in a 40 oz. size, what is the estimated cost?
- If the laundry detergent were sold in a 90 oz. size, what is the estimated cost?
- Does it appear that a line is the best way to fit the data? Why or why not?
- Are there any outliers in the given data?
- Is the least-squares line valid for predicting what a 300 oz. size of the laundry detergent would cost? Why or why not?
- What is the slope of the least-squares (best-fit) line? Interpret the slope.
76.
- Complete Table 12.33 for the cost per ounce of the different sizes of laundry detergent.
- Using size as the independent variable and cost per ounce as the dependent variable, draw a scatter plot of the data.
- Does it appear from inspection that there is a relationship between the variables? Why or why not?
- Calculate the least-squares line. Put the equation in the form ŷ = a + bx.
- Find the correlation coefficient. Is it significant?
- If the laundry detergent were sold in a 40 oz. size, what is the estimated cost per ounce?
- If the laundry detergent were sold in a 90 oz. size, what is the estimated cost per ounce?
- Does it appear that a line is the best way to fit the data? Why or why not?
- Are there any outliers in the the data?
- Is the least-squares line valid for predicting what a 300 oz. size of the laundry detergent would cost per ounce? Why or why not?
- What is the slope of the least-squares (best-fit) line? Interpret the slope.
77.
According to a flyer published by Prudential Insurance Company, 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
12.34
- Decide which variable should be the independent variable and which should be the dependent variable.
- Draw a scatter plot of the data.
- Does it appear from inspection that there is a relationship between the variables? Why or why not?
- Calculate the least-squares line. Put the equation in the form ŷ = a + bx.
- Find the correlation coefficient. Is it significant?
- Find the estimated total cost for a net taxable estate of $1,000,000. Find the cost for $2,500,000.
- Does it appear that a line is the best way to fit the data? Why or why not?
- Are there any outliers in the data?
- Based on these results, what would be the probate fees and taxes for an estate that does not have any assets?
- What is the slope of the least-squares (best-fit) line? Interpret the slope.
78.
The following are advertised sale prices of color televisions at Anderson’s:
| Size (inches) | Sale Price ($) |
|---|---|
| 9 | 147 |
| 20 | 197 |
| 27 | 297 |
| 31 | 447 |
| 35 | 1,177 |
| 40 | 2,177 |
| 60 | 2,497 |
Table
12.35
- Decide which variable should be the independent variable and which should be the dependent variable.
- Draw a scatter plot of the data.
- Does it appear from inspection that there is a relationship between the variables? Why or why not?
- Calculate the least-squares line. Put the equation in the form ŷ = a + bx.
- Find the correlation coefficient. Is it significant?
- Find the estimated sale price for a 32-inch television. Find the cost for a 50-inch television.
- Does it appear that a line is the best way to fit the data? Why or why not?
- Are there any outliers in the data?
- What is the slope of the least-squares (best-fit) line? Interpret the slope.
79.
Table 12.36 shows the average heights for American boys in 1990.
| Age (years) | Height (centimeters) |
|---|---|
| Birth | 50.8 |
| 2 | 83.8 |
| 3 | 91.4 |
| 5 | 106.6 |
| 7 | 119.3 |
| 10 | 137.1 |
| 14 | 157.5 |
Table
12.36
- Decide which variable should be the independent variable and which should be the dependent variable.
- Draw a scatter plot of the data.
- Does it appear from inspection that there is a relationship between the variables? Why or why not?
- Calculate the least-squares line. Put the equation in the form ŷ = a + bx.
- Find the correlation coefficient. Is it significant?
- Find the estimated average height for a 1-year-old. Find the estimated average height for an 11-year-old.
- Does it appear that a line is the best way to fit the data? Why or why not?
- Are there any outliers in the data?
- Use the least-squares line to estimate the average height for a 62-year-old man. Do you think that your answer is reasonable? Why or why not?
- What is the slope of the least-squares (best-fit) line? Interpret the slope.
80.
| State | No. of Letters in Name | Year Entered the Union | Rank for Entering the Union | Area (square miles) |
|---|---|---|---|---|
| Alabama | 7 | 1819 | 22 | 52,423 |
| Colorado | 8 | 1876 | 38 | 104,100 |
| Hawaii | 6 | 1959 | 50 | 10,932 |
| Iowa | 4 | 1846 | 29 | 56,276 |
| Maryland | 8 | 1788 | 7 | 12,407 |
| Missouri | 8 | 1821 | 24 | 69,709 |
| New Jersey | 9 | 1787 | 3 | 8,722 |
| Ohio | 4 | 1803 | 17 | 44,828 |
| South Carolina | 13 | 1788 | 8 | 32,008 |
| Utah | 4 | 1896 | 45 | 84,904 |
| Wisconsin | 9 | 1848 | 30 | 65,499 |
Table
12.37
We are interested in whether there is a relationship between the ranking of a state and the area of the state.
- What are the independent and dependent variables?
- What do you think the scatter plot will look like? Make a scatter plot of the data.
- Does it appear from inspection that there is a relationship between the variables? Why or why not?
- Calculate the least-squares line. Put the equation in the form ŷ = a + bx.
- Find the correlation coefficient. What does it imply about the significance of the relationship?
- Find the estimated areas for Alabama and for Colorado. Are they close to the actual areas?
- Use the two points in Part F to plot the least-squares line on your graph from Part B.
- Does it appear that a line is the best way to fit the data? Why or why not?
- Are there any outliers?
- Use the least-squares line to estimate the area of a new state that enters the Union. Can the least-squares line be used to predict it? Why or why not?
- Delete Hawaii and substitute Alaska for it. Alaska is a state with an area of 656,424 square miles.
- Calculate the new least-squares line.
- Find the estimated area for Alabama. Is it closer to the actual area with this new least-squares line or with the previous one that included Hawaii? Why do you think that’s the case?
- Do you think that, in general, newer states are larger than the original states?