Content check — Statistics, Chapter 12: Linear Regression and Correlation

12-1 §12.1 source ↗

Use the following information to answer the next three exercises. A vacation resort rents scuba equipment to certified divers. The resort charges an up-front fee of $25 and another fee of $12.50 an hour.

What are the dependent and independent variables?

Answer

dependent variable: fee amount

independent variable: time

12-3 §12.1 source ↗

Use the following information to answer the next three exercises. A vacation resort rents scuba equipment to certified divers. The resort charges an up-front fee of $25 and another fee of $12.50 an hour.

Graph the equation from Exercise 12.2.

Answer

This is a graph of the equation y = 25 + 12.50x. The x-axis is labeled in intervals of 1 from 0 - 7; the y-axis is labeled in intervals of 25 from 0 - 100. The equation's graph is a line that crosses the y-axis at 25 and is sloped up and to the right, rising 12.50 units for every one unit of run. — Figure 12.25 in the book ↗

12-5 §12.1 source ↗

Use the following information to answer the next two exercises. A credit card company charges $10 when a payment is late and $5 a day each day the payment remains unpaid.

Graph the equation from Exercise 12.4.

Answer

This is a graph of the equation y = 10 + 5x. The x-axis is labeled in intervals of 1 from 0 - 7; the y-axis is labeled in intervals of 10 from 0 - 50. The equation's graph is a line that crosses the y-axis at 10 and is sloped up and to the right, rising 5 units for every one unit of run. — Figure 12.26 in the book ↗

12-7 §12.1 source ↗

Use the following information to answer the next two exercises. A credit card company charges $10 when a payment is late and $5 a day each day the payment remains unpaid.

Which of the following equations are linear?

a. y = 6x + 8

b. y + 7 = 3x

c. y – x = 8x²

d. 4y = 8

Answer

y = 6x + 8, 4y = 8, and y + 7 = 3x are all linear equations.

12-9 §12.1 source ↗

Use the following information to answer the next exercise. Table 12.13 contains real data for the first two decades of flu reporting.

Use the columns Year and Number of Flu Cases Diagnosed. Why is year the independent variable and number of flu cases diagnosed the dependent variable (instead of the reverse)?

Table 12.13 in the book ↗

Year	Number of Flu Cases Diagnosed	Number of Flu Deaths
Pre-1981	91	29
1981	319	121
1982	1,170	453
1983	3,076	1,482
1984	6,240	3,466
1985	11,776	6,878
1986	19,032	11,987
1987	28,564	16,162
1988	35,447	20,868
1989	42,674	27,591
1990	48,634	31,335
1991	59,660	36,560
1992	78,530	41,055
1993	78,834	44,730
1994	71,874	49,095
1995	68,505	49,456
1996	59,347	38,510
1997	47,149	20,736
1998	38,393	19,005
1999	25,174	18,454
2000	25,522	17,347
2001	25,643	17,402
2002	26,464	16,371
Total	802,118	489,093

Answer

The number of flu cases depends on the year. Therefore, year becomes the independent variable and the number of flu cases is the dependent variable.

12-11 §12.1 source ↗

Use the following information to answer the next two exercises. A specialty cleaning company charges an equipment fee and an hourly labor fee. A linear equation that expresses the total amount of the fee the company charges for each session is y = 50 + 100x.

What is the y-intercept, and what is the slope? Interpret them using complete sentences.

Answer

The y-intercept is 50 (a = 50). At the start of the cleaning, the company charges a one-time fee of $50 (this is when x = 0). The slope is 100 (b = 100). For each session, the company charges $100 for each hour they clean.

12-13 §12.1 source ↗

Use the following information to answer the next three questions. As a result of erosion, a river shoreline is losing several thousand pounds of soil each year. A linear equation that expresses the total amount of soil lost per year is y = 12,000x.

How many pounds of soil does the shoreline lose in a year?

Answer

12,000 lb of soil

12-15 §12.1 source ↗

Use the following information to answer the next two exercises. The price of a single issue of stock can fluctuate throughout the day. A linear equation that represents the price of stock for Shipment Express is y = 15 – 1.5x, where x is the number of hours passed in an eight-hour day of trading.

What are the slope and y-intercept? Interpret their meaning.

Answer

The slope is –1.5 (b = –1.5). This means the stock is losing value at a rate of $1.50 per hour. The y-intercept is $15 (a = 15). This means the price of stock before the trading day was $15.

12-17 §12.2 source ↗

Table 12.16 below represents the relationship between the number of hours spent studying and final exam grades.

Fill in the following chart as a first step in finding the line of best fit, using the median–median approach.

Table 12.16 in the book ↗

x	y	x	y
0	2	5	12
3	8	4	9
2	7	3	9
1	3	0	3
5	13	4	10

Table 12.14 in the book ↗

x (number of hours spent studying)	y (final exam grades)
3	50
5	72
1	45
2	51
6	80
8	96
4	65
7	90

Table 12.15 in the book ↗

Group	x (no. of hours spent studying)	y (final exam grades)	Median x Value	Median y Value
1
2
3

Answer

Group	x (no. of hours spent studying)	y (final exam grades)	Median x value	Median y value
1	1 2 3	45 50 51	2	50
2	4 5	65 72	4.5	68.5
3	6 7 8	80 90 96	7	90

Table 12.38

Table 12.38 in the book ↗

Group	x (no. of hours spent studying)	y (final exam grades)	Median x value	Median y value
1	1 2 3	45 50 51	2	50
2	4 5	65 72	4.5	68.5
3	6 7 8	80 90 96	7	90

12-19 §12.2 source ↗

Use the following information to answer the next five exercises. A random sample of 10 professional athletes produced the following data, where x is the number of endorsements the player has and y is the amount of money made, in millions of dollars.

Use regression to find the equation for the line of best fit.

Table 12.16 in the book ↗

x	y	x	y
0	2	5	12
3	8	4	9
2	7	3	9
1	3	0	3
5	13	4	10

Answer

ŷ = 2.23 + 1.99x

12-21 §12.2 source ↗

Use the following information to answer the next five exercises. A random sample of 10 professional athletes produced the following data, where x is the number of endorsements the player has and y is the amount of money made, in millions of dollars.

What is the slope of the line of best fit? What does it represent?

Table 12.16 in the book ↗

x	y	x	y
0	2	5	12
3	8	4	9
2	7	3	9
1	3	0	3
5	13	4	10

Answer

The slope is 1.99 (b = 1.99). It means that for every endorsement deal a professional player gets, he gets an average of another $1.99 million in pay each year.

12-23 §12.2 source ↗

Use the following information to answer the next five exercises. A random sample of 10 professional athletes produced the following data, where x is the number of endorsements the player has and y is the amount of money made, in millions of dollars.

What does an r value of zero mean?

Table 12.16 in the book ↗

x	y	x	y
0	2	5	12
3	8	4	9
2	7	3	9
1	3	0	3
5	13	4	10

Answer

It means that there is no correlation between the data sets.

12-25 §12.2 source ↗

Use the following information to answer the next five exercises. A random sample of 10 professional athletes produced the following data, where x is the number of endorsements the player has and y is the amount of money made, in millions of dollars.

When n = 100 and r = –0.89, is there a significant correlation? Explain.

Table 12.16 in the book ↗

x	y	x	y
0	2	5	12
3	8	4	9
2	7	3	9
1	3	0	3
5	13	4	10

Answer

Yes. There are enough data points and the value of r is strong enough to show there is a strong negative correlation between the data sets.

12-27 §12.3 source ↗

When testing the significance of the correlation coefficient, what is the alternative hypothesis?

Answer

H_a: ρ ≠ 0

12-29 §12.4 source ↗

Use the following information to answer the next two exercises. An electronics retailer used regression to find a simple model to predict sales growth in the first quarter of the new year (January through March). The model is good for 90 days, where x is the day. The model can be written as ŷ = 101.32 + 2.48x, where ŷ is in thousands of dollars.

What would you predict the sales to be on day 60?

Answer

$250,120

12-31 §12.4 source ↗

Use the following information to answer the next three exercises. A landscaping company is hired to mow the grass for several large properties. The total area of the properties is 1,345 acres. The rate at which one person can mow is ŷ = 1350 – 1.2x, where x is the number of hours and ŷ represents the number of acres left to mow.

How many acres are left to mow after 20 hours of work?

Answer

1326 acres

12-33 §12.4 source ↗

Use the following information to answer the next three exercises. A landscaping company is hired to mow the grass for several large properties. The total area of the properties is 1,345 acres. The rate at which one person can mow is ŷ = 1350 – 1.2x, where x is the number of hours and ŷ represents the number of acres left to mow.

How many hours does it take to mow all the lawns, or when is ŷ = 0?

Answer

1125 hours, or when x = 1125

12-35 §12.4 source ↗

Use the following information to answer the next 14 exercises. Table 12.17 contains real data for the first two decades of flu reporting.

Perform a linear regression. What is the linear equation? Round to the nearest whole number. Find the following:

Write the equations:

• Linear equation: __________

• a = ________

• b = ________

• r = ________

• n = ________

Table 12.17 Adults and Adolescents Only, United States in the book ↗

Year	Number of Flu Cases Diagnosed	Number of Flu Deaths
Pre-1981	91	29
1981	319	121
1982	1,170	453
1983	3,076	1,482
1984	6,240	3,466
1985	11,776	6,878
1986	19,032	11,987
1987	28,564	16,162
1988	35,447	20,868
1989	42,674	27,591
1990	48,634	31,335
1991	59,660	36,560
1992	78,530	41,055
1993	78,834	44,730
1994	71,874	49,095
1995	68,505	49,456
1996	59,347	38,510
1997	47,149	20,736
1998	38,393	19,005
1999	25,174	18,454
2000	25,522	17,347
2001	25,643	17,402
2002	26,464	16,371
Total	802,118	489,093

Answer

Check student solution.

12-36 §12.4 source ↗

Use the following information to answer the next 14 exercises. Table 12.17 contains real data for the first two decades of flu reporting.

Solve.

a. When x = 1985, ŷ = _____.

b. When x = 1990, ŷ = _____.

c. When x = 1970, ŷ = _____. Why doesn’t this answer make sense?

Table 12.17 Adults and Adolescents Only, United States in the book ↗

Year	Number of Flu Cases Diagnosed	Number of Flu Deaths
Pre-1981	91	29
1981	319	121
1982	1,170	453
1983	3,076	1,482
1984	6,240	3,466
1985	11,776	6,878
1986	19,032	11,987
1987	28,564	16,162
1988	35,447	20,868
1989	42,674	27,591
1990	48,634	31,335
1991	59,660	36,560
1992	78,530	41,055
1993	78,834	44,730
1994	71,874	49,095
1995	68,505	49,456
1996	59,347	38,510
1997	47,149	20,736
1998	38,393	19,005
1999	25,174	18,454
2000	25,522	17,347
2001	25,643	17,402
2002	26,464	16,371
Total	802,118	489,093

Answer

a. When x = 1985, ŷ = 25,52.

b. When x = 1990, ŷ = 34,275.

c. When x = 1970, ŷ = –725. Why doesn’t this answer make sense? The range of x values was 1981 to 2002; the year 1970 is not in this range. The regression equation does not apply, because predicting for the year 1970 is extrapolation, which requires a different process. Also, a negative number does not make sense in this context, when we are predicting flu cases diagnosed.

12-38 §12.4 source ↗

Use the following information to answer the next 14 exercises. Table 12.17 contains real data for the first two decades of flu reporting.

What does the correlation imply about the relationship between time (years) and the number of diagnosed flu cases reported in the United States?

Table 12.17 Adults and Adolescents Only, United States in the book ↗

Year	Number of Flu Cases Diagnosed	Number of Flu Deaths
Pre-1981	91	29
1981	319	121
1982	1,170	453
1983	3,076	1,482
1984	6,240	3,466
1985	11,776	6,878
1986	19,032	11,987
1987	28,564	16,162
1988	35,447	20,868
1989	42,674	27,591
1990	48,634	31,335
1991	59,660	36,560
1992	78,530	41,055
1993	78,834	44,730
1994	71,874	49,095
1995	68,505	49,456
1996	59,347	38,510
1997	47,149	20,736
1998	38,393	19,005
1999	25,174	18,454
2000	25,522	17,347
2001	25,643	17,402
2002	26,464	16,371
Total	802,118	489,093

Answer

Also, the correlation r = 0.4526. If r is compared with the value in the 95 Percent Critical Values of the Sample Correlation Coefficient Table, because r > 0.423, r is significant, and you would think that the line could be used for prediction. But, the scatter plot indicates otherwise.

12-39 §12.4 source ↗

Use the following information to answer the next 14 exercises. Table 12.17 contains real data for the first two decades of flu reporting.

Plot the two points on the graph. Then, connect the two points to form the regression line.

Table 12.17 Adults and Adolescents Only, United States in the book ↗

Year	Number of Flu Cases Diagnosed	Number of Flu Deaths
Pre-1981	91	29
1981	319	121
1982	1,170	453
1983	3,076	1,482
1984	6,240	3,466
1985	11,776	6,878
1986	19,032	11,987
1987	28,564	16,162
1988	35,447	20,868
1989	42,674	27,591
1990	48,634	31,335
1991	59,660	36,560
1992	78,530	41,055
1993	78,834	44,730
1994	71,874	49,095
1995	68,505	49,456
1996	59,347	38,510
1997	47,149	20,736
1998	38,393	19,005
1999	25,174	18,454
2000	25,522	17,347
2001	25,643	17,402
2002	26,464	16,371
Total	802,118	489,093

Answer

Check student’ solution.

12-40 §12.4 source ↗

Use the following information to answer the next 14 exercises. Table 12.17 contains real data for the first two decades of flu reporting.

Write the equation: ŷ = ____________.

Table 12.17 Adults and Adolescents Only, United States in the book ↗

Year	Number of Flu Cases Diagnosed	Number of Flu Deaths
Pre-1981	91	29
1981	319	121
1982	1,170	453
1983	3,076	1,482
1984	6,240	3,466
1985	11,776	6,878
1986	19,032	11,987
1987	28,564	16,162
1988	35,447	20,868
1989	42,674	27,591
1990	48,634	31,335
1991	59,660	36,560
1992	78,530	41,055
1993	78,834	44,730
1994	71,874	49,095
1995	68,505	49,456
1996	59,347	38,510
1997	47,149	20,736
1998	38,393	19,005
1999	25,174	18,454
2000	25,522	17,347
2001	25,643	17,402
2002	26,464	16,371
Total	802,118	489,093

Answer

ŷ

= 3,448,225 + 1750x

12-42 §12.4 source ↗

Use the following information to answer the next 14 exercises. Table 12.17 contains real data for the first two decades of flu reporting.

Does the line seem to fit the data? Why or why not?

Table 12.17 Adults and Adolescents Only, United States in the book ↗

Year	Number of Flu Cases Diagnosed	Number of Flu Deaths
Pre-1981	91	29
1981	319	121
1982	1,170	453
1983	3,076	1,482
1984	6,240	3,466
1985	11,776	6,878
1986	19,032	11,987
1987	28,564	16,162
1988	35,447	20,868
1989	42,674	27,591
1990	48,634	31,335
1991	59,660	36,560
1992	78,530	41,055
1993	78,834	44,730
1994	71,874	49,095
1995	68,505	49,456
1996	59,347	38,510
1997	47,149	20,736
1998	38,393	19,005
1999	25,174	18,454
2000	25,522	17,347
2001	25,643	17,402
2002	26,464	16,371
Total	802,118	489,093

Answer

There was an increase in flu cases diagnosed until 1993. From 1993 through 2002, the number of flu cases diagnosed declined each year. It is not appropriate to use a linear regression line to fit to the data.

12-44 §12.4 source ↗

Use the following information to answer the next 14 exercises. Table 12.17 contains real data for the first two decades of flu reporting.

What does the correlation imply about the relationship between time (years) and the number of diagnosed flu cases reported in the United States?

Table 12.17 Adults and Adolescents Only, United States in the book ↗

Year	Number of Flu Cases Diagnosed	Number of Flu Deaths
Pre-1981	91	29
1981	319	121
1982	1,170	453
1983	3,076	1,482
1984	6,240	3,466
1985	11,776	6,878
1986	19,032	11,987
1987	28,564	16,162
1988	35,447	20,868
1989	42,674	27,591
1990	48,634	31,335
1991	59,660	36,560
1992	78,530	41,055
1993	78,834	44,730
1994	71,874	49,095
1995	68,505	49,456
1996	59,347	38,510
1997	47,149	20,736
1998	38,393	19,005
1999	25,174	18,454
2000	25,522	17,347
2001	25,643	17,402
2002	26,464	16,371
Total	802,118	489,093

Answer

Because there is no linear association between year and number of flu cases diagnosed, it is not appropriate to calculate a linear correlation coefficient. When there is a linear association and it is appropriate to calculate a correlation, we cannot say that one variable causes the other variable.

12-46 §12.4 source ↗

Use the following information to answer the next 14 exercises. Table 12.17 contains real data for the first two decades of flu reporting.

Enter your data into your calculator or computer. The pre-1981 data should not be included. Why is that so?

Write the linear equation, rounding to four decimal places.

Table 12.17 Adults and Adolescents Only, United States in the book ↗

Year	Number of Flu Cases Diagnosed	Number of Flu Deaths
Pre-1981	91	29
1981	319	121
1982	1,170	453
1983	3,076	1,482
1984	6,240	3,466
1985	11,776	6,878
1986	19,032	11,987
1987	28,564	16,162
1988	35,447	20,868
1989	42,674	27,591
1990	48,634	31,335
1991	59,660	36,560
1992	78,530	41,055
1993	78,834	44,730
1994	71,874	49,095
1995	68,505	49,456
1996	59,347	38,510
1997	47,149	20,736
1998	38,393	19,005
1999	25,174	18,454
2000	25,522	17,347
2001	25,643	17,402
2002	26,464	16,371
Total	802,118	489,093

Answer

We don’t know if the pre-1981 data were collected from a single year. So, we don’t have an accurate x value for this figure.

Regression equation: ŷ (number of flu cases) = –3,448,225 + 1749.777 (year).

	Coefficients
Intercept	–3,448,225
x Variable 1	1,749.777

Table 12.39

Table 12.39 in the book ↗

	Coefficients
Intercept	–3,448,225
x Variable 1	1,749.777

12-47 §12.4 source ↗

Use the following information to answer the next 14 exercises. Table 12.17 contains real data for the first two decades of flu reporting.

Calculate the following:

• a = _____

• b = _____

• correlation = _____

• n = _____

Table 12.17 Adults and Adolescents Only, United States in the book ↗

Year	Number of Flu Cases Diagnosed	Number of Flu Deaths
Pre-1981	91	29
1981	319	121
1982	1,170	453
1983	3,076	1,482
1984	6,240	3,466
1985	11,776	6,878
1986	19,032	11,987
1987	28,564	16,162
1988	35,447	20,868
1989	42,674	27,591
1990	48,634	31,335
1991	59,660	36,560
1992	78,530	41,055
1993	78,834	44,730
1994	71,874	49,095
1995	68,505	49,456
1996	59,347	38,510
1997	47,149	20,736
1998	38,393	19,005
1999	25,174	18,454
2000	25,522	17,347
2001	25,643	17,402
2002	26,464	16,371
Total	802,118	489,093

Answer

• a = –3,488,225

• b = 1,750

• correlation = 0.4526

• n = 22

12-48 §12.5 source ↗

Marcus states that all outliers are influential points. Is he correct? Explain.

Answer

No, he is not correct. An outlier is only an influential point if it significantly impacts the slope of the least-squares regression line and the correlation coefficient, r. If omission of this data point from the calculation of the regression line does not show much impact on the slope or r-value, then the outlier is not considered an influential point. For different reasons, it still may be determined that the data point must be omitted from the data set.

12-49 §12.5 source ↗

Use the following information to answer the next four exercises. The scatter plot shows the relationship between hours spent studying and exam scores. The line shown is the calculated line of best fit. The correlation coefficient is 0.69.

Do there appear to be any outliers?

A graph is shown. The y-axis has tick marks at 0, 20, 40, 60, 80, and 100. The x axis has tick marks at 0, 2, 4, 6, 8, 10, 12, 14. The line goes up steadily from 68 and ends around 100. — Figure 12.24 in the book ↗

Answer

Yes. There appears to be an outlier at (6, 58).

12-51 §12.5 source ↗

Use the following information to answer the next four exercises. The scatter plot shows the relationship between hours spent studying and exam scores. The line shown is the calculated line of best fit. The correlation coefficient is 0.69.

What effect did the potential outlier have on the line of best fit?

Answer

The potential outlier flattened the slope of the line of best fit because it was below the data set. It made the line of best fit less accurate as a predictor for the data.

12-53 §12.5 source ↗

Use the following information to answer the next four exercises. The scatter plot shows the relationship between hours spent studying and exam scores. The line shown is the calculated line of best fit. The correlation coefficient is 0.69.

The sum of squared errors (SSE) for a data set of 18 numbers is 49. What is the standard deviation?

Answer

s = 1.75

12-55 §12.1 source ↗

For each of the following situations, state the independent variable and the dependent variable.

a. A study is done to determine whether elderly drivers are involved in more motor vehicle fatalities than other drivers. The number of fatalities per 100,000 drivers is compared with the age of drivers.

b. A study is done to determine whether 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 whether a higher education reduces the crime rate in a population.

Answer

a. independent variable: age; dependent variable: fatalities

b. independent variable: number of family members; dependent variable: grocery bill

c. independent variable: age of applicant; dependent variable: insurance premium

d. independent variable: power consumption; dependent variable: utility

e. independent variable: higher education (years); dependent variable: crime rates

12-58 §12.2 source ↗

Explain what it means when a correlation has an r² value of .72.

Answer

It means that 72 percent of the variation in the dependent variable (y) can be explained by the variation in the independent variable (x).

12-60 §12.2 source ↗

The table below represents the relationship between SAT scores on the math portion of the test and high school grade point averages (GPAs).

Use the median–-median line approach to find the equation for the line of best fit.

Table 12.19 in the book ↗

x (SAT math scores)	y (GPAs)
624	90
544	86
363	70
373	71
350	65
741	98
262	60
587	87
327	62
364	67
261	50

Answer

x (SAT math scores)	y (GPAs)
261	50
262	60
327	62
350	65
363	70
364	67
373	71
544	86
587	87
624	90
741	98

Table 12.40

We must remember to check the order of the y values within each group as well. We notice that the y values in the second group are not in order from the least value to the greatest value; these values thus must be reordered, meaning the median y value for that group is 70.

Group	x (SAT math scores)	y (GPAs)	Median x value	Median y value
1	261 262 327 350	50 60 62 65	294.5	61
2	363 364 373	67 70 71	364	70
3	544 587 624 741	86 87 90 98	605.5	88.5

Table 12.41

The ordered pairs are (294.5, 61), (364, 70), and (605.5, 88.5).

The slope can be calculated using the formula $m =$ $\frac{y_{3} - y_{1}}{x_{3} - x_{1}} .$ Substituting the median x and y values, from the first and third groups gives $m =$ $\frac{88.5 - 61}{605.5 - 294.5},$ which simplifies to $m \approx 0.09 .$

The y-intercept may be found using the formula $b = \frac{\sum y - m \sum x}{3} .$ The sum of the median x values is 1264, and the sum of the median y values is 219.5. Substituting these sums and the slope into the formula gives $b =$ $\frac{219.5 - 0.09 (1264)}{3},$ which simplifies to $b \approx 35.25 .$

The line of best fit is represented as $y = m x + b$ . Thus, the equation can be written as $y =$ $0.09 x +$ $35.25 .$

Table 12.40 in the book ↗

x (SAT math scores)	y (GPAs)
261	50
262	60
327	62
350	65
363	70
364	67
373	71
544	86
587	87
624	90
741	98

Table 12.41 in the book ↗

Group	x (SAT math scores)	y (GPAs)	Median x value	Median y value
1	261 262 327 350	50 60 62 65	294.5	61
2	363 364 373	67 70 71	364	70
3	544 587 624 741	86 87 90 98	605.5	88.5

12-61 §12.4 source ↗

Recently, the annual numbers of driver deaths per 100,000 people for the selected age groups are as follows:

a. For each age group, pick the midpoint of the interval for the x value. For the 75+ group, use 80.

b. Using age as the independent variable and number of driver deaths per 100,000 people 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 ŷ = a + bx.

d. Find the correlation coefficient. Is it significant?

e. Predict the number of deaths for ages 40 years and 60 years.

f. Based on the given data, is there a linear relationship between age of a driver and driver fatality rate?

g. What is the slope of the least-squares (best-fit) line? Interpret the slope.

Table 12.20 in the book ↗

Age (years)	Number of Driver Deaths (per 100,000 people)
16–19	38
20–24	36
25–34	24
35–54	20
55–74	18
75+	28

Answer

b. Check student solution.

c. ŷ = 35.5818045 – 0.19182491x

d. r = –0.57874
For four degrees of freedom and alpha = 0.05, the LinRegTTest gives a p value of 0.2288, so we do not reject the null hypothesis; there is not a significant linear relationship between deaths and age.
Using the table of critical values for the correlation coefficient, with four degrees of freedom, the critical value is 0.811. The correlation coefficient r = –0.57874 is not less than –0.811, so we do not reject the null hypothesis.

f. There is not a linear relationship between the two variables, as evidenced by a p value greater than 0.05.

12-63 §12.4 source ↗

Table 12.22 in the book ↗

Page Number	Maximum Value ($)
4	16
14	19
25	15
32	17
43	19
57	15
72	16
85	15
90	17

The maximum discount value of the Entertainment^® card for the Fine Dining section, 10th edition, for various pages is given in Table 12.22.

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 ŷ = a + bx.

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. 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?

g. Suppose there are 200 pages of restaurants. What do you estimate to be the maximum value for a restaurant listed on page 200?

h. Is the least-squares line valid for page 200? Why or why not?

i. What is the slope of the least-squares (best-fit) line? Interpret the slope.

Answer

a. We wonder if the better discounts appear earlier in the book, so we select page as x and discount as y.

b. Check student solution.

c. ŷ = 17.21757 – 0.01412x

d. r = – 0.2752
For seven degrees of freedom and alpha = 0.05, LinRegTTest gives a p value = 0.4736, so we do not reject; there is a not a significant linear relationship between page and discount.
Using the table of critical values for the correlation coefficient, with seven gives degrees of freedom, the critical value is 0.666. The correlation coefficient xi = –0.2752 is not less than 0.666, so we do not reject.

f. There is not a significant linear correlation so it appears there is no relationship between the page and the amount of the discount.

As the page number increases by one page, the discount decreases by $0.01412.

12-65 §12.4 source ↗

Table 12.24 in the book ↗

State	No. of Letters in Name	Year Entered the Union	Rank for Entering the Union	Area in 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

We are interested in whether the number of letters in a state name depends on the year the state entered the Union.

a. Decide which variable should be the independent variable and which should be the dependent variable.

b. Draw 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 ŷ = a + bx.

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 name would have if it entered the Union in 1900. Find the estimated number of letters a state name would have if it entered the Union in 1940.

g. Does it appear that a line is the best way to fit the data? Why or why not?

h. Use the least-squares line to estimate the number of letters for a new state that enters the Union this year. Can the least-squares line be used to predict it? Why or why not?

Answer

a. Year is the independent or x variable; the number of letters is the dependent or y variable.

b. Check student’s solution.

c. No.

d. ŷ = 47.03 – 0.0216x

e. –0.4280. The r value indicates that there is not a significant correlation between the year the state entered the Union and the number of letters in the name.

g. No. The relationship does not appear to be linear; the correlation is not significant.

12-66 §12.5 source ↗

Given the information in Table 12.30, which represents the relationship between final exam math grades and final exam history grades, decide whether point (56, 95) is an influential point. Explain how you arrived at your decision. Show all work.

Table 12.30 in the book ↗

School	Mid-Career Salary (in thousands of U.S. dollars)	Yearly Tuition (in U.S. dollars)
Princeton	137	28,540
Harvey Mudd	135	40,133
CalTech	127	39,900
U.S. Naval Academy	122	0
West Point	120	0
MIT	118	42,050
Lehigh University	118	43,220
NYU-Poly	117	39,565
Babson College	117	40,400
Stanford	114	54,506

Table 12.25 in the book ↗

x (final exam math grades)	y (final exam history grades)
54	60
56	68
77	82
74	78
63	69
51	55
88	97
72	77
69	78
56	95

Answer

Using LinRegTTest, the output for the original least-squares regression line is $y =$ $26.14 +$ $0.7539 x$ and $r = 0.6657 .$

The output for the new least-squares regression line, after omitting the outlier of (56, 95), is $ŷ =$ $6.36 +$ $1.0045 x$ and $r = 0.9757 .$

The slope of the new line is quite a bit different from the slope of the original least-squares regression line, but the larger change is shown in the r-values, such that the new line has an r-value that has increased to a value that is almost equal to one.

Thus, it may be stated that the outlier (56, 95) is also an influential point.

12-68 §12.5 source ↗

Ornithologists (scientists who study birds) tag sparrow hawks in 13 different colonies to study their population. They gather data for the percentage of new sparrow hawks in each colony and the percentage of those that have returned from migration.

Percent return: 74, 66, 81, 52, 73, 62, 52, 45, 62, 46, 60, 46, 38
Percent new: 5, 6, 8, 11, 12, 15, 16, 17, 18, 18, 19, 20, 20

a. Enter the data into a calculator and make a scatter plot.

b. Use the calculator’s regression function to find the equation of the least-squares regression line. Add this to your scatter plot from Part A.

c. Explain what the slope and y-intercept of the regression line tell us.

d. How well does the regression line fit the data? Explain your response.

e. Which point has the largest residual? Explain what the residual means in context. Is this point an outlier? An influential point? Explain.

f. An ecologist wants to predict how many birds will join another colony of sparrow hawks to which 70 percent of the adults from the previous year have returned. What is the prediction?

Answer

a. and b. Check student solution.

c. The slope of the regression line is –0.3031 with a y-intercept of 31.93. In context, the y-intercept indicates that when there are no returning sparrow hawks, there will be almost 32 percent new sparrow hawks, which doesn’t make sense, because if there are no returning birds, then the new percentage would have to be 100% (this is an example of why we do not extrapolate). The slope tells us that for each percentage increase in returning birds, the percentage of new birds in the colony decreases by 30.3 percent.

d. If we examine r₂, we see that only 57.52 percent of the variation in the percentage of new birds is explained by the model and the correlation coefficient, r = –.7584 only indicates a somewhat strong correlation between returning and new percentages.

e. The ordered pair (66, 6) generates the largest residual of 6.0. This means that when the observed return percentage is 66 percent, our observed new percentage, 6 percent, is almost 6 percent less than the predicted new value of 11.98 percent. If we remove this data pair, we see only an adjusted slope of –0.2789 and an adjusted intercept of 30.9816. In other words, although these data generate the largest residual, it is not an outlier, nor is the data pair an influential point.

f. If there are 70 percent returning birds, we would expect to see y =– 0.2789(70) + 30.9816 = 0.114 or 11.4 percent new birds in the colony.

12-70 §12.5 source ↗

The following table consists of one student athlete’s time (in minutes) to swim 2,000 yards and the student’s heart rate (beats per minute) after swimming on a random sample of 10 days.

a. Enter the data into a calculator and make a scatter plot.

b. Use the calculator’s regression function to find the equation of the least-squares regression line. Add this to your scatter plot from Part A.

c. Explain what the slope and y-intercept of the regression line tell us.

d. How well does the regression line fit the data? Explain your response.

e. Which point has the largest residual? Explain what the residual means in context. Is this point an outlier? An influential point? Explain.

Table 12.28 in the book ↗

Swim Time	Heart Rate
34.12	144
35.72	152
34.72	124
34.05	140
34.13	152
35.73	146
36.17	128
35.57	136
35.37	144
35.57	148

Answer

a. Check student solution.

b. Check student solution.

c. We have a slope of –1.4946 with a y-intercept of 193.88. The slope, in context, indicates that for each additional minute added to the swim time, the heart rate decreases by 1.5 beats per minute. If the student is not swimming at all, the y-intercept indicates that his heart rate will be 193.88 beats per minute. Although the slope has meaning (the longer it takes to swim 2000 m, the less effort the heart puts out), the y-intercept does not make sense. If the athlete is not swimming (resting), then his heart rate should be very low.

d. Because only 1.5 percent of the heart rate variation is explained by this regression equation, we must conclude that this association is not explained with a linear relationship.

e. Point (34.72, 124) generates the largest residual: –11.82. This means that our observed heart rate is almost 12 beats less than our predicted rate of 136 beats per minute. When this point is removed, the slope becomes –2.953, with the y-intercept changing to 247.1616. Although the linear association is still very weak, we see that the removed data pair can be considered an influential point in the sense that the y-intercept becomes more meaningful.

12-72 §12.5 source ↗

Use the data in the Table 12.35 to determine the linear regression line equation with the outliers removed. Is there a linear correlation for the data set with outliers removed? Justify your answer.

Table 12.30 in the book ↗

School	Mid-Career Salary (in thousands of U.S. dollars)	Yearly Tuition (in U.S. dollars)
Princeton	137	28,540
Harvey Mudd	135	40,133
CalTech	127	39,900
U.S. Naval Academy	122	0
West Point	120	0
MIT	118	42,050
Lehigh University	118	43,220
NYU-Poly	117	39,565
Babson College	117	40,400
Stanford	114	54,506

Table 12.35 in the book ↗

Size (inches)	Sale Price ($)
9	147
20	197
27	297
31	447
35	1,177
40	2,177
60	2,497

Answer

If we remove the two service academies (the tuition is $0.00), we construct a new regression equation of y = –0.0009x + 160, with a correlation coefficient of 0.71397 and a coefficient of determination of 0.50976. This allows us to say there is a fairly strong linear association between tuition costs and salaries if the service academies are removed from the data set.

12-73 §12.BIT source ↗

Table 12.31 in the book ↗

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

The average number of people in a family who attended college for various years is given in Table 12.31.

a. Using year as the independent variable and number of family members attending college as the dependent variable, draw a scatter plot of the data.

b. Calculate the least-squares line. Put the equation in the form ŷ = a + bx.

c. Does the y-intercept, a, have any meaning here?

d. Find the correlation coefficient. Is it significant?

e. Pick two years between 1969 and 1991 and find the estimated number of family members attending college.

f. 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?

g. 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.

h. Are there any outliers in the data?

i. 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.

j. What is the slope of the least-squares (best-fit) line? Interpret the slope.

Answer

c. No. The y-intercept would occur at year 0, which doesn’t exist.

12-74 §12.BIT source ↗

Table 12.32 in the book ↗

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

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.

a. Using year as the independent variable and percent of workers paid hourly rates as the dependent variable, draw 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. Does the y-intercept, a, have any meaning here?

d. Calculate the least-squares line. Put the equation in the form ŷ = a + bx.

e. Find the correlation coefficient. Is it significant?

f. Find the estimated percentages for 1991 and 1988.

g. 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?

h. Are there any outliers in the data?

i. 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.

j. What is the slope of the least-squares (best-fit) line? Interpret the slope.

Answer

a. Check student's solution.

b. Yes.

c. No, the y-intercept would occur at year 0, which doesn’t exist.

d. ŷ = −266.8863 + 0.1656x.

e. 0.9448, yes.

f. 62.8233, 62.3265.

g. Yes.

h. No, (1987, 62.7).

i. 72.5937, no.

j. Slope = 0.1656. As the year increases by one, the percent of workers paid hourly rates tends to increase by 0.1656.

12-76 §12.BIT source ↗

Table 12.33 in the book ↗

Size (ounces)	Cost ($)	Cost per Ounce
16	3.99
32	4.99
64	5.99
200	10.99

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.

a. Complete Table 12.33 for the cost per ounce of the different sizes of laundry detergent.

b. Using size as the independent variable and cost per ounce as the dependent variable, draw 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 ŷ = a + bx.

e. Find the correlation coefficient. Is it significant?

f. If the laundry detergent were sold in a 40 oz. size, what is the estimated cost per ounce?

g. If the laundry detergent were sold in a 90 oz. size, what is the estimated cost per ounce?

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 the data?

j. 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?

k. What is the slope of the least-squares (best-fit) line? Interpret the slope.

Answer

a.

Size (ounces)	Cost ($)	Cost per ounce
16	3.99	24.94
32	4.99	15.59
64	5.99	9.36
200	10.99	5.50

Table 12.42

b. Check student solution.

c. There is a linear relationship for the sizes 16 through 64, but that linear trend does not continue to the 200-oz size.

d. ŷ = 20.2368 – 0.0819x

e. r = –.8086

f. 40-oz: 16.96 cents/oz

g. 90-oz: 12.87 cents/oz

h. The relationship is not linear; the least-squares line is not appropriate.

i. There are no outliers.

j. No. You would be extrapolating. The 300-oz size is outside the range of x.

k. X = –0.08194. For each additional ounce in size, the cost per ounce decreases by 0.082 cents.

Table 12.42 in the book ↗

Size (ounces)	Cost ($)	Cost per ounce
16	3.99	24.94
32	4.99	15.59
64	5.99	9.36
200	10.99	5.50

12-78 §12.BIT source ↗

Table 12.33 in the book ↗

Size (ounces)	Cost ($)	Cost per Ounce
16	3.99
32	4.99
64	5.99
200	10.99

Table 12.35 in the book ↗

Size (inches)	Sale Price ($)
9	147
20	197
27	297
31	447
35	1,177
40	2,177
60	2,497

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.

The following are advertised sale prices of color televisions at Anderson’s:

a. Decide which variable should be the independent variable and which should be the dependent variable.

b. Draw 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 ŷ = a + bx.

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. Does it appear that a line is the best way to fit the data? Why or why not?

h. Are there any outliers in the data?

i. What is the slope of the least-squares (best-fit) line? Interpret the slope.

Answer

a. Size is x, the independent variable, and price is y, the dependent variable.

b. Check student solution.

c. The relationship does not appear to be linear.

d. ŷ = –745.252 + 54.75569x.

e. r = .8944 and yes, it is significant.

f. 32-inch: $1006.93, 50-inch: $1992.53.

g. No, the relationship does not appear to be linear. However, r is significant.

h. No, the 60-inch TV.

i. For each additional inch, the price increases by $54.76.

12-80 §12.BIT source ↗

Table 12.33 in the book ↗

Size (ounces)	Cost ($)	Cost per Ounce
16	3.99
32	4.99
64	5.99
200	10.99

Table 12.37 in the book ↗

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

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.

We are interested in whether there is a relationship between the ranking of a state and the area of the state.

a. What are the independent and dependent variables?

b. What do you think the scatter plot will look like? 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 ŷ = a + bx.

e. Find the correlation coefficient. What does it imply about the significance of the relationship?

f. Find the estimated areas for Alabama and for Colorado. Are they close to the actual areas?

g. Use the two points in Part F to plot the least-squares line on your graph from Part 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?

j. 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?

k. Delete Hawaii and substitute Alaska for it. Alaska is a state with an area of 656,424 square miles.

l. Calculate the new least-squares line.

m. 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?

n. Do you think that, in general, newer states are larger than the original states?

Answer

a. Rank is the independent variable and area is the dependent variable.

b. Check student solution.

c. There appears to be a linear relationship, with one outlier.

d. ŷ (area) = 24177.06 + 1010.478x

e. r = .50047. r is not significant, so there is no relationship between the variables.

f. Alabama: 46,407.576 square miles, Colorado: 62,575.224 square miles.

g. The Alabama estimate is closer than the Colorado estimate.

h. If the outlier is removed, there is a linear relationship.

i. There is one outlier (Hawaii).

j. rank 51: 75,711.4 square miles, no.

k.

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 12.43

l. ŷ = –87065.3 + 7828.532x.

m. Alabama: 85,162.404; the prior estimate was closer. Alaska is an outlier.

n. Yes, with the exception of Hawaii.

Table 12.43 in the book ↗

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