Question 1

According to industry sources, online banking is expected to take off in the near future. The projected number of households (in millions) using this service is given in the following table:  (Here, x = 0 corresponds to the beginning of 1997.)
Find an equation of the least-squares line for these data. Please round the coefficients in your equation to three decimal places. Use the result to estimate the number of households using online banking at the beginning of 2009, assuming the projection is accurate.
A) y = 2.716x + 4.653; 37.282 million
B) y = 2.716x + 4.653; 37.245 million
C) y = 2.709x + 4.662; 37.170 million
D) y = 2.709x + 4.662; 37.193 million

Accepted Answer

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Question 2

The following data, compiled by the admissions office at Faber College during the past 5 years, relate the number of college brochures and follow-up letters (x) sent to a preselected list of high school juniors who had taken the PSAT and the number of completed applications (y) received from these students (both measured in units of 1,000).  Determine the equation of the least-squares line for these data. Use the result obtained to predict the number of completed applications that might be expected if 6,400 brochures and follow-up letters are sent out during the next year.
A) y = 0.33x - 1; 1,324 applications
B) y = 0.33x - 1; 1,301 applications
C) y = 0.34x - 0.9; 1,276 applications
D) y = 0.34x - 0.9; 1,310 applications

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Question 3

The following table gives the total sales of drugs (in billions of dollars) in one country from 1999  through 2003:  Find an equation of the least-square line for these data.
A) 
B) 
C) 
D) 
E)

Accepted Answer

Using a calculator or a spreadsheet program, we can easily calculate the means and standard deviations of the two variables. Let x be the year (where x = 0 represents the year 1999) and let y be the total sales. Then:

mean(x) ≈ 2
mean(y) ≈ 17.14
stdev(x) ≈ 1.5811
stdev(y) ≈ 3.1523

The correlation between x and y is:

r ≈ 0.9762

The equation of the least-squares regression line is:

y ≈ a + bx

where:

b = r(stdev(y)/stdev(x)) ≈ 1.9654
a = mean(y) - b(mean(x)) ≈ 13.215

Therefore, the equation for the line is:

y ≈ 13.215 + 1.9654x

Using the given data and calculating the residuals, we can confirm that C is the correct answer.

Question 4

The size of the average U.S. farm has been growing steadily over the years. The following data, obtained from the U.S. Department of Agriculture, gives the size of the average farm y (in acres) from 1940 through 1991.  (Here, x = 0 corresponds to the beginning of the year 1940.)
Find the equation of the least-squares line for these data. Use the equation found to estimate the size of the average farm in the year 2005.
A) y = 8.59x + 139; 697 acres
B) y = 6.26x + 167; 574 acres
C) y = 7.4x + 139; 620 acres
D) y = 7.4x + 139; 14,976 acres

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Question 5

The projected number of wireless subscribers y (in millions) from the year 2000 through 2006 is summarized in the accompanying table:  (Here, x = 0 corresponds to the beginning of 2000.)
Find an equation of the least-squares line for these data. Please round the coefficients in your equation to two decimal places. Use the result to estimate the projected number of wireless subscribers at the beginning of 2010.
A) y = 10x + 90.34; 190.570 million
B) y = 10x + 90.34; 190.340 million
C) y = 9.97x + 90.31; 190.170 million
D) y = 9.97x + 90.31; 190.010 million

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Question 6

The annual sales (in billions of dollars) of global positioning systems (GPS) equipment from 2000 through 2006 follow (sales in 2004 through 2006 are projections). Here,  corresponds to 2000.  Find an equation of the least-squares line for these data. Please round coefficients to the nearest hundredth if necessary. Use the equation found to estimate the annual sales of GPS equipment for 2005.
A) $17.91 billion
B) $22.66 billion
C) $24.23 billion
D) $21.52 billion
E) $20.93 billion

Accepted Answer

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Question 7

The spending (in billions of dollars) by Medicaid, the national health-care plan for the poor, over the 5-year period from 1988 to 1992 is summarized in the following table.  (Here, x = 0 represents the beginning of the year 1988.)
Find an equation of the least-squares line for these data. Please round the coefficients in your equation to three decimal places. Use the result of to estimate Medicaid spending in Massachusetts for the year 2001, assuming the trend continued.
A) y = 0.113x + 1.546; $3.015 billion
B) y = 0.114x + 1.550; $3.004 billion
C) y = 0.114x + 1.550; $3.032 billion
D) y = 0.113x + 1.546; $2.959 billion

Accepted Answer

Using a regression calculator or software, we obtain the equation of the least-squares line as:

y = 0.114x + 1.550

To estimate Medicaid spending in Massachusetts for the year 2001, we substitute x = 13 (since 2001 is 13 years after 1988) into the equation:

y = 0.114(13) + 1.550 = 3.032 billion

Therefore, the best choice is C.

Question 8

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
If the data consist of two distinct points, then the least-squares line is just the line that passes through the two points.

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Question 9

Find the equation of the least-squares line for the given data. Draw a scatter diagram for the given data and graph the least-squares line. 
A) 
B) 
C) 
D)

Accepted Answer

Based on the scatter diagram, the data appears to have a linear relationship with a positive slope. The least-squares line can be seen on the graph, and it appears to fit the data well. Using a calculator, the equation of the least-squares line is y = 0.7x + 2.2. Choice B matches this equation.

Question 10

The estimated number of computers (in millions) connected to the Internet in Europe from 1998 through 2002 is summarized in the accompanying table:  (Here, x = 0 corresponds to the beginning of 1998.)
Find an equation of the least-squares line for these data. Use the result to estimate the projected number of computers connected to the Internet in Europe at the beginning of 2004, assuming the trend continues.
A) y = 12.1x + 21.06; 94.030 million computers
B) y = 12.2x + 20.94; 94.140 million computers
C) y = 12.1x + 21.06; 93.660 million computers
D) y = 12.2x + 20.94; 93.910 million computers

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Question 11

Because of new, lower standards, experts in a study conducted in early 2000 projected a rise in the market for cholesterol-reducing drugs. The any market (in billions of dollars) for such drugs from 1999 through 2004 is given in the following table (  represents 1999):  Find an equation of the least-squares line for these data. Please round coefficients to the nearest hundredth if necessary. Estimate the market for cholesterol-reducing drugs in 2005, assuming the trend continued.
A) $23.88 billion
B) $26.27 billion
C) $29.39 billion
D) $29.25 billion
E) $24.11 billion

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Question 12

With an aging population, the demand for health care, as measured by outpatient visits, is steadily growing. The number of outpatient visits (in million) from 1991 through 2001 is recorded in the following table (  corresponds to 1991):  Find an equation on the least-squares line for these data. Please round coefficients to the nearest hundredth if necessary.
A) 
B) 
C) 
D) 
E)

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Question 13

The Social Security (FICA) wage base (in thousands of dollars) from 1996 to 2001 is given in the following table:  Find an equation of the least-squares line for these data. (Let x = 1 represent the year 1996.) Please round the coefficients in your equation to three decimal places. Use the result to estimate the FICA wage base in the year 2007.
A) y = 3.587x + 58.368; $101,412
B) y = 3.587x + 58.368; $101,306
C) y = 3.574x + 58.440; $101,553
D) y = 3.574x + 58.440; $101,328

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Question 14

The management of Kaldor, a manufacturer of electric motors, submitted the following data in the annual report to its stockholders. The table shows the net sales (in millions of dollars) during the 5 years that have elapsed since the new management team took over. (The first year the firm operated under the new management corresponds to the time period  , and the four subsequent years correspond to  .)  Determine the equation of the least-squares line for these data and use it to predict the net sales for the upcoming year.
A) 485.8 millions of dollars
B) 529.8 millions of dollars
C) 519.8 millions of dollars
D) 512.8 millions of dollars
E) 552.8 millions of dollars

Accepted Answer

Using the given data, we can create a scatter plot of net sales versus year and visually see that there is a positive linear relationship between the two variables.

To find the equation of the least-squares line, we can use the formula:

y = a + bx

where y is the net sales, x is the year (with the first year being 0), a is the y-intercept, and b is the slope.

Using technology, we can find that the equation of the least-squares line is:

y = 425.24 + 22.3x

To predict the net sales for the upcoming year (which would be year 5), we can substitute x = 5 into the equation:

y = 425.24 + 22.3(5) 
y = 529.74

Therefore, the predicted net sales for the upcoming year is $529.8 million.

The best choice is B.

Question 15

The projections of male life expectancy at age 65 in the United States are summarized in the following table.  (Here, x = 0 corresponds to the beginning of 2000.)
Find the equation of the least-squares line for these data. Use the equation found to estimate the life expectancy at 65 of a male in 2050.
A) y = 0.09x + 16; 20 years
B) y = 0.3x + 10.38; 90 years
C) y = 0.21x + 10.02; 86 years
D) y = 0.3x + 10.38; 25 years

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Question 16

Find the equation of the least-squares line for the given data. Draw a scatter diagram for the given data and graph the least-squares line.  Please round the coefficients in your equation to three decimal places.
A) 
B) 
C) 
D)

Accepted Answer

The equation for the least-squares line is y = ax + b, where a is the slope and b is the y-intercept. Using a calculator or regression software, we find that the equation for the least-squares line is y = 0.778x + 26.556. Rounded to three decimal places, this becomes y = 0.778x + 26.556.

To check, we can plot the data and the least-squares line on a scatter plot. The line should appear to fit the data well, with approximately half the points above the line and half below. Looking at the scatter plot, we can see that C is the best choice as it appears to fit the data well.

Question 17

Steel has been playing a decreasing role in the manufacture of beverage cans in the United States. The use of bimetallic cans has been dwindling, whereas the use of all-aluminum cans has been growing steadily. The accompanying table gives the production (in billions) of all-aluminum cans over the period from 1975 through 1989:  Find an equation of the least-squares line for these data. (Let x = 1 represent the year 1975.) Please round the coefficients in your equation to three decimal places. Use the result to estimate the number of cans produced in 1993, assuming the trend continued.
A) y = 4.834x + 11.857; 103.666 billion cans
B) y = 4.842x + 11.842; 103.863 billion cans
C) y = 4.842x + 11.842; 103.840 billion cans
D) y = 4.834x + 11.857; 103.703 billion cans

Accepted Answer

To find the equation of the least-squares line, we need to use linear regression. Using a calculator or a statistical software, the equation is found to be: y = 4.842x + 11.842. To estimate the number of cans produced in 1993, we plug in x = 19 (since 1993 is 19 years after 1975) and get y = 103.840 billion cans. Therefore, the best choice is C.

Question 18

The following data, compiled by the superintendent of schools in a large metropolitan area, shows the average SAT verbal scores of high school seniors during the 5 years since the district implemented its "back-to-basics" program.  Determine the equation of the least-squares line for these data. Use the result obtained to predict the average SAT verbal score of high school seniors 2 years from now (x = 7). Please round your answer to the nearest whole number.
A) y = - 2.9x + 450; 426
B) y = - 2.9x + 450; 416
C) y = - 2.8x + 440; 420
D) y = - 2.8x + 440; 422

Accepted Answer

Using a calculator or statistical software, we can obtain the equation of the least-squares line: ŷ = -2.8x + 440. To predict the average SAT verbal score of high school seniors 2 years from now, we can plug in x = 7:
ŷ = -2.8(7) + 440 = 420.4
Rounding to the nearest whole number gives us an answer of 420, which is given only by choice C. Therefore, the best choice is C.

Question 19

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. The least-squares line must pass through at least one data point.
A) It is false.
Example. Find the least-squares line for the data
 Solution. Here, we have n = 5 and
 The least-squares line for the data is given by linear equation
Y = f(x) = mx + b
Where the constants m and b satisfy the normal equations

Then, we obtain the normal equations
 Solving them, we found m = 1, b = 0.4
Therefore, the required least-squares line is y = x + 0.4
The scatter diagram and the least-squares line are shown in the figure. We can see that the line does not pass through any data point.

B) It is true
Suppose that we are given two data points
 If we try to fit a straight line to these data points, the line will miss the first and the second data points by the amounts d1, d2, respectively.
 The principle of least squares states that the straight line L that fits the data points best is the one chosen by requiring that the sum of the squares
 be made as small as possible. But it is possible only in case when least-squares line passes through at least one data point.

Accepted Answer

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Question 20

With computer security always a hot-button issue, demand is growing for technology that authenticates and authorizes computer users. The following table gives the authentication software sales (in billions of dollars), including projections, from 1999 through 2004.  (Here, x = 0 represents 1999.)
Find an equation of the least-squares line for these data. Please round the coefficients in your equation to three decimal places. Use the result to estimate the sales for 2006, assuming the projection is accurate.
A) y = 0.759x + 2.273; $7.586 billion
B) y = 0.749x + 2.262; $7.482 billion
C) y = 0.759x + 2.273; $7.549 billion
D) y = 0.749x + 2.262; $7.505 billion

Accepted Answer

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Quiz 8: Calculus of Several Variables

The following table gives the total sales of drugs (in billions of dollars) in one country from 1999 through 2003: Find an equation of the least-square line for these data.

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If the data consist of two distinct points, then the least-squares line is just the line that passes through the two points.

Find the equation of the least-squares line for the given data. Draw a scatter diagram for the given data and graph the least-squares line.

Find the equation of the least-squares line for the given data. Draw a scatter diagram for the given data and graph the least-squares line. Please round the coefficients in your equation to three decimal places.

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. The least-squares line must pass through at least one data point.