Algebra 2
Mr. Prykuta
Worksheet #90- Linear Regression
If you have a set of points, you can use the calculator to find a graph of equation that represents that data.
Once you have developed a graph, the calculator can find the linear regression line, or line of best fit.
Model Example:
The table below gives the length of the right foot and the height of 10 males, all in centimeters.
Right foot
(cm)
25
26
27
27
28
28
28
29
29
30
Height
(cm)
165
170
175
179
181
182
183
184
185
186
a.
Create the scatter plot that represents these data.
b.
Determine the line of best fit for the given data.
c.
What is the correlation coefficient for
the size of the right foot vs. height?
d.
Predict the approximate height of a person
whose right foot measures:
(1) 24 cm
(2) 32 cm
e.
Predict the approximate length of the right foot of a
person whose height measures 190 centimeters.
Example #1:
The number of sets of twins born in the
United States has been increasing as shown in the table below:Year Number of Sets of Twins
1990
93,865
1991
94,779
1992
95,372
1993
96,445
1994
97,064
Note: When years are given as part of the data, in order to make the calculations easier, we usually set the first year as 0, so in this problem, enter 0, 1, 2, 3, and 4 as the independent variable year in L1.
a.
Find the line of best fit for these data.
b.
What is the correlation coefficient for these data?
Personal expenditures for clothing in the United States since 1992 are listed below. The expenditures are reported in billions of dollars.
Year
Clothing Expenditures
1992
283
1993
298
1994
312
1995
323
1996
338
1997
353
1998
367
a.
Use the data to form a scatter plot.
b.
Use linear regression to find the equation of the line
of best fit for annual expenditures on clothes since 1992.
c.
What is the correlation coefficient of your equation?
d.
According to this model, how much will be spent in the U.S. on clothes in 2000?
Example #3:
The following table represents the number of United States radio stations on the air since 1950.
Use
Year
Number of radio Stations
0
1950
2,773
5
1955
3,211
10
1960
4,133
15
1965
5,249
20
1970
6,760
25
1975
8,844
30
1980
8,566
35
1985
10,359
40
1990
10,788
45
1995
11,834
46
1996
12,295
47
1997
12,482
48
1998
12,642
a.
Create a scatter plot using these data.
b.
Use liner regression to find the equation of the line for
the number of radio stations on the air as a function of the year since 1950.
c.
What is the correlation coefficient for the line of best fit?
The following table lists the average sale price of new one-family homes in the United States since 1975.
Year
Average Sale Price of Houses
1975
$42,600
1980
$76,400
1985
$122,800
1990
$149,800
1995
$158,700
1996
$166,400
1997
$176,200
1998
$181,900
1999
$195,800
a.
Create a scatter plot using these data.
b.
If
H
(
x
) represents the average cost of a home in the United States as a function of the year of
purchase, find the line of best fit for the data. (
Round a and b to the nearest hundredth
).
c.
What is the correlation coefficient? Does this indicate a strong correlation?
d.
If Doug and Marissa want to buy a one-family house in 2005 and are able to get one for the national
average price, what should they expect to pay for it? Based on the behavior of the latter data points,
is this a reasonable estimate? (
Round your answer to the nearest dollar
).
A real estate agent plans to compare the price of a cottage,
y
, in a town on the seashore to the number of
blocks,
x
, the cottage is from the beach. The accompanying table shows a random sample of sales and
location data.
Write a linear regression equation that relates the price of a cottage to its distance from the beach.
Use the equation to predict the price of a cottage, to the nearest dollar, located three blocks from the
beach.
Number of
Blocks from
the Beach
(x)
Price of a Cottage
(y)
5
$132,000
0
$310,000
4
$204,000
2
$238,000
1
$275,000
7
$60,000
Example #6:
Two different tests were designed to measure understanding of
a topic. The two tests were given to ten students with the following results:Test x
75
78
88
92
95
67
58
72
74
81
Test y
81
73
85
88
89
73
66
75
70
78
Construct a scatter plot for these scores, and then write an equation for the line of best fit (round slope and
intercept to the
nearest hundredth
).
a.
Find the correlation coefficient
to the
nearest hundredth
.
The accompanying table illustrates the number of movie theaters showing a popular film and the film’s weekly gross earnings, in millions of dollars.
Number of
Theaters
(x)
Gross Earnings (y)
(millions of dollars)
443
2.57
455
2.65
493
3.73
530
4.05
569
4.76
657
4.76
723
5.15
1,064
9.35
Write the linear regression equation for this set of data, rounding values to
five decimal places
.
Using this linear regression equation, find the approximate gross earnings, in millions of dollars, generated
by 610 theaters. (
Round your answer to two decimal places)
Algebra 2
Mr. Prykuta
Worksheet #90.2- Exponential Regression
This equation has the format:
1. A box containing 1,000 coins is shaken, and the coins are emptied onto a table. Only the coins that land heads up are returned to the box, and then the process is repeated. The accompanying table shows the number of trials and the number of coins returned to the box after each trial.
Write an exponential regression equation, rounding the calculated values to the nearest ten-thousandth.
Use the equation to predict how many coins would be returned to the box after the eighth trial.
2. Jean invested $380 in stocks. Over the next 5 years, the value of her investments grew, as shown in the accompanying table.
Years Since Investment
(x)
Value of Stock in Dollars (y)
0 380
1 395
2 411
3 427
4 445
5 462
Write the exponential regression equation for this set of data, rounding all values to two decimal places.
Using this equation, find the value of her stock, to the nearest dollar, 10 years after her initial purchase.
Trials Coins
Returned
0 1000
1 610
3 220
4 132
Year Fare ($)
55 0.10
60 0.15
65 0.20
70 0.30
75 0.40
80 0.60
85 0.80
90 1.15
95 1.50
On the accompanying grid, construct a scatter plot where the independent variable is years.
State the exponential regression equation with the coefficient and base rounded to the nearest thousandth.
Algebra 2
Mr. Prykuta
Worksheet #90.3- Power Regression
This equation has the format:
1. The accompanying table shows the number of new cases reported by the Nassau and Suffolk County Police Crime Stoppers program for the years 2000 through 2003.
Year (x) New Cases (y)
2000 457
2001 369
2002 353
2003 331
If x =1 represents the year 2000, and y represents the number of new cases, find the equation of best fit using a power regression, rounding all values to the nearest thousandth.
Using this equation, find the estimated number of new cases, to the nearest whole number, for the year 2007.
2. The data represents some possible dimensions of picture frames in the I've Been Framed Photo Shop.
Length Width
1 24
2 12
3 8
4 6
6 4
8 3
12 2
24 1
Find the equation of best fit using a power regression.
Using this equation, find the width of a frame if the length is 16 inches.
Algebra 2
Mr. Prykuta
Worksheet #90.4- Logarithmic Regression
LOGARITHMIC REGRESSION: y = a + b(ln x)
1. What type of function would best model the following data?
(1) linear (2) exponential (3) logarithmic (4) none of the above
x
y
1 6
2 9.5
3 13
4 15
5 16.5
6 17.5
7 18.5
8 19
9 19.5
10 19.7
11 19.8
2. A pediatrician has the following table that lists the head circumferences for a group of 12 baby girls from the same extended family. The circumference is given in centimeters.
a. Make a scatter plot of the data
b. Choose what appears to be the curve that best fits the data. c. Find the regression equation of this model.
Age in Months 2 2 5 4 1 17 11 14 7 11 10 19
months.
Month 2 4 6 8 10 12 14 16 18
Height in
inches 22.7 26.1 27.5 28.9 31.7 32.1 32.7 33.1 34.0
Month 20 22 24 26 28 30 32 34 36
Height in
inches 34.4 34.6 34.9 35.2 35.6 36.0 36.6 37.2 37.6
a. Find the logarithmic regression equation for the data with the coefficients rounded to the three decimal places.
