Math 2355 – Least Square Example Problems
1. Since 1993, California has generated new jobs to replace those lost, mainly in the defense and aerospace
industries. Many of these new jobs are in the computer field. The results of a survey in which x represented
the last two digits of the year and y represented the number of new jobs (in thousands) generated in
computers, gave the following summations.
n6
 x 2  51,355  x  555  xy  69,869  y  754  y 2  95,740
(a)
(b)
(c)
(d)
Find an equation for the least squares line.
Predict the number of new jobs in 1997.
If this growth in jobs continues linearly, when will the number of new jobs reach 170,000?
Find and interpret the coefficient of correlation.
2. Biologists have observed a linear relationship between the temperature and the frequency with which a
cricket chirps. The following data were measured for the striped ground cricket.
Temperature (x)
88.6
71.6
93.3
84.3
80.6
75.2
69.7
82.0
69.4
83.3
79.6
82.6
80.6
83.4
76.3
Chirps per second (y)
20.0
16.0
19.8
18.4
17.1
15.5
14.7
17.1
15.4
16.2
15.0
17.2
16.0
17.0
14.4
(a) Find the equation for the least squares line for the data.
(b) Use the results of part (a) to determine how many chirps per second you would expect to hear from
the striped ground cricket if the temperature were 73o.
(c) Use the results from part (a) to determine what the temperature is when the striped ground crickets
are chirping at a rate of 18 times per second.
(d) Find the coefficient of correlation and interpret it.
3. The following table gives the high school and college grade point averages for seven students, where the
highest possible grade point in both cases is 4.0.
High School (x)
2.50
3.00
3.25
3.50
3.60
3.75
4.00
College (y)
1.80
2.50
3.50
3.00
2.20
3.20
3.50
(a) Write the equation of the least squares line for the data.
(b) Find the coefficient of correlation between high school and college grade point averages & interpret.
4. The following table gives the average cost for a product at different levels of production.
Units(x)
Average Cost (y)
400
30
500
28
600
27
700
27
800
26
(a) Write the linear regression equation that predicts average cost from production.
(b) Find the correlation between production and average cost and interpret it.
5. In a study to determine the linear relationship between the size (in decimeters) of an ear of corn (y) and the
amount (in tons per acre) of fertilizer used (x), the following data were collected.
n = 10
xy = 75
x = 30
x2 = 100
y = 24
y2 = 80
(a) Find an equation for the least squares line.
(b) Find the coefficient of correlation.
(c) If 3 tons of fertilizer are used per acre, what length (in decimeters) would the equation in part a
predict for an ear of corn?
(d) Based on the coefficient of correlation in part b, does there seem to be a linear relationship between
the size of an ear of corn and the amount of fertilizer used?
6. In an experiment to determine the linear relationship between temperatures on the Celsius scale (y) and on
the Fahrenheit scale (x), a student got the following results.
n=5
xy = 28,050
x = 376
x2 = 62,522
y = 120
y2 = 13,450
(a) Find an equation for the least squares line.
(b) Find the reading on the Celsius scale that corresponds to a reading of 120 Fahrenheit, using the
equation from part a.
(c) Find the coefficient of correlation and interpret it.
7. Suppose that a company purchased a new software package to handle their word processing, payroll,
inventory, and accounting needs. After a few employees had received some training with the new system,
their work with it was monitored and the following data were gathered.
Hours of Training (x)
1
2
10
8
6
12
7
6
4
4
Number of Errors (y)
8
7
2
2
8
0
3
3
4
6
(a) Find an equation for the least squares line.
(b) Use the equation developed in part a to determine the number of hours of training needed for the
predicted number of errors to equal 0.
(c) Find the coefficient of correlation and interpret it.