Math3080-001
Spring 2009
Exam #2
1.)
Name:_______________________________
In simple linear regression analysis, _________________, measures the amount
of variation in observed y values that is not explained by the model - that is, the amount
of y-data variation that cannot be attributed to a linear relationship, and
____________________ measures the total amount of variation in observed y values.
2.) An experiment to measure the macroscopic magnetic relaxation time in
crystals ( µ sec) as a function of the strength of the external biasing magnetic field (KG)
yielded the following data
x
y
11.0
187
12.5
225
15.2
305
17.2
318
19.0
367
x
y
22.0
400
24.2
435
25.3
450
27.0
506
29.0
558
The summary statistics are
∑x
i
= 223.2, ∑ yi = 4116, ∑ xi2 = 4877.50, ∑ xi yi = 90, 096.1,
∑y
a.) Compute the equation of the estimated regression line.
2
i
20.8
365
= 1, 666, 782.
b.) Suppose it is possible to make a single observation at each of the n=20 values
x1 = 11.0, x2 = 11.5 , . . . , x20 = 20.5 . If a major objective is to estimate β1 as accurately
as possible, would the experiment with n=20 be preferable to the one with n=11?
Explain.
3.) Suppose that in a certain chemical process the expected value of y = reaction time
(hours) is related to x = temperature (o F) in the chamber in which the reaction takes place
according to a simple linear regression model with equation y = 5.00 - .01x and σ = .075.
a.) What is the expected change in reaction time for a 10 o F increase in temperature?
b.)Let Y1 and Y2 denote observations of reaction time when x = x1 and x = x2
respectively. If x2 − x1 = 1 o F, then what is P(Y2 > Y1 ) ?
4.) An investigation of the relationship between x = traffic flow (1000’s of cars per 24
hours) and y = lead content in bark on trees near the highway ( µ g / g dry wt) yielded the
data in the accompanying table.
8.3
227
x
y
8.3
312
12.1 12.1 17.0
362 521 640
17.0
539
17.0
728
24.3
945
24.3
738
= 5,390,382,
∑x y
24.3 33.6
759 1263
The summary statistics are:
n = 11,
∑x
i
= 198.3,
∑y
i
= 7034 ,
∑x
2
i
= 4198.03,
∑y
2
i
i
i
= 149,354.4
SSE = 76,492.54, and SST = 892,458.73
In addition, the least squares estimates are given by: βˆ0 = −12.84159, and βˆ1 = 36.18385
a.) What proportion of observed variation in lead content can be explained by the
approximate linear relationship between the variables?
b.) Does it appear that there is a useful linear relationship between the two variables?
Conduct a test at significance level α = .01.
5.) The simple linear regression model provides a very good fit to a data set on rainfall
volume and runoff volume.
The equation of the least squares line is
2
yˆ = −1.128 + .82697 x , r = .975 , s = 5.24 , and n=15.
a.) Use the fact that sYˆ = 1.44 when rainfall volume is 50 m 3 to calculate a 95% CI for true
average runoff volume
b.) Calculate a 95% PI for runoff when rainfall volume is 50 m 3 . How does this interval
compare to the one you found in (a)? Why?