Stat 512-2
Solutions to Homework #1
Dr. Simonsen
Due Wednesday August 31, 2005.
A reminder – Please do not hand in any unlabelled or unedited SAS output.
Include in your write-up only
those results that are necessary to present a complete solution.
In particular, questions must be answered
in order (including graphs), and all graphs must be fully labeled (main title should include question
number and all axes should be labelled).
Don’t forget to put all necessary information (see course
policies) on the first page.
Include the SAS input for all questions at the very end of your homework.
You
will often be asked to continue problems on successive homework assignments, so save all your SAS code.
1.
A regression analysis relating test scores (Y) to training hours (X) produced the following
fitted equation:
.
ˆ
15
0.9
y
x
=
+
(a)
What is the fitted value of the response variable corresponding to x = 6?
(
)( )
ˆ
15
0.9
6
15
5.4
20.4
y
=
+
=
+
=
(b)
What is the residual corresponding to the data point with x = 5 and y = 17?
(
)( )
ˆ
15
0.9
5
15
4.5
19.5
17
19.5
2.5
i
y
e
=
+
=
+
=
=
−
= −
(c)
If x increases 3 units, how does
change?
ˆ
y
For each increase of 1 in x,
changes by the slope.
Therefore if x increases by 3 units,
will
increase by 3 times the slope, i.e. by (3)(0.9) = 2.7
ˆ
y
ˆ
y
(d)
Consider the data point in part (b).
An additional test score is to be obtained for a new
observation at x = 5.
Would the test score for the new observation necessarily be 17?
Explain.
Not necessarily.
The new observation is a random variable from a normal distribution with
estimated mean 19.5.
So you would not likely see 17 a second time.
(e)
The error sums of squares (SSE) for this model was found to be 8.
If there were n = 18
observations, provide the best estimate for
2
σ
.
2
8
0.5
2
16
E
SSE
SSE
s
MSE
df
n
=
=
=
=
=
−
(f)
Rewrite the regression equation in terms of x
*
where x
*
is training time measured in
minutes.
Show that your answer makes sense, i.e. gives the same predictions as the
original equation (an example is sufficient).

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