13 Lecture Administrative
1. Lab 3 tomorrow.
MA 2612 - Applied Statistics II
D 2004
2. Exam 1 on Monday.
Today
1. Example using SLR model. 2. Inference about regression parameters.
Exam 1: Monday 11:00am (60 minutes, no extra time)
Chapters 6 and 9 Open book/notes Some problems similar to quizzes Some true/false or multiple choice questions on concepts
Example i Xi 1 5 2 7 3 2 4 7 5 9 Yi 12 11 12 9 6
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Example i Xi 1 5 2 7 3 2 4 7 5 9 Sum 30 SS Yi (Xi X) (Yi Y ) (Xi X)(Yi Y ) Xi Yi ei Yi 12 -1 2 -2 60 10.79 1.209 11 1 1 1 77 9.22 1.780 12 -4 2 -8 24 13.15 -1.149 9 1 -1 -1 63 9.22 -0.220 6 3 -4 -12 54 7.65 -1.649 50 0 0 -22 278 28 26 8.714
In lecture 12 we saw that Pn Xi X Yi Y 22 i=1 1 = = = 0.79 2 Pn 28 i=1 Xi X 50 22 30 + = 14.74 0 = Y 1X = 5 28 5 Therefore, Yi = 0 + 1 Xi = 14.74 0.79Xi We can also ll in the ANOVA table for this model. Source Regression Error Total df SS MS
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Inference about the parameters of the SLR model Condence intervals Recall from MA 2611:
A condence interval is a range of likely values for the true value of
the parameter that we compute based on the data.
Whenever we estimate a parameter it is more informative to give a con-
dence interval than to simply state the point estimate.
This is because our estimate is subject to sampling variability. In situations where there is a lot of sampling variability, it is more di-
cult to accurately estimate the parameter and so the condence interval is wider.
Similarly, when there is very little sampling variability, it is more likely
that our estimate is close to the true value of the parameter and so the condence interval is narrower.
The form of most condence intervals we encounter is
ESTIMATE MARGIN OF ERROR where margin the of error has two components
1. The estimated standard error of the estimate (of the parameter). 2. A constant that is related to the sampling distribution of the esti-
mate.
For example, in MA 2611 you learned that 95% condence interval for
the mean of a normal population based on n observations is SY Y tn1,0.975 n
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Condence intervals for 0 and 1 We have already seen that our estimates of 0 and 1 are 0 = Y 1X 1 Pn Xi X Yi Y i=1 = Pn 2 Xi X i=1
The estimated standard errors for these estimates are v ! u u 2 X 1 + Pn ( 0 ) = tMSE 2 n i=1 Xi X s MSE ( 1 ) = Pn 2 Xi X i=1 And the sampling distributions of both 0 0 ( 0 ) is the tn2 distribution. Therefore, the 100(1 )% condence intervals for 0 and 1 are: 0 tn2,1/2 ( 0 ) 1 tn2,1/2 ( 1 ) and 1 1 ( 1 )
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Hypothesis tests 0 and 1
Scientic hypothesis Statistical model
We observe (X1 , Y1 ), (X2 , Y2 ), . . . (Xn , Yn ) which we assume are described by Yi = 0 + 1 Xi + i where
1, 2, . . . n
N(0, 2 ) or equivalently, Y...