6 Lecture Administrative
MA 2612 - Applied Statistics II
D 2004
1. Lab 1 Handout is due today. 2. This is the last lecture for Chapter 6. 3. Homework for Chapter 6 is due on Tuesday. 4. Quiz 2 is tomorrow.
Today
1. Quick Review and Example of Comparing two population means (PNC
6.8)
2. Comparing two population proportions (PNC 6.9) 3. Power (PNC 6.11)
Comparing the means of two independent populations Statistical model Y1,1 , Y1,2 , . . . , Y1,n1 N(1 , 2 ) 1 Y2,1 , Y2,2 , . . . , Y2,n2 N(2 , 2 ) 2 Statistical hypothesis H0 : 1 2 = 0 Ha : 1 2 < 0 H0 : 1 2 = 0 Ha : 1 2 6= 0 H0 : 1 2 = 0 Ha : 1 2 > 0
For most scientic hypotheses 0 = 0. This corresponds to null hypothesis of no dierence. Test statistic (Y1 Y2 ) 0 s(Y1 Y2 ) is the standard error of (Y1 Y2 ). t =
where s(Y1 Y2 )
The standard error is another name for the standard deviation of the
sampling distribution.
Thus, s(Y1 Y2 ) will depend on any assumptions we make in the statistical
model for Y1,1 , . . . , Y1,n1 and Y2,1 , . . . , Y2,n2 . clusions we make about H0 .
This will also aect the sampling distribution of t as well as any con-
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Computation of s(Y1 Y2 ) and the sampling distribution of t Recall the statistical model: Y1,1 , Y1,2 , . . . , Y1,n1 N(1 , 2 ) 1 Y2,1 , Y2,2 , . . . , Y2,n2 N(2 , 2 ) 2 s
n
1. With no further assumptions
s(Y1 Y2 ) = where
2 S1
2 2 S1 S2 + n1 n2
1 1 X 2 Y1,i Y1 = n1 1 i=1 2 1 X 2 = Y2,i Y2 n2 1 i=1
n
2 S2
(Y1 Y2 ) 0 s(Y1 Y2 ) follows a t distribution where is the largest integer less than or equal to 2 2 2 S1 S2 n1 + n2 t =
2 (S1 /n1 ) n1 1 2 2 + (Sn2/n2 ) 1 2 2
and
2. However, if it is reasonable to assume that 2 = 2 then 1 2
s(Y1 Y2 ) = where
2 Sp
s
2 Sp
1 1 + n1 n2
2 2 (n1 1)S1 + (n2 1)S2 = n1 + n2 2 and t follows a tn1 +n2 2 distribution.
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Example 1: Researchers wish to discover whether the enzyme Glucose 6Phosphate Dehydrogenase (G6PD) is involved in the disease Rheumatoid Arthritis (RA). To study this, they randomly select 14 adults with RA and 17 adults as controls. The control subjects do not have RA or any other arthritic condition. In each adult, the level of G6PD is measured with a blood test. Test results are given in units/g Hgb (Hgb = hemoglobin). The data are summarized in the following table: RA Control mean 17.80 12.30 s.d. 3.20 2.84 Scientic hypothesis
Statistical model
Statistical hypothesis
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If we assume 1 = 2 Compute s(Y1 Y2 ) .
If we cant assume 1 = 2
Compute t .
Find the degrees of freedom.
Find the critical value and draw a conclusion about H0 .
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Comparing two population proportions (PNC 6.9) Examples of where this might be interesting:
1. Compare proportion of subjects in treatment and control group that
experience side-eects.
2. Compare cancer rates for people who live close/far from power lines. 3. Compare proportion defective of items from two dierent production
lines. Statistical model
We observe Y1 , the number out of the n1 units in our sample from the
rst population that have some specied property.
We observe Y2 , the number out of the n2 units in our sample from the
second population that have some specied property.
The proportion of units in the rst population that have the specied
property is some (unknown) value p1 between 0 and 1. property is some (unknown) value p2 between 0 and 1.
The proportion of units in the second population that have the specied Therefore, Y1 bin(n1 , p1 ) and Y2 bin(n2 , p2 ).
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Statistical hypothesis H0 : p1 p2 = 0 Ha : p1 p2 < 0 H0 : p1 p2 = 0 Ha : p1 p2 6= 0 H0 : p1 p2 = 0 Ha : p1 p2 > 0
For most scientic hypotheses 0 = 0. This corresponds to null hypothesis of no dierence. Test statistic z = q (1 p2 ) 0 p
p1 (12 ) p n1
p + p2 (12 ) n2
where p1 = Y1 /n1 and p2 = Y2 /n2 . Sampling distribution When Y1 , Y2 , (n1 Y1 ) and (n2 Y2 ) are all greater than 5, z has approximately a N(0,1) distribution. We use the sampling distribution to make conclusions about H0 in the usual way(s).
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Example: Two drugs, zidovudine and didanosine, were tested for their effectiveness in preventing progression of HIV in children. In a double-blind clinical trial 276 children with HIV were given zidovudine, 281 were given didanosine. Using the data in the table below, test the scientic hypothesis that zidovudine is less eective than didanosine. Zidovudine Didanosine Total Died 17 7 24 Survived 259 274 533 Total 276 281 557
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Power (PNC 6.11)
Whenever we conduct a hypothesis test, there is the possibility that we
will reach the wrong conclusion.
Obviou...