Statistics
Two proportions
April 9, 2009
Outline
1. the 2 × 2 table.
HeartDisease
Bald
Hairy
Not
127
548
1224
7611
One way to think about it: there are two parameters p1 and p2 associated with the rows.
(p1 is the proportion of the population determined by the first row that is classified into
the first column.)
A
B
Success
p1
p2
Parameters
Failure
1 − p1
1 − p2
A
B
Data
Success
x1
x2
Failure
n 1 − x1
n 2 − x2
2. Confidence interval for p1 − p2 :
(a) Standard interval
p̂1 =
x1
n1
p̂2 =
s
x2
n2
(p̂1 − p̂2 ) ± z ∗
p̂1 (1 − p̂1 ) p̂2 (1 − p̂2 )
+
n1
n2
(b) Plus-4 interval (add one success and one failure to each group)
s
x1 + 1
x2 + 1
p̃1 (1 − p̃1 ) p̃2 (1 − p̃2 )
p̃1 =
p̃2 =
(p̃1 − p̃2 ) ± z ∗
+
n1 + 2
n2 + 2
n1 + 2
n2 + 2
3. Odds ratios and relative risk.
1 4
1 6
Odds for each population (rows)
pi
1 − pi
p̂1 /(1 − p̂1 )
(1 means no difference)
p̂2 /(1 − p̂2 )
p̂1
Relative risk:
(1 means no difference)
p̂2
Odds ratio:
Statistics
Two proportions
April 9, 2009
Test 3 - Tuesday, April 14
1. The test covers Chapters 15, 16, the hypothesis test portion of Chapter 18, Chapter 20,
and Chapter 9.
2. The outlines covered are those from March 9 (outline 20) to April 3 (Outline 28)
3. The most important idea on this test is the logic of hypothesis testing. Know such things
as
(a) What kinds of hypotheses are appropriate null hypothesis
(b) How to interpret a P -value
(c) How the language of statistical significance is used
(d) If decisions need to be made, what are the possible decisions and the possible errors
(outline 24)
4. The vocabulary of experimental design is important. Know such terms as explanatory
and response variables, confounding, lurking variables, and the three R’s.
5. The following table might be useful for the test. You will get a copy of this (as well as
Table C) with the test.
Parameter
Estimate
µ
x̄
r
p
x
p̂ =
n
x+2
p̃ =
n+4
r
p
SE
s
√
n
Confidence Interval
Null
x̄ ± t∗ SE
µ = µ0
p̂(1 − p̂)
n
p̂ ± z ∗ SE
p = p0
p̃(1 − p̃)
n+4
p̃ ± z ∗ SE
Test Statistic
x̄ − µ0
SE
Dist.
t(n − 1)
p̂ − p0
q
p0 (1−p0 )
n
6. There is a copy of last semester’s Test 3 and Test 4 on the course webpage. Neither is a
very good example of this test – Test 3 covered a lot of this stuff and Test 4 a little but
both covered other stuff as well.
z