Inference for p1 − p2
M243
April 12
Does cell phone use while driving cause accidents?
1. The two-by-two table.
2. the 2 × 2 table.
Error
Distracted
Not Distracted
No Error
17
22
7
2
3. Same three data collection paradigms as in the case of two means.
Parameters
A
B
Success
p1
p2
Data
Failure
1 − p1
1 − p2
Success
x1
x2
A
B
Failure
n1 − x1
n2 − x2
4. Confidence interval for p1 − p2 :
(a) Standard interval (R command prop.test)
x1
p̂1 =
n1
x2
p̂2 =
n2
s
(p̂1 − p̂2 ) ± z
∗
p̂1 (1 − p̂1 ) p̂2 (1 − p̂2 )
+
n1
n2
(b) Plus-4 interval (add one success and one failure to each group)
x1 + 1
p̃1 =
n1 + 2
Bald
Hairy
HeartDisease
127
548
x2 + 1
p̃2 =
n2 + 2
s
(p̃1 − p̃2 ) ± z
Not
∗
p̃1 (1 − p̃1 ) p̃2 (1 − p̃2 )
+
n1 + 2
n2 + 2
> prop.test( c(127,548), c(127+1124,548+7611) ,correct=F)
1224
7611
2-sample test for equality of proportions without
continuity correction
data: c(127, 548) out of c(127 + 1124, 548 + 7611)
X-squared = 19.2249, df = 1, p-value = 1.162e-05
alternative hypothesis: two.sided
95 percent confidence interval:
0.01675861 0.05194877
sample estimates:
prop 1
prop 2
0.1015188 0.0671651
5. Odds ratios and relative risk.
(a) Odds for each population (rows)
pi
1 − pi
p1 /(1 − p1 )
(1 means no difference)
p2 /(1 − p2 )
p1
(c) Relative risk:
(1 means no difference)
p2
(b) Odds ratio:
M243
Inference for p1 − p2
April 12
Homework
1. Read Chapter 22, pages 526–530.
2. For practice do problem 22.3.
3. To turn in (Due Tuesday, April 20) do Problem 22.6.
Test Friday. Take-home portion handed out Tuesday and due Friday. In-class portion covers Chaters 18,19,20,21,23,24.
More details Tuesday.