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Lecture 5

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University of Ottawa

Administration

ADM2304

Tony Quon

Fall

Description

Comparing Two Population Proportions:
The Distribution of the Difference between Two Sample Proportions
Conﬁdence Interval for the Difference
—> Since we know the distribution of the difference of two proportions we can calculate the
conﬁdence interval in exactly the same way —> The general structure of a conﬁdence interval is (estimate ± (z- or t-value)×(SD or SE of
estimate)
—> Thus for the difference in the proportions for two independent populations, a 100(1-α)%
conﬁdence interval is given by:
⎛ p (1− p ) ˆ p (1− p ) ˆ ⎞
⎜(p − p )±z 1 1 + 2 2 ⎟
⎜ 1 2 α/2 ⎟
⎝ n 1 n 2 ⎠
Rule of Thumb
—> When can we use the two sample proportion test?
- If n1p1, n2p2, n1(1 - p1) and n2(1 - p2) are all greater than 10
—> Note that there is no need to restrict ourselves to testing p1 - p2 = 0. We can test for any
non-zero difference, p1 - p2 = p0, as well in which case the test statistic is:
(p − p )− p
z = 1 2 0 ~ N(0,1)
p (1− p ) p (1− p )
1 1 + 2 2
n 1 n 2
Pooling:
Under hypothesis H0: p1 - p2 = 0, we are assuming that the proportions are actually X + X (p − p )−0
the same thing. So in estimating the SE we can use a pool estimate of proportion:

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