Chapter 22 Comparing two
proportions
Math2200
Are men more intelligent?
• Gallup poll
• A random sample of 520 women and 506
men
• 28% of the men thought men were more
intelligent
• 14% of the women agreed
• Is there a gender gap in opinions about
which sex is smarter?
Difference between two proportions
• To assess the significance of the
difference of the two sample proportions,
we need its sd or se
• Recall that ‘the variance of the sum of
difference of two independent random
variables is the sum of their variances’
– If X and Y are independent,
Var(X-Y) = Var(X) + Var(Y)
SD of the difference between two
sample proportions
Assumptions and conditions
• Independence assumptions
– Randomization
• Data in each group should be drawn independently and at
random from a homogeneous population or generated by a
randomized comparative experiment
– 10% condition
• When the data are sampled without replacement, the sample
should not exceed 10% of the population
– Independent samples
• The two groups must be independent of each other
• Sample size condition
– Success/failure condition for each sample
Sampling distribution of
• Using normal approximation,
–
–
–
is normal
is normal
is also normal because of
independence
• Mean:
• Standard deviation
A two-proportion z-interval
Are men more intelligent? (cont’)
• Goal: estimate the gap
• Parameter of interest
• Conditions
– Randomization
– 10% condition
– Independent samples
– Success/failure condition
• 95% confidence interval
Example: ZZzzzz
• Study on snoring by the National Sleep
Foundation
• Out of 995 respondents,
– Overall, 37% reported they snored at least a
few nights a week
– 26% of 184 people with age under 30
– 39% of 811 people with age above 30
• Is the difference of 13% real, or due to
natural fluctuations in the sample?
Two-proportion z-test
•
– Additional information under the null
hypothesis: two proportions are equal!
– A pooled estimate of this equal proportion is
– The corresponding se is
Two-proportion z-test
• P-value is then decided using standard
normal (but also depends on one-sided or
two-sided alternatives!)
ZZzzzz (cont’)
• Hypotheses
• Conditions
–
–
–
–
Randomization
10% condition
Independent sample
Success/failure condition
• Two-proportion z-test
• P-value = 0.0008
ZZzzzz (cont’)
STAT TESTS 6
Two proportion z-test
x1: 48 (# of younger people snore)
n1: 184 (# of younger respondents)
x2: 318 (# of older people snore)
n2: 811 (# of older respondents)
p1: ≠ p2 (two sided alternative)
Calculate
------------2- PropZTEST
p1: ≠ p2
Z = -3.3329
P = 8.5944146E-4
P1_hat = .2608695652 (x1/n1)
p2_hat = .367839196 (x2/n2)
P_hat =.36783916 ((x1+x2)/(n1+n2))
n1= 184
n2= 811
What Can Go Wrong?
• Don’t use two-sample proportion methods when
the samples aren’t independent.
– These methods give wrong answers when the
independence assumption is violated.
• Don’t apply inference methods when there was
no randomization.
– Our data must come from representative random
samples or from a properly randomized experiment.
• Don’t interpret a significant difference in
proportions causally.
– Be careful not to jump to conclusions about causality.
What have we learned?
• We’ve now looked at the difference in two
proportions.
• Perhaps the most important thing to
remember is that the concepts and
interpretations are essentially the same—
only the mechanics have changed slightly.
What have we learned?
• Hypothesis tests and confidence intervals
for the difference in two proportions are
based on Normal models.
– Both require us to find the standard error of
the difference in two proportions.
• We do that by adding the variances of the two
sample proportions, assuming our two groups are
independent.
• When we test a hypothesis that the two
proportions are equal, we pool the sample data; for
confidence intervals we don’t pool.