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 6Two proportion z-testx1: 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- PropZTESTp1: ≠ p2Z = -3.3329P = 8.5944146E-4P1_hat = .2608695652 (x1/n1)p2_hat = .367839196 (x2/n2)P_hat =.36783916 ((x1+x2)/(n1+n2))n1= 184n2= 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.