AP Statistics Inference Part 2 Unit 9 Day 3
10-2: Inference for Two Proportions

  1. Confidence Intervals for the difference of two proportions(2-propZ-Int)

P – define parameter, population, and process
A– check Assumptions & Conditions
-Random – both samples must be random
-Normal – check that the number of successes & failures for each sample is at least 10 (there should be 4 numbers to verify).
-Independent – for experiments, state independence if experiment was done properly; for samples, check the 10% condition (add a 0 to each sample size and check population size).
S–Statistical Work (2-Prop Z-Int)

-by hand: Formula: Calculator:
*point estimate=; critical value = ; standard error =
*margin of error = (critical value)(standard error)
*the critical value, z*, comes from the Normal curve. The Confidence Level is the middle percent, so you subtract from 1 and divide by 2 to put the “area to the left into 2ndVars3: InvNorm(area to left)

S– Conclusion: “We are ____% confident that the interval between ______and ______will capture the true difference in proportion of parameter 1 and parameter 2 (in context).

Example 1)Just before the presidential election in November 2008, a local newspaper conducted a poll of residents in a medium-sized city and found that 120 out of a simple random sample of 250 men intended to vote for BarackObama and 132 out of an SRS of 240 women intended to vote for Obama. Construct and interpret a 95% confidence interval for the difference in proportion of women and men who supported Obama in this city.

P –2-proportion Z-interval
true proportion of male voters who supported Obama in this city
true proportion of female voters who supported Obama in this city
A -Random – both samples are an SRS
-Normal – men: 120 successes (votes for Obama) and 130 failures (not voting for Obama) are both at least 10; women: 132 successes, 108 failures are both at least 10
N -Independent – there are at least 2500 men and 2400 women in this medium-sized city
S–

S– We are 95% confident the interval between -.1583 and .0183 will capture the true difference in proportions of men and women who support Obama in this medium-sized city.

**formulas for confidence intervals use the sample statistic, , in the calculations because the goal is to estimate the population proportion p. That’s why the normal condition counts successes and failures instead of np and n(1-p), because we don’t know the value of p.

  1. Significance Tests for the difference of two proportions (2-proportionZ-Test)

P – state hypothesis, define parameters & populations, and identify procedure

Null: *the null will ALWAYS be that there is no difference between proportions
Alternative: *choose the appropriate inequality symbol based on the problem a and always state first.

A– check Assumptions & Conditions & state alpha level
-all conditions for significance tests for two-proportions are the same as confidence intervals for two proportions
S– Statistical Work (2-Prop Z-Test)
-by hand – Formula: , then use tcdf to find p-value -Calculator:
pooled sample proportion; the combined sample proportion of the two samples

is used in the formula for standard deviation of the test statistic

S– Conclusion: Because our p-value of ______is less than [greater than]α = 0.05, we [fail to] reject the null hypothesis and [cannot] conclude ______(alternative in context).
**if the p-value is greater than alpha, then insert the words in brackets in the conclusion**

Example 2)Is there convincing evidence that there was a gender difference in Obama’s support in this city? (scenario from example 1) Support your conclusion with a test of significance using α = 0.05

A–α=0.05
Random, Normal, & Independent were checked & verified in Example 1.

S–

S–Because our p-value of 0.1212 is greater than α = 0.05, we fail to reject the null hypothesis and cannot conclude that there is a difference in proportion of men and women supporters for Obama in this city.

**Notice how the confidence interval captured 0 as a possible parameter value for the true difference in proportions for men and women supporters of Obama in that city and we failed to reject the null hypothesis & could not conclude there was a difference***

*confidence intervals & significance tests go hand-in-hand