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Lesson 10.1 - Comparing Two Proportions
two populations  two samples  two statistics
The Sampling Distribution of a Difference between Two Proportions

Choose an SRS of size 𝑛1 from Population 1 with proportion of successes 𝑝1 and an independent SRS of size 𝑛2 from
Population 2 with proportion of successes 𝑝2 . The sampling distribution of 𝑝̂1 − 𝑝̂2 has the following properties:
 Shape:
 Center:
 Spread:
Example: American made cars
Nathan and Kyle both work for the Department of Motor Vehicles (DMV), but they live in different states. In Nathan’s
state, 80% of the registered cars are made by American manufacturers. In Kyle’s state, only 60% of the registered cars
are made by American manufacturers. Nathan selects a random sample of 100 cars in his state and Kyle selects a
random sample of 70 cars in his state. Let 𝑝̂𝑁 − 𝑝̂𝐾 be the difference in the sample proportion of cars made by
American manufacturers.
(a) What is the shape of the sampling distribution of 𝑝̂𝑁 − 𝑝̂𝐾 ? Why?
(b) Find the mean of the sampling distribution. Show your work.
(c) Find the standard deviation of the sampling distribution. Show your work.
Confidence intervals for 𝒑𝟏 − 𝒑𝟐
 Inference method: ________________________________________________
 Conditions for constructing a confidence interval about a difference in proportions:
o
______________________:
o
______________________:
o
______________________:
 Because we don’t know the values of the parameters 𝑝1 and 𝑝2 , we replace them in the standard deviation
formula with the sample proportions. The result is the __________________________ ____________________:
 When the conditions are met, an approximate 𝐶% confidence interval for 𝑝̂1 − 𝑝̂2 is
where 𝑧 ∗ is the critical value for the standard Normal curve with 𝐶% of its area between −𝑧 ∗ and 𝑧 ∗ .
Example: Presidential approval
Many news organizations conduct polls asking adults in the United States if they approve of the job the president is
doing. How did President Obama’s approval rating change from October 2012 to October 2013? According to a Gallup
poll of 1500 randomly selected U.S. adults on October 2–4, 2012, 52% approved of Obama’s job performance. A Gallup
poll of 1500 randomly selected U.S. adults on October 5–7, 2013, showed that 46% approved of Obama’s job
performance.
(a) Calculate the standard error of the sampling distribution of the difference in the sample proportions
(2013 – 2012).
(b) Use the results of these polls to construct and interpret a 90% confidence interval for the change in Obama’s
approval rating among all U.S. adults from October 2012 to October 2013.
(c) Based on your interval, is there convincing evidence that Obama’s job approval rating has changed?
Significance tests for 𝒑𝟏 − 𝒑𝟐
 Inference method: ________________________________________________

Remember: Significance tests help us determine if an observed difference between two sample proportions reflects
an actual difference in the parameters or if it’s due to chance variation in random sampling or random assignment.

The null hypothesis has the general form:

The alternative hypothesis says what kind of difference we expect (<, >, 𝑜𝑟 ≠).

Test statistic:

Find the 𝑃-value by calculating the probability of getting a 𝑧 statistic this large or larger in the direction specified by
the alternative hypothesis 𝐻𝑎 :
Don’t you LOVE Statistics?!?!?!?!
I mean, really… Isn’t this way more FUN than planning for
Spring Break???!!!!!
Example: Hearing loss
Are teenagers going deaf? In a study of 3000 randomly selected teenagers in 1988–1994, 15% showed some hearing
loss. In a similar study of 1800 teenagers in 2005–2006, 19.5% showed some hearing loss.
(a) Do these data give convincing evidence that the proportion of all teens with hearing loss has increased?
Inference for Experiments
Example: Preschool
To study the long-term effects of preschool programs for poor children, researchers de- signed an experiment. They
recruited 123 children who had never attended preschool from low-income families in Michigan. Researchers randomly
assigned 62 of the children to attend preschool (paid for by the study budget) and the other 61 to serve as a control
group who would not go to preschool. One response variable of interest was the need for social services as adults. Over
a 10-year period, 38 children in the preschool group and 49 in the control group have needed social services.4
Does this study provide convincing evidence that preschool reduces the later need for social services? Justify your
answer.