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Differences of sample proportions — Probability examples

Practice using shape, center (mean), and variability (standard deviation) to calculate probabilities of various results when we're dealing with sampling distributions for the differences of sample proportions.

Intro and review

In this article, we'll practice applying what we've learned about sampling distributions for the differences in sample proportions to calculate probabilities of various sample results.
Skip ahead if you want to go straight to some examples.
Here's a review of how we can think about the shape, center, and variability in the sampling distribution of the difference between two proportions p^1p^2:

Shape

The shape of a sampling distribution of p^1p^2 depends on whether both samples pass the large counts condition.
  • If we expect at least 10 successes and at least 10 failures in both samples, then the sampling distribution of p^1p^2 will be approximately normal.
  • If one or more of these counts is less than 10, then the sampling distribution won't be approximately normal.

Center

The mean difference is the difference between the population proportions:
μp^1p^2=p1p2

Variability

The standard deviation of the difference is:
σp^1p^2=p1(1p1)n1+p2(1p2)n2
(where n1 and n2 are the sizes of each sample).
This standard deviation formula is exactly correct as long as we have:
  • Independent observations between the two samples.
  • Independent observations within each sample*.
*If we're sampling without replacement, this formula will actually overestimate the standard deviation, but it's extremely close to correct as long as each sample is less than 10% of its population.
Let's try applying these ideas to a few examples and see if we can use them to calculate some probabilities.

Example 1

Yuki is a candidate is running for office, and she wants to know how much support she has in two different districts. Yuki doesn't know it, but 45% of the 8,000 voters in District A support her, while 40% of the 6,500 voters in District B support her.
Yuki hires a polling firm to take separate random samples of 100 voters from each district. The firm will then look at the difference between the proportions of voters who support her in each sample (p^Ap^B).
Question 1.1
What are the mean and standard deviation of the sampling distribution of p^Ap^B?
Round to three decimal places.
Choose 1 answer:

Example 2

A company has two offices, one in Mumbai, and the other in Delhi.
  • Each office has about 600 total employees.
  • 85% of the employees at the Mumbai office are younger than 40 years old.
  • 81% of the employees at the Delhi office are younger than 40 years old.
The company plans on taking separate random samples of 50 employees from each office. They'll look at the difference between the proportions of employees in each sample that are younger than 40 years old (p^Mp^D).
The company wonders how likely it is that the difference between the two samples is greater than 10 percentage points.
Question 2.1
Why is it inappropriate to use a normal distribution to calculate this probability?
Choose 1 answer:

Want to join the conversation?

  • orange juice squid orange style avatar for user eric.jackson2
    They call me Mr.Math
    (19 votes)
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  • blobby green style avatar for user Omar
    Example 1, Question 1.1 - Solving for Standard Deviation

    In the solution, "100" is used for both "n" values. I believe this is an error as the "n" values should correspond with the number of voters in each district, which would make them 8,000 and 6,000 for districts A and B, respectively.
    (0 votes)
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