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Wize AP Statistics Textbook > Inference for Two Population Proportions

Confidence Interval for Two Proportions

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Confidence Interval for Differences in Population Proportions

To make inferences about the difference in the population proportions p1−p2p_1-p_2 (parameter), we use the difference in the sample proportions p^1−p^2\hat{p}_1-\hat{p}_2.

Wize Tip
Review Confidence Intervals if you need a refresher. (See: Estimating with Confidence Intervals)


We do not know the actual value of the difference in population proportions p1−p2p_1-p_2 but we can provide a reasonable range to estimate it, by constructing confidence intervals given a certain confidence level C\colorFour C (e.g. 90%, 95%, 99%). Like all confidence interval, it is constructed using a point estimate plus or minus a margin of error.

[point estimate]±[MOE]\left[point\ estimate\right]\pm\left[MOE\right]

When comparing two proportions, the point estimate (or statistic) is p^1−p^2\hat{p}_1-\hat{p}_2.

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Example
We wish to estimate the difference in the percentage of international students at U of A (Population 1) and U of B (Population 2).
  • The parameter that we are trying to estimate is p1−p2p_1-p_2.
  • p1p_1 is unkown
  • p2p_2 is unkown
  • Therefore, the difference p1−p2p_1-p_2 is unkown.

Suppose we draw samples from each population. Results:


  • The proportion of international students in Sample 1 is p^1=0.12\hat p_1=0.12
  • The proportion of international students in Sample 2 is p^2=0.09\hat p_2=0.09
  • The point estimate or statistic is p^1−p^2=0.12−0.09=0.03\hat{p}_1-\hat{p}_2=0.12-0.09=0.03
  • We estimate that the true difference in proportions is 0.03 (or 3%).
The above is just a point estimate. Suppose we construct a 95% confidence interval. Results:
[0.01,0.05][0.01, 0.05]
  • 0.01 is the lower confidence level (LCL)
  • 0.05 is the upper confidence level (UCL)
  • We are 95% confident that the true difference in proportion of international students is between 0.01 and 0.05.
  • We do not know the value of the true difference in proportions but we are very confident that it is somewhere within that confidence interval.



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Confidence Interval for Two Proportions


(p^1−p^2)±z∗p^1q^1n1+p^2q^2n2\displaystyle\boxed{\left(\hat p_1-\hat p_2\right)\pm z^{\ast}\sqrt{\frac{\hat p_1\hat q_1}{n_1}+\frac{\hat p_2\hat q_2}{n_2}}}
where,
  • q^i=1−p^i\hat q_i=1-\hat p_i
  • the standard error of the statistic p^1−p^2\hat p_1-\hat p_2 is p^1q^1n1+p^2q^2n2\displaystyle{\sqrt{\frac{\hat p_1\hat q_1}{n_1}+\frac{\hat p_2\hat q_2}{n_2}}}


We make inferences with confidence intervals.


Wize Concept
If the confidence interval does not contain "0", then there is evidence that the two population proportions differ.
If the interval contains "0", then there is no evidence that the two population proportions differ.


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Example: Confidence Interval for Two Proportions

Based on a random sample of 50 male professors, 26 of them voted in favor of electing a new president. Based on a random sample of 40 female professors, 12 of them voted in favor.

(a) Solve for p^1\hat p_1and p^2\hat p_2:
p^1=x1n1\displaystyle{\hat p _1=\frac{x_1}{n_1}}
x1=26n1=50x_1=26\\n_1=50
p^1=2650=0.52\hat p_1=\frac{26}{50}=0.52

p^2=x2n2\displaystyle{\hat p _2=\frac{x_2}{n_2}}
x2=12n2=40x_2=12\\n_2=40
p^2=1240=0.30\hat p_2=\frac{12}{40}=0.30


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(b) Construct a 95% confidence interval for the difference in the proportions of male and female professors who are in favor of electing a new president. Interpret it (in plain English).

We use:
(p^1−p^2)±z∗p^1q^1n1+p^2q^2n2\displaystyle\boxed{\left(\hat p_1-\hat p_2\right)\pm z^{\ast}\sqrt{\frac{\hat p_1\hat q_1}{n_1}+\frac{\hat p_2\hat q_2}{n_2}}}
We need to find z⋆z^\star :

z⋆=1.96z^\star = 1.96

(0.52−0.30)±1.96(0.52)(0.48)50+(0.3)(0.7)40\displaystyle{(0.52-0.30)\pm1.96\sqrt{\frac{\left(0.52\right)\left(0.48\right)}{50}+\frac{\left(0.3\right)\left(0.7\right)}{40}}}

0.22±0.19840.22\pm0.1984

[0.0216,0.4184][0.0216,0.4184]

We are 95% confident that the difference in the proportions of male and female professors who are in favor of electing a new president is between 2% and 42%. Specifically, there is a greater proportion of male professors who are in favor.

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(c) Does the interval contain "0"? Interpret that that means.

The interval does not contain "0". This means that one group (specifically the male professors) is consistently different (specifically greater) than the other group (specifically the female professors) for being in favor of electing a new president.



Based on a random sample of 40 students who live on campus (Population 1), 16 of them say get enough sleep. Based on a random sample of 32 students who live off campus (Population 2), 10 of them say get enough sleep.

Construct a 99% confidence interval to estimate the difference in the proportions of students who live on campus and off campus who say they get enough sleep.
(i) What is the point estimate?