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Two-proportion confidence intervals · Tutorial 502 of 1000

Point Estimate for the Difference of Two Proportions

Calculate \(\hat{p}_1-\hat{p}_2\) and explain how this single sample statistic estimates the difference between two population proportions.

Intermediate 9 min read

What You'll Learn

  • Calculate each sample proportion from its success count and sample size.
  • Subtract the sample proportions in a specified group order.
  • Identify \(\hat{p}_1-\hat{p}_2\) as a point estimate of \(p_1-p_2\).
  • Interpret a positive, negative, or zero difference in context.
  • State a difference in proportions as a difference in percentage points.
  • Distinguish a percentage-point difference from a relative percent change.

From Two Sample Proportions to One Estimate

In Comparing Two Proportions: Setting and Notation, you learned to define the groups, label their sample proportions, and choose a subtraction order. Now use those sample proportions to calculate a single statistic: the difference between them. This statistic provides a point estimate of the difference between the two population proportions.

A point estimate is one value calculated from sample data and used to estimate an unknown population parameter. Here, the parameter of interest is \(p_1-p_2\), the true proportion in Group 1 minus the true proportion in Group 2. Its point estimate is \(\hat{p}_1-\hat{p}_2\). The hats indicate that these are sample statistics, not the unknown population proportions themselves.

Definition: The point estimate for the difference in two population proportions is the difference between the sample proportions, calculated in the same group order: \(\hat{p}_1-\hat{p}_2\) estimates \(p_1-p_2\).

Calculate each sample proportion using its own group’s count and sample size: \(\hat{p}_1=x_1/n_1\) and \(\hat{p}_2=x_2/n_2\). Then subtract the second sample proportion from the first. Do not subtract the counts alone or combine the sample sizes: the target is the difference between proportions, not the difference between numbers of successes.

$$ \hat{p}_1-\hat{p}_2 = \frac{x_1}{n_1}-\frac{x_2}{n_2} \qquad\text{estimates}\qquad p_1-p_2 $$

The subtraction order controls the sign and meaning. If Group 1’s sample proportion is larger, the estimate is positive; if Group 1’s is smaller, the estimate is negative. A positive estimate does not mean the difference is known exactly to be positive in the populations. It describes the direction and size of the difference observed in these samples, as an estimate of the population difference.

Interpreting the Difference in Proportions

Sample proportions are often written as decimals or percentages. When subtracting proportions, the result is naturally described as a difference in percentage points. For example, a difference of \(0.13\) is 13 percentage points, because \(0.13\) of the whole is 13 out of 100. This is not the same as saying “13 percent higher,” which could refer to a relative change calculated using a different denominator.

A careful interpretation names both groups and the characteristic. If \(\hat{p}_1-\hat{p}_2=0.13\), then the sample proportion for Group 1 is 13 percentage points higher than the sample proportion for Group 2. As a point estimate, this suggests that the true proportion in Group 1 is 13 percentage points higher than the true proportion in Group 2. The word suggests matters: the statistic comes from samples and is used to estimate the population difference.

Interpretation guide: Describe \(\hat{p}_1-\hat{p}_2\) as the observed sample proportion in Group 1 minus the observed sample proportion in Group 2. Then say how many percentage points higher or lower Group 1’s sample proportion is. Identify it as an estimate of the corresponding population difference \(p_1-p_2\).

A point estimate gives one number, not a range of plausible values. It does not by itself say how much the estimate might vary across samples or how precise it is. The key task here is to calculate and interpret the sample difference without treating it as the exact population difference.

Worked Examples

Worked Example: Compare Two Groups with 100 Observations Each

Setting: In a fictional survey, 48 of 100 sampled customers in Group 1 say they would use a new parcel-pickup option. In an independent sample, 35 of 100 customers in Group 2 say they would use it. Let \(p_1\) and \(p_2\) be the true proportions of customers in the two groups who would use the option. The requested order is Group 1 minus Group 2.

Calculate the sample proportions: For Group 1, \(x_1=48\) and \(n_1=100\). For Group 2, \(x_2=35\) and \(n_2=100\).

$$ \hat{p}_1=\frac{x_1}{n_1}=\frac{48}{100}=0.48, \qquad \hat{p}_2=\frac{x_2}{n_2}=\frac{35}{100}=0.35 $$

Find the point estimate: Subtract Group 2’s sample proportion from Group 1’s, matching the defined order.

$$ \hat{p}_1-\hat{p}_2 = 0.48-0.35 = 0.13 $$

The subtraction checks out in percentage form as well: \(48\%-35\%=13\) percentage points. Therefore, the point estimate of \(p_1-p_2\) is \(0.13\), or 13 percentage points.

Interpret in context: In these samples, the proportion of Group 1 customers who would use the parcel-pickup option is 13 percentage points higher than the proportion of Group 2 customers who would use it. The estimated difference in the two population proportions, Group 1 minus Group 2, is 13 percentage points. The sample result estimates the population difference; it does not establish its exact value.

Worked Example: Calculate a Difference with Unequal Sample Sizes

Setting: A fictional recreation program surveys two independent groups about whether participants attended at least one outdoor session. In Group 1, 36 of 80 sampled participants attended. In Group 2, 42 of 120 sampled participants attended. Group 1 minus Group 2 is the comparison of interest.

Calculate each proportion separately: The denominators differ, so each success count must be divided by its own group’s sample size.

$$ \hat{p}_1=\frac{36}{80}=0.45, \qquad \hat{p}_2=\frac{42}{120}=0.35 $$

The decimal calculations can be checked as percentages: \(36\) is \(45\%\) of \(80\), and \(42\) is \(35\%\) of \(120\). Now calculate the ordered difference.

$$ \hat{p}_1-\hat{p}_2 = 0.45-0.35 = 0.10 $$

Since \(0.10\) is 10 out of 100, the estimated difference is 10 percentage points. In these samples, the proportion attending at least one outdoor session was 10 percentage points higher in Group 1 than in Group 2. Thus \(0.10\) is the point estimate of the true Group 1-minus-Group 2 difference in attendance proportions.

Notice that subtracting the success counts, \(36-42\), would give \(-6\), which does not answer the question. The samples are different sizes, and the comparison is about the proportions, not which group had more attendees in total.

Worked Example: Make the Group Order Visible

Setting: A fictional environmental club surveys residents in two areas about whether they compost food scraps. In Lakeside, 19 of 50 sampled residents compost. In Hillview, 52 of 100 sampled residents compost. Define Group 1 as Lakeside and Group 2 as Hillview.

Calculate the sample proportions:

$$ \hat{p}_1=\frac{19}{50}=0.38, \qquad \hat{p}_2=\frac{52}{100}=0.52 $$

Subtract in the stated order: The comparison is Lakeside minus Hillview, so subtract \(0.52\) from \(0.38\).

$$ \hat{p}_1-\hat{p}_2 = 0.38-0.52 = -0.14 $$

As a percentage-point calculation, \(38\%-52\%=-14\) percentage points. The negative sign means the sample proportion for Lakeside is lower: in these samples, it is 14 percentage points lower than the proportion for Hillview. The point estimate of the Lakeside-minus-Hillview population difference is \(-0.14\), or \(-14\) percentage points.

If the question instead asked for Hillview minus Lakeside, the estimate would be \(0.52-0.38=0.14\), or 14 percentage points. Reversing the group order changes the sign, not the underlying sample information. Both expressions describe the same gap from opposite directions.

Common Mistakes and AP Exam Tip

  • Subtracting counts instead of proportions: The quantities \(x_1\) and \(x_2\) are numbers of successes, not rates. Calculate \(x_1/n_1\) and \(x_2/n_2\) first, especially when sample sizes differ.
  • Using the wrong denominator: Each success count belongs to its own sample size. Write \(\hat{p}_1=x_1/n_1\) and \(\hat{p}_2=x_2/n_2\) before subtracting.
  • Reversing the subtraction: If the question specifies Group 1 minus Group 2, calculate \(\hat{p}_1-\hat{p}_2\), not the reverse. A reversed answer has the opposite sign and describes the opposite ordered difference.
  • Confusing the statistic with the parameter: \(\hat{p}_1-\hat{p}_2\) is calculated from the samples. \(p_1-p_2\) is the unknown population difference that it estimates. Do not claim the sample statistic is the exact population value.
  • Calling a difference of \(0.13\) “13 percent higher” without clarification: State “13 percentage points higher” when subtracting proportions. A relative percent increase uses a reference proportion as a denominator and is a different calculation.
  • Dropping a negative sign: A negative Group 1-minus-Group 2 estimate means Group 1’s sample proportion is lower. Preserve the sign in the calculation, then explain its direction in words.
AP Exam Tip: Show both sample proportions before subtracting, include the group order in your interpretation, and report the difference in percentage points. A full-credit explanation identifies the statistic as an estimate of the population difference, not as a known population truth.

Key Takeaway

The point estimate for the difference between two population proportions is the difference between the corresponding sample proportions. Calculate each proportion with its own sample size, preserve the specified order, and interpret the sign and size in context. A difference such as \(0.13\) means 13 percentage points, not automatically a 13 percent relative increase.

Key takeaway: \(\hat{p}_1-\hat{p}_2\) estimates \(p_1-p_2\). The hats mark sample statistics, the subscripts preserve group order, and the result is interpreted as a difference in percentage points.

Check Your Understanding

For each question, calculate the ordered difference and interpret it as a point estimate in context.

  1. In a fictional survey, 44 of 100 sampled commuters in Group 1 and 31 of 100 sampled commuters in Group 2 prefer a new bus schedule. Find \(\hat{p}_1-\hat{p}_2\) and interpret the estimate.
  2. A fictional sports program finds that 27 of 60 sampled players in Group 1 attended a practice, while 32 of 80 sampled players in Group 2 attended. Calculate the two sample proportions and their Group 1-minus-Group 2 difference.
  3. A sample proportion is \(0.42\) in Group 1 and \(0.50\) in Group 2. What is the Group 1-minus-Group 2 point estimate, and what does its sign indicate?
  4. Explain why subtracting the numbers of successes is not generally a valid way to estimate the difference between two population proportions.
  5. If the point estimate for Group 1 minus Group 2 is \(0.08\), express it in percentage points and write one sentence interpreting it in context.

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