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3.9 Differences Between Sample Proportions

A Difference Has an Order

Suppose 60% of the 15,000 members at library A and 50% of the 12,000 members at library B have borrowed audiobooks. These are known population proportions in our teaching model. Take independent SRSs of 150 members from A and 200 from B, without replacement within each library. Define

D=p^A−p^B.D=\hat p_A-\hat p_B.

A positive D means the sample audiobook proportion is higher at A. A negative D means it is higher at B. Reversing the order changes the sign of the mean and the requested event; it does not change the standard deviation. Label the two populations before entering any numbers.

For two independent populations and independently sampled values, the mean and SD are

μp^1−p^2=p1−p2,σp^1−p^2=p1(1−p1)n1+p2(1−p2)n2.\mu_{\hat p_1-\hat p_2}=p_1-p_2,\qquad \sigma_{\hat p_1-\hat p_2}=\sqrt{\frac{p_1(1-p_1)}{n_1}+\frac{p_2(1-p_2)}{n_2}}.

Subtract the means, but add the two variance contributions inside the square root. Differences fluctuate because both sample proportions fluctuate. Subtracting the two SDs would confuse direction with variability. With small sampling fractions, we use the usual approximation to independence for sampling without replacement; the stated SD formula omits a finite-population correction.

Verify Both Samples and All Four Counts

The sampling design must provide two independent random samples. Using the same people twice would require a different model; labeling the groups “A” and “B” alone does not establish independence. For sampling without replacement, check each sample against its own population:

n1≤0.10N1,n2≤0.10N2.n_1\leq 0.10N_1,\qquad n_2\leq 0.10N_2.

Here 150≤1,500150\leq 1{,}500 and 200≤1,200200\leq 1{,}200. An approximately normal difference also requires four expected counts, not only a large combined sample size:

n1p1≥10,n1(1−p1)≥10,n2p2≥10,n2(1−p2)≥10.n_1p_1\geq 10,\quad n_1(1-p_1)\geq 10,\quad n_2p_2\geq 10,\quad n_2(1-p_2)\geq 10.
PopulationExpected audiobook borrowersExpected nonborrowers
Library A, n = 150 and p = 0.609060
Library B, n = 200 and p = 0.50100100

All four are at least 10, so a normal approximation for D is appropriate. If one population has a rare outcome, even a substantial combined sample can leave that group’s expected success count below 10. Use each true population proportion for this theoretical distribution rather than silently substituting observed sample proportions.

Random Assignment Is a Different Route

In an experiment, randomly assigning experimental units to two treatments supplies the randomization condition. The sampling-without-replacement 10% checks are not required merely because units were assigned to treatments. Random assignment helps compare treatments fairly; it does not make a convenience group representative of a wider population.

When a probability model for binary treatment responses is supplied, inspect its two expected success and two expected failure counts before using a normal approximation. Random assignment addresses how treatment groups are formed; it does not automatically make rare-event distributions normal. Keep the experimental setting distinct from drawing two population samples.

Worked Example: Could the Observed Order Reverse?

For the two libraries,

μD=0.60−0.50=0.10,σD≈0.60(0.40)150+0.50(0.50)200=0.05339.\mu_D=0.60-0.50=0.10,\qquad \sigma_D\approx\sqrt{\frac{0.60(0.40)}{150}+\frac{0.50(0.50)}{200}}=0.05339.

The difference has mean 0.10, or 10 percentage points, in the A-minus-B order. Its typical sample-to-sample deviation from 0.10 is about 5.34 percentage points. To estimate the chance that the two random samples show A’s proportion no higher than B’s, use the event D at most 0:

P(D≤0)≈P(Z≤0−0.100.05339)=P(Z≤−1.8732)≈0.0305.P(D\leq 0)\approx P\left(Z\leq\frac{0-0.10}{0.05339}\right)=P(Z\leq -1.8732)\approx 0.0305.

About 3.1% of repeated pairs of these independent samples would show A’s sample proportion at most B’s, even though A’s population proportion is 10 percentage points higher. This conclusion keeps the same subtraction order, sample sizes and area direction as the calculation. It describes chance variation in paired sample results, not a change in the fixed population proportions.

Try It Yourself

The following original practice applies the ideas above to a fresh setting. Show the reasoning requested in each part. Answers and task-specific rubrics are on the separate solutions page.

Original generated practice for the Fall 2026 course.

Two Towns, Two Sampling Designs

Town A has 10,000 households and a known population recycling proportion p_A=0.55. Town B has 15,000 households and known p_B=0.45. Take independent SRSs of 200 distinct A households and 300 distinct B households without replacement. Let D=sample proportion in A minus sample proportion in B. Use the usual 10% approximation and a normal approximation without a continuity correction where justified.

Part a

Calculate the mean and SD of D. Show both variance contributions and explain why the subtraction order affects the mean but not which contributions are added under the square root.

Part b

Justify the original difference sampling model. Verify two independent random samples, both numerical 10% checks and all four expected-success/failure counts, then state the normal-model conclusion.

Part c

If Town B instead had known p_B=0.01 with the same n_B=300, which normality condition would fail? Would increasing only Town A’s sample fix that failure? Separately, explain why asking the same 200 households about two different months would not be two independent population samples.

Part d

A separate experiment has 200 volunteer households randomly assigned, 100 to message A and 100 to message B. The supplied independent-response model has recycling probabilities 0.55 under message A and 0.45 under message B. Explain which randomization requirement is satisfied and whether two population SRSs or a 10% population-size check are required for this assignment comparison. Calculate all four expected counts and evaluate the normal approximation. Explain the limit on generalizing to all town households.

Part e

For the original town SRSs, estimate P(D>0.15)P(D>0.15). Show the standardized cutoff and area direction. Interpret D’s mean, SD and this probability in terms of the two populations and their specified repeated sample sizes.

Worked answer and rubric

Glossary and deeper reading · 14 refreshers

These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.

  • Population proportion

    A population proportion is the fraction of the entire population with a specified characteristic. It is a parameter, commonly written p, that a sample proportion can estimate. For example, the fraction of all enrolled students who walk to school is the population proportion of walkers.

    Read more: Population proportion in the captured Statistics reading →

  • Sampling without replacement

    Sampling without replacement does not return selected items before later selections. Selections from a finite population are dependent, although treating them as approximately independent can be reasonable when the sample is a small fraction of the population. For example, selecting students without replacement ensures that no student is chosen twice.

    Read more: Sampling without replacement in the captured Statistics reading →

  • Sample

    A sample is the subset of a population actually studied. How it is selected matters for the conclusions that can be drawn about the population. For example, 60 selected students may be a sample from a school’s full enrollment.

    Read more: Sample in the captured Statistics reading →

  • Mean

    The mean is a measure of center often called an average. For a finite data set, the arithmetic mean is the sum of the values divided by their number; for a random variable, the mean is its probability-weighted expected value when that value exists. Means can be pulled toward extreme values and are not generally resistant. For example, the arithmetic mean of 2, 4, and 9 is 5.

    Read more: Mean in the captured Statistics reading →

  • Population

    A population is the full set of individuals or measurements a study aims to describe. It must be defined clearly before deciding whether a sample supports generalization. For example, all students enrolled in a particular school this year form a population.

    Read more: Population in the captured Statistics reading →

  • Variance

    Variance is the square of the standard deviation. It measures spread in squared units, with the precise averaging rule depending on whether it describes a sample or a probability distribution. For example, a standard deviation of 3 centimeters corresponds to a variance of 9 square centimeters.

    Read more: Variance in the captured Statistics reading →

  • Sample proportion

    A sample proportion is the number of sampled individuals with a specified characteristic divided by the sample size. It is commonly written p-hat and estimates the corresponding population proportion. For example, 18 walkers among 60 sampled students give a sample proportion of 0.30.

    Read more: Sample proportion in the captured Statistics reading →

  • Statistical independence

    Statistical independence means that learning one outcome does not change the probability distribution of another. For two events, it implies that the probability of both occurring equals the product of their probabilities. For example, independent draws with replacement keep the second draw’s color probabilities unchanged by the first.

    Read more: Statistical independence in the captured Statistics reading →

  • Random sampling

    Random sampling uses a specified chance process to select individuals from a population. It supports population inference when the sampling frame and data collection are appropriate, but does not guarantee perfect balance or eliminate every source of bias. For example, a random-number generator can select student identifiers from an enrollment list.

    Read more: Random sampling in the captured Statistics reading →

  • Expected count

    An expected count is the count predicted for a category or cell under the null model. Goodness-of-fit expectations use the total count times the null proportion, while independence-table expectations use row total times column total divided by the grand total. For example, a null proportion of 0.2 in 50 trials gives an expected count of 10.

    Read more: Expected count in the current two-way chi-square lesson →

  • Experiment

    An experiment deliberately imposes treatments and measures responses. Random assignment, comparison, and replication help distinguish treatment effects from other variation. For example, assigning plants to fertilizer treatments is an experiment.

    Read more: Experiment in the captured Statistics reading →

  • Random assignment

    Random assignment uses chance to allocate experimental units to treatment conditions. It tends to balance other influences across groups and supports causal inference in a well-conducted experiment, without guaranteeing identical groups. For example, shuffled labels can assign plants to two fertilizer treatments.

    Read more: Random assignment in the captured Statistics reading →

  • Probability model

    A probability model specifies possible outcomes and probabilities for events formed from those outcomes. Probabilities are nonnegative, and the probability of the whole sample space is 1. In a discrete model, individual outcome probabilities sum to 1; in a continuous density model, total area is 1 even though each single value has probability 0. For example, a fair four-sided die can be modeled with four outcomes each assigned probability one quarter.

    Read more: Probability model in the captured Statistics reading →

  • Treatment

    A treatment is a specific condition imposed on an experimental unit. In a multifactor experiment, a treatment is a combination of factor levels. For example, low fertilizer with eight hours of light is one treatment combination.

    Read more: Treatment in the captured Statistics reading →

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