# Estimators

> AP Statistics · AP 2024-2027 Statistics
> Source: https://www.owlsprep.com/study/ap-statistics-u12-estimators/

This module covers core properties of point estimators, how to calculate estimator bias, and how to select optimal estimators for population proportions, aligned to AP categorical inference requirements.

**Prerequisites:** [Understanding of sampling distributions for sample proportions](https://www.owlsprep.com/study/ap-statistics-u12-sampling-distributions-proportions/); [Basic distinction between population parameters and sample statistics](https://www.owlsprep.com/study/ap-statistics-u11-intro-to-inference/)

## Learning objectives

- Define point estimators and distinguish them from fixed population parameters
- Verify if an estimator is unbiased by calculating its expected value
- Compare estimators using properties of variance and consistency
- Select the optimal estimator for population proportion in AP exam contexts

## Introduction to Point Estimators

A point estimator is any sample statistic used to approximate an unknown fixed population parameter. For inference for proportions, the most common estimator is the sample proportion $\hat{p}$, which estimates the true population proportion $p$. A single calculated value of an estimator from one specific sample is called a point estimate.

**Point Estimator** — A function of sample data that produces a single value intended to estimate an unknown fixed population parameter $\theta$

*Notation:* $\hat{\theta}$ (general form)

*Example:* Sample proportion $\hat{p} = \frac{\text{number of successes in sample}}{\text{sample size } n}$ is the standard estimator for population proportion $p$.

**Worked example:** Identify which of the following qualify as estimators for population proportion p: (A) Fixed sample size n, (B) Raw count of successes X in sample, (C) Sample proportion X/n

1. First, eliminate any values that are not functions of random sample outcomes designed to estimate p
2. Sample size n is fixed by study design, not a function of sample results, so it is not an estimator
3. Raw count X does not normalize by sample size, so it cannot estimate p directly, so it is not an estimator
4. Sample proportion X/n is a function of sample data designed to approximate p, so it is the only valid estimator here

**Check your understanding**

Test your basic understanding of estimator definitions

1. Which of the following is a valid estimator for population mean $\mu$?

   - A) Population standard deviation $\sigma$
   - B) Sample mean $\bar{x}$
   - C) Fixed sample size n

   *Why:* Sample mean is calculated from sample data to estimate the unknown population mean, making it a valid point estimator.

## Unbiasedness of Estimators

An estimator is defined as unbiased if the expected value of the sampling distribution of the estimator equals the true value of the population parameter it is estimating. This means that the average value of the estimator across all possible random samples of the same size will exactly match the population parameter, with no systematic over- or under-estimation.

$$E[\hat{\theta}] = \theta$$

**Derivation:** Prove that sample proportion $\hat{p}$ is an unbiased estimator of population proportion p

*Starting from:* We know the count of successes X in a binomial sample has expectation E[X] = np

1. The sample proportion is defined as $\hat{p} = X/n$
2. Apply linearity of expectation: $E[\hat{p}] = E[X/n] = \frac{1}{n} E[X]$
3. Substitute E[X] = np: $E[\hat{p}] = \frac{1}{n} \times np = p$

*Conclusion:* The expected value of $\hat{p}$ is exactly equal to p, so sample proportion is an unbiased estimator of population proportion.

**Worked example:** A student proposes a new estimator for population proportion p: $\hat{p}_{adj} = \frac{X+1}{n+2}$. If the true p is 0.3 and n=10, calculate the bias of this new estimator.

1. Bias is defined as $E[\hat{\theta}] - \theta$, so first calculate $E[\hat{p}_{adj}]$
2. $E[\hat{p}_{adj}] = E\left[\frac{X+1}{n+2}\right] = \frac{E[X] +1}{n+2} = \frac{np +1}{n+2}$
3. Substitute n=10, p=0.3: $E[\hat{p}_{adj}] = \frac{10*0.3 +1}{10+2} = \frac{4}{12} \approx 0.333$
4. Calculate bias: $0.333 - 0.3 = 0.033$, so the estimator has a positive bias of ~0.033

**Exam command terms**

AP exam questions use specific command terms for estimator properties that you must follow exactly:

- **Show that the estimator is unbiased** — You must explicitly calculate the expected value of the estimator and prove it equals the target parameter to earn full points

- **Calculate the bias of this estimator** — You do not need to prove unbiasedness, just compute the difference between expected value and true parameter

## Minimum Variance and Consistency

Unbiasedness alone is not enough to make an estimator optimal. You also want the estimator to have low sampling variability, measured by the variance of its sampling distribution. An estimator that has the smallest possible variance among all unbiased estimators for a parameter is called a minimum variance unbiased estimator (MVUE).

> **tip**
>
> For population proportion p, the standard sample proportion $\hat{p}$ is the MVUE, so it is always the default estimator you should use for AP inference procedures unless explicitly told otherwise.

An estimator is consistent if as the sample size n grows to infinity, the probability that the estimator is arbitrarily close to the true parameter approaches 1. All unbiased estimators for proportions are consistent, as increasing sample size reduces their standard error.

**Worked example:** Two unbiased estimators for p are proposed: Estimator A has variance $\frac{p(1-p)}{n}$, Estimator B has variance $\frac{2p(1-p)}{n}$. Which estimator is better, and why?

1. Both estimators are unbiased, so we compare their variances to select the optimal one
2. Estimator A has half the variance of Estimator B for any fixed sample size n
3. This means Estimator A will produce estimates that are, on average, closer to the true p than Estimator B
4. Therefore Estimator A is the better minimum variance unbiased estimator

## Estimator Selection for Proportion Inference

All confidence interval and hypothesis test procedures for population proportion on the AP exam use the sample proportion $\hat{p}$ as the point estimator. You will never be asked to use an alternative estimator for standard inference questions, but you may be tested on why $\hat{p}$ is chosen over other possible estimators.

> **warning**
>
> A common mistake is using the hypothesized p0 value to calculate the point estimate for a confidence interval, or using $\hat{p}$ to calculate standard error for a proportion hypothesis test. These values are not interchangeable.

**Worked example:** A researcher surveys 200 randomly selected US adults and finds 120 support a new public health policy. What is the correct point estimator for the true population proportion of support?

1. Identify the number of successes X = 120, sample size n = 200
2. Calculate the sample proportion: $\hat{p} = 120 / 200 = 0.6$
3. This value is the unbiased minimum variance estimator for the true population proportion of support

## Common pitfalls

- **Wrong:** Claiming all sample statistics are unbiased estimators
  - Why it fails: Many common statistics (e.g., sample range, sample standard deviation s) are biased estimators for their corresponding population parameters
  - Correct: Only confirm an estimator is unbiased if you can prove its expected value equals the target parameter
- **Wrong:** Prioritizing low variance over unbiasedness when selecting an estimator
  - Why it fails: A biased estimator with very low variance will systematically produce wrong estimates, even if they are consistent
  - Correct: First confirm the estimator is unbiased, then select the one with the smallest variance
- **Wrong:** Confusing a point estimate (single calculated value from a sample) with a point estimator (the general formula)
  - Why it fails: AP graders will deduct points if you refer to a specific 0.6 sample value as the estimator, rather than the formula X/n
  - Correct: Label the formula as the estimator, and the single calculated number as the point estimate
- **Wrong:** Calculating bias as the difference between your sample point estimate and the true parameter
  - Why it fails: Bias is a property of the estimator's sampling distribution, not a single sample result
  - Correct: Bias is defined as the difference between the expected value of the estimator across all samples and the true parameter

## Cheatsheet

| Property | Definition | Requirement for $\hat{p}$ (estimator of p) |
| --- | --- | --- |
| Unbiasedness | $E[\hat{\theta}] = \theta$ | Satisfied, $E[\hat{p}] = p$ |
| Minimum Variance | Smallest variance among all unbiased estimators | Satisfied for simple random samples |
| Consistency | Estimator converges to $\theta$ as $n \to \infty$ | Satisfied, standard error shrinks as n increases |
| Bias | $E[\hat{\theta}] - \theta$ | Bias of $\hat{p}$ = 0 |

## What's next

Mastering the properties of estimators is the critical foundation you need to build for all subsequent inference for proportions topics. The unbiasedness of $\hat{p}$ is the core justification for why one-proportion z-confidence intervals and z-hypothesis tests produce valid, reliable results that do not systematically over- or under-estimate the true population proportion. You will see estimator properties tested directly in 1-2 multiple choice questions on almost every AP exam, and they are frequently required as justification steps in free response inference questions to earn full reasoning points. Next, you will apply these estimator properties to calculate the standard error of the sample proportion, then move on to constructing formal confidence intervals for population proportions.

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