# Statistical Inference

> CIE A-Level Further Mathematics · Further Probability & Statistics Unit 4
> Source: https://www.owlsprep.com/study/cie-9231-u4-statistical-inference/

Statistical inference uses sample data to draw evidence-based conclusions about unknown population parameters. This module covers core concepts of unbiased estimation, confidence interval construction, and correct interpretation of results.

**Prerequisites:** [Knowledge of sampling distributions](https://www.owlsprep.com/study/cie-9231-u4-sampling-distributions/); [Understanding of normal and t-distributions](https://www.owlsprep.com/study/cie-9231-u4-continuous-probability-distributions/)

## Learning objectives

- Distinguish between population parameters and sample statistics
- Identify and calculate unbiased estimators for population parameters
- Construct confidence intervals for means and proportions under different conditions
- Correctly interpret the results of statistical inference in context

## Core Inference Concepts & Terminology

**Statistical Inference** — The process of drawing generalised conclusions about an entire population, using only data collected from a smaller random sample of that population.

*Example:* Estimating the average exam score of all 11th graders in England from a sample of 500 11th graders

**Parameter vs Statistic** — A parameter is a fixed, unknown number that describes the whole population. A statistic is a known number calculated from sample data, used to estimate the unknown parameter.

*Example:* True average height of all adult men (parameter) vs average height of 100 sampled men (statistic)

**Worked example:** A bakery produces 2000 loaves of bread daily. A sample of 40 loaves has an average mass of 402.1 g. Identify the parameter and statistic in this scenario.

1. The parameter of interest is the true average mass of all 2000 loaves produced that day. This is unknown and fixed for the entire population.
2. The statistic is the sample average mass of 402.1 g, calculated from the 40 sampled loaves. This is known and used to estimate the unknown parameter.

## Unbiased Estimators

**Unbiased Estimator** — A statistic $\hat{\theta}$ is an unbiased estimator of the population parameter $\theta$ if its expected value equals the true parameter: $E(\hat{\theta}) = \theta$.

*Example:* The sample mean $\bar{X}$ is always an unbiased estimator of the population mean $\mu$

> **warning**
>
> Unbiasedness does not mean that a single sample estimate will equal the true parameter. It only means that the average of all possible sample estimates will equal the true parameter.

**Worked example:** Show that $s^2 = \frac{1}{n-1}\sum_{i=1}^n (X_i - \bar{X})^2$ is an unbiased estimator of population variance $\sigma^2$.

1. Expand the expectation of the sum of squared deviations:
2. $$E\left[\sum (X_i - \bar{X})^2\right] = E\left[\sum ((X_i - \mu) - (\bar{X} - \mu))^2\right]$$
3. Simplify the expression using $E(X_i - \mu)^2 = \sigma^2$ and $E(\bar{X} - \mu)^2 = \frac{\sigma^2}{n}$:
4. $$\sum E(X_i - \mu)^2 - n E(\bar{X} - \mu)^2 = n\sigma^2 - n \cdot \frac{\sigma^2}{n} = (n-1)\sigma^2$$
5. Rearrange to find the expected value of $s^2$:
6. $$E(s^2) = \frac{1}{n-1} (n-1)\sigma^2 = \sigma^2$$
7. This confirms $s^2$ is an unbiased estimator of $\sigma^2$.

*Calculator:* allowed

## Confidence Interval Construction

**Confidence Interval** — A range of values constructed from sample data that is designed to capture the true unknown population parameter, with a specified confidence level (e.g. 95%).

*Example:* A 95% confidence interval for mean battery life: (11.5 hours, 13.5 hours)

1. Population mean, $\sigma^2$ known: $ \bar{x} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}} $
2. Population mean, $\sigma^2$ unknown, large $n$: $ \bar{x} \pm z_{\alpha/2} \frac{s}{\sqrt{n}} $
3. Population mean, $\sigma^2$ unknown, small $n$: $ \bar{x} \pm t_{n-1, \alpha/2} \frac{s}{\sqrt{n}} $
4. Population proportion, large $n$: $ \hat{p} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} $

**Worked example:** A sample of 25 batteries has an average lifetime of 12.5 hours, with sample standard deviation 2.4 hours. Construct a 95% confidence interval for the true mean lifetime.

1. Identify the case: $\sigma^2$ is unknown, sample size $n=25 < 30$, so use t-distribution with $df = 25-1 = 24$.
2. Find the 95% two-tailed critical value: $t_{24, 0.025} = 2.064$.
3. Calculate the standard error: $ \frac{s}{\sqrt{n}} = \frac{2.4}{\sqrt{25}} = 0.48 $.
4. Calculate the margin of error: $ 2.064 \times 0.48 = 0.99 $.
5. Final 95% confidence interval: $ 12.5 \pm 0.99 = (11.51, 13.49) $ hours.

*Calculator:* allowed

## Interpreting Inference Results

**Exam command terms**

CIE exams require specific wording for interpretation questions, as shown below:

- **Construct a 95% confidence interval** — Calculate lower and upper bounds, show all working steps *(Round final answers to 2-3 decimal places unless stated)*

- **Interpret your confidence interval** — Explain the meaning of the confidence level in context *(We are 95% confident that the true mean lifetime lies between 11.5 and 13.5 hours)*

> **warning**
>
> Do NOT say 'there is a 95% probability the true mean is in the interval'. In frequentist inference, the true mean is fixed, so the interval either contains it or it does not. The 95% refers to the method of constructing intervals, not a single interval.

**Check your understanding**

Test your understanding of interpretation:

1. Which of the following is the correct interpretation of a 95% confidence interval?

   - 95% of the population data lies in the interval
   - There is a 95% probability the true parameter is in the interval
   - 95% of intervals constructed this way will contain the true parameter
   - The sample statistic is in the interval 95% of the time

   *Why:* Correct: The confidence level describes the long-run performance of the interval construction method, not a single interval or the parameter.

## Common pitfalls

- **Wrong:** Interpreting a 95% confidence interval as '95% of the sample data lies in the interval'
  - Why it fails: This confuses the spread of sample data with the range of uncertainty for the unknown population parameter
  - Correct: The 95% confidence level describes the probability that the interval construction method will produce an interval containing the true parameter
- **Wrong:** Using a z-critical value instead of a t-critical value for small samples with unknown variance
  - Why it fails: The t-distribution accounts for extra uncertainty from estimating population variance, which the z-distribution does not
  - Correct: Always use the t-distribution with $n-1$ degrees of freedom for small samples with unknown population variance
- **Wrong:** Dividing the sum of squared deviations by $n$ to get an unbiased sample variance
  - Why it fails: Dividing by $n$ produces a biased downward estimate of population variance, leading to incorrect intervals
  - Correct: Divide by $n-1$ to get an unbiased sample variance for inference
- **Wrong:** Claiming that an unbiased estimator will always give a value equal to the true parameter
  - Why it fails: Unbiasedness describes the expected value over all possible samples, not any single sample
  - Correct: A single sample estimate will almost always differ from the true parameter, so we use interval estimates to account for uncertainty
- **Wrong:** Using the normal approximation for a proportion confidence interval when $n\hat{p} < 5$ or $n(1-\hat{p}) < 5$
  - Why it fails: The normal approximation is invalid for very small expected counts, leading to inaccurate intervals
  - Correct: Always check that both $n\hat{p}$ and $n(1-\hat{p})$ are at least 5 before using the normal approximation

## Cheatsheet

| Parameter | Conditions | Confidence Interval Formula |
| --- | --- | --- |
| Population mean $\mu$ | $\sigma^2$ known, any $n$ | $\bar{x} \pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}$ |
| Population mean $\mu$ | $\sigma^2$ unknown, large $n$ | $\bar{x} \pm z_{\alpha/2}\frac{s}{\sqrt{n}}$ |
| Population mean $\mu$ | $\sigma^2$ unknown, small $n$ | $\bar{x} \pm t_{n-1, \alpha/2}\frac{s}{\sqrt{n}}$ |
| Population proportion $p$ | Large $n$, $n\hat{p}, n(1-\hat{p}) \geq 5$ | $\hat{p} \pm z_{\alpha/2}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$ |

## What's next

Statistical inference is the foundation for hypothesis testing, the next core topic in CIE A-Level Further Probability & Statistics. The concepts of unbiased estimators, sampling variation, and interval estimation you learned here underpin all hypothesis testing methods you will encounter. They help you understand why hypothesis tests work, how p-values quantify evidence against a claim, and how to draw valid conclusions about unknown population parameters. After mastering the basics of inference, you will be ready to tackle one- and two-sample tests for means, proportions, and differences between population parameters, which make up a large proportion of marks on 9231 papers. Correct interpretation of inference also helps you avoid common costly exam mistakes.

- [Generating Functions](https://www.owlsprep.com/study/cie-9231-u4-generating-functions/)
- [Non-parametric tests](https://www.owlsprep.com/study/cie-9231-u4-non-parametric-tests/)

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