Statistical Inference

CIE A-Level Further Mathematics· 7 min read

1. Core Inference Concepts & Terminology★★☆☆☆⏱ 15 min

📘 Definition

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

📘 Definition

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.

2. Unbiased Estimators★★★☆☆⏱ 20 min

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📘 Definition

Unbiased Estimator

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

Example:

The sample mean $\overset{ˉ}{X}$ is always an unbiased estimator of the population mean $μ$

📐 Worked Example

Show that $s^{2} = \frac{1}{n - 1} \sum_{i = 1}^{n} (X_{i} - \overset{ˉ}{X})^{2}$ is an unbiased estimator of population variance $σ^{2}$ .

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 $E (X_{i} - μ)^{2} = σ^{2}$ and $E (\overset{ˉ}{X} - μ)^{2} = \frac{σ ^{2}}{n}$ :
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 $σ^{2}$ .

3. Confidence Interval Construction★★★☆☆⏱ 25 min

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📘 Definition

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)

Population mean, $σ^{2}$ known: $\overset{x}{ˉ} \pm z_{α /2} \frac{σ}{n}$
Population mean, $σ^{2}$ unknown, large $n$ : $\overset{x}{ˉ} \pm z_{α /2} \frac{s}{n}$
Population mean, $σ^{2}$ unknown, small $n$ : $\overset{x}{ˉ} \pm t_{n - 1, α /2} \frac{s}{n}$
Population proportion, large $n$ : $\overset{p}{^} \pm z_{α /2} \frac{p ^ ( 1 - p ^ )}{n}$

📐 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: $σ^{2}$ 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: $\frac{s}{n} = \frac{2.4}{25} = 0.48$ .
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.

4. Interpreting Inference Results★★★★☆⏱ 15 min

✓ Quick check

Test your understanding of interpretation:

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
Reveal answer
95% of intervals constructed this way will contain the true parameter —
Correct: The confidence level describes the long-run performance of the interval construction method, not a single interval or the parameter.

5. Common Pitfalls

Wrong move:

Interpreting a 95% confidence interval as '95% of the sample data lies in the interval'

Why:

This confuses the spread of sample data with the range of uncertainty for the unknown population parameter

Correct move:

The 95% confidence level describes the probability that the interval construction method will produce an interval containing the true parameter

Wrong move:

Using a z-critical value instead of a t-critical value for small samples with unknown variance

Why:

The t-distribution accounts for extra uncertainty from estimating population variance, which the z-distribution does not

Correct move:

Always use the t-distribution with $n - 1$ degrees of freedom for small samples with unknown population variance

Wrong move:

Dividing the sum of squared deviations by $n$ to get an unbiased sample variance

Why:

Dividing by $n$ produces a biased downward estimate of population variance, leading to incorrect intervals

Correct move:

Divide by $n - 1$ to get an unbiased sample variance for inference

Wrong move:

Claiming that an unbiased estimator will always give a value equal to the true parameter

Why:

Unbiasedness describes the expected value over all possible samples, not any single sample

Correct move:

A single sample estimate will almost always differ from the true parameter, so we use interval estimates to account for uncertainty

Wrong move:

Using the normal approximation for a proportion confidence interval when $n \overset{p}{^} < 5$ or $n (1 - \overset{p}{^}) < 5$

Why:

The normal approximation is invalid for very small expected counts, leading to inaccurate intervals

Correct move:

Always check that both $n \overset{p}{^}$ and $n (1 - \overset{p}{^})$ are at least 5 before using the normal approximation

6. Quick Reference Cheatsheet

Parameter	Conditions	Confidence Interval Formula
Population mean $μ$	$σ^{2}$ known, any $n$	$\overset{x}{ˉ} \pm z_{α /2} \frac{σ}{n}$
Population mean $μ$	$σ^{2}$ unknown, large $n$	$\overset{x}{ˉ} \pm z_{α /2} \frac{s}{n}$
Population mean $μ$	$σ^{2}$ unknown, small $n$	$\overset{x}{ˉ} \pm t_{n - 1, α /2} \frac{s}{n}$
Population proportion $p$	Large $n$ , $n \overset{p}{^}, n (1 - \overset{p}{^}) \geq 5$	$\overset{p}{^} \pm z_{α /2} \frac{p ^ ( 1 - p ^ )}{n}$

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.