Hypothesis Testing and Contingency Tables (AQA A-Level Further Maths): Revision Notes

Confidence Intervals

Introduction

When we need to find the mean or variance of an entire population, measuring every individual is usually impractical or impossible. Instead, we take a sample from the population and calculate the sample mean and sample variance. These sample statistics help us estimate the corresponding population parameters.

A confidence interval provides a range of values within which we expect the true population mean to lie, with a specified level of confidence.

Sample statistics

Sample mean

For a set of $n$ data values $\{x_i\}$, the sample mean is:

$\bar{x} = \frac{\sum x_i}{n}$

The sample mean is calculated by adding all data values and dividing by the number of observations.

Sample variance

The sample variance is:

$s^2 = \frac{1}{n-1}\sum(x_i - \bar{x})^2$

Unbiased estimators

An unbiased estimator is a statistic whose expected value equals the population parameter it estimates. Both the sample mean $\bar{x}$ and sample variance $s^2$ are unbiased estimators of the population mean $\mu$ and population variance $\sigma^2$ respectively.

However, being unbiased does not guarantee the estimate will be close to the true parameter value, especially for small samples. The estimate improves as sample size increases.

Distribution of the sample mean

For large samples drawn from a population with mean $\mu$ and variance $\sigma^2$, the distribution of the sample mean follows a normal distribution:

$\bar{X} \sim N\left(\mu, \frac{\sigma^2}{n}\right)$

This means:

• The expected value of the sample mean equals the population mean $\mu$
• The variance of the sample mean is $\frac{\sigma^2}{n}$ (smaller than the population variance)
• As sample size $n$ increases, the sample mean becomes more concentrated around $\mu$

The accuracy of the sample mean as an estimate of the population mean depends on the population variance, which is often unknown.

What is a confidence interval?

A $p\%$-confidence interval is a range of values generated from sample data. Before the sample is taken, we expect the population mean $\mu$ to fall within this interval with probability $p\%$.

Key point: A $p\%$-confidence interval means that if we repeatedly took samples and calculated confidence intervals, we would expect approximately p% of those intervals to contain the true population mean.

Calculating confidence intervals when variance is estimated

When the population variance is unknown and must be estimated from the sample, the $p\%$-confidence interval for the population mean $\mu$ is:

$\bar{x} - z \times \frac{s}{\sqrt{n}} < \mu < \bar{x} + z \times \frac{s}{\sqrt{n}}$

where:

• $\bar{x}$ is the sample mean
• $s$ is the sample standard deviation (square root of sample variance)
• $n$ is the sample size
• $z$ is the critical value from the standard normal distribution

Standard error

The standard error is the standard deviation of the sample mean:

$\text{Standard error} = \frac{s}{\sqrt{n}}$

The standard error measures how much the sample mean varies from sample to sample. A smaller standard error indicates more precise estimates.

Finding the z-value

The critical value $z$ depends on the confidence level $p$. For common confidence levels:

Confidence level ($p$)	z-value
90%	1.645
95%	1.96
98%	2.326
99%	2.576

These values can be calculated using the inverse normal function:

$z = \Phi^{-1}\left(\frac{1+p}{2}\right)$

where $\Phi^{-1}$ is the inverse of the standard normal cumulative distribution function. Most calculators have this function built in.

Interpretation

The confidence interval tells us that we can be $p\%$ confident the true population mean lies between the lower and upper bounds. The interval width increases as:

• The confidence level $p$ increases (we need more certainty)
• The sample variance $s^2$ increases (more variable data)
• The sample size $n$ decreases (less information)

Calculating confidence intervals when population variance is known

When the population variance $\sigma^2$ is known (rare in practice), we can use it directly without needing to estimate it from the sample. The $p\%$-confidence interval becomes:

$\bar{x} - z \times \frac{\sigma}{\sqrt{n}} < \mu < \bar{x} + z \times \frac{\sigma}{\sqrt{n}}$

This formula uses the known population standard deviation $\sigma$ instead of the sample standard deviation $s$.

Worked examples

Exam tips and common traps

Sample size and interval width

Increasing the sample size decreases the width of the confidence interval because the standard error contains $\sqrt{n}$ in the denominator. A larger sample provides a more accurate estimate of the population mean.

When to use each formula

• Use sample variance ($s^2$): When the population variance is unknown (most common situation)
• Use population variance ($\sigma^2$): When explicitly stated that the population variance is known

Confidence level interpretation

Representative samples

Always ensure your sample is representative of the population to avoid bias. A biased sample (such as measuring only basketball players when estimating average human height) will produce misleading confidence intervals, even if calculated correctly.

Trade-off between confidence and precision

You can never be certain that your interval contains the population mean. The more confident you want to be, the larger the interval becomes ($z$ increases as $p$ increases). This represents a fundamental trade-off between confidence and precision.

Checking your work

When calculating confidence intervals:

1. Identify whether $\sigma$ or $s$ should be used
2. Calculate the standard error correctly
3. Find the appropriate $z$-value for the confidence level
4. Apply the formula systematically
5. Give your final answer to an appropriate degree of accuracy (usually 3 significant figures)

Remember!