Sampling, estimates and resampling

Welcome to Statistical Inference!

Welcome to one of the most exciting parts of your Statistics A Level! Think of Statistical Inference as being a data detective. In the real world, we rarely know everything about a whole group (the population). Instead, we take a small piece (a sample) and use it to make a "best guess" about the whole.

In this chapter, we are going to learn how to bridge the gap between a small sample and the big picture. Don't worry if it feels a bit abstract at first—we’ll break it down step-by-step!

1. The Basics: Parameters vs. Statistics

Before we can make inferences, we need to know the difference between the "whole" and the "part."

What is a Parameter?

A parameter is a numerical value that describes an entire Population. Because populations are usually huge, we rarely know the true value of a parameter.
• Example: The average height of every single teenager in the UK.
• Memory Aid: Parameter = Population.

What is a Statistic?

A statistic is a numerical value that comes from a Sample. We use statistics to estimate parameters. A statistic is a function only of the values in the sample and contains no unknown parameters.
• Example: The average height of 50 teenagers you measured at a local school.
• Memory Aid: Statistic = Sample.

Key Terms to Know

• Unbiased: We call a statistic "unbiased" if, on average, it equals the true population parameter we are trying to find. It’s like an archer who might not hit the bullseye every time, but their shots are centered perfectly around it.
• Standard Error: This is just a special name for the standard deviation of a sample statistic. It tells us how much we expect our sample "guess" to vary from sample to sample. The smaller the standard error, the more reliable our estimate is!

Quick Review Box: Symbols

• Population Mean (Parameter): \( \mu \)
• Population Variance (Parameter): \( \sigma^2 \)
• Sample Mean (Statistic): \( \bar{x} \)
• Sample Variance (Statistic): \( s^2 \)

Key Takeaway: We use Statistics (from our sample) to make an educated guess about Parameters (from the population).

2. The Central Limit Theorem (CLT)

This is arguably the most important rule in all of statistics! It’s like a "magic wand" that lets us use the Normal Distribution even when our data doesn't look normal at all.

What is the CLT?

The Central Limit Theorem states that if you take a large enough random sample (usually \(n \ge 30\)), the distribution of the sample mean (\(\bar{X}\)) will be approximately Normal, regardless of the shape of the original population.

Why is this amazing?

Imagine a population of data that is very "wonky"—maybe it's all bunched up on one side (skewed). If you pick one person, they will likely be part of that wonky distribution. However, if you pick 40 people and calculate their average, that average is very likely to be near the true center. If you did this over and over, those averages would form a beautiful, symmetrical Bell Curve (Normal Distribution).

The Formula

If the original population has a mean \( \mu \) and a variance \( \sigma^2 \), then the sample mean follows:
\( \bar{X} \sim N(\mu, \frac{\sigma^2}{n}) \)

Important Point: Notice that the variance of the sample mean is \( \frac{\sigma^2}{n} \). This means that as your sample size (\(n\)) gets bigger, the "spread" of your averages gets smaller. Your guess becomes much more precise!

When do we use it?

• You MUST use the CLT if the original population is NOT normally distributed.
• You DON'T need it if the original population is already Normal (because the sample mean of a normal population is always normal, even for small samples).
• Rule of thumb: Always check if your sample size \(n \ge 30\).

Did you know? Even if you are sampling "Yes/No" data (Binomial) or "Counting" data (Poisson), if your sample is large enough, the average results will still follow a Normal Distribution!

Common Mistake to Avoid

Don't confuse the distribution of the individual data points with the distribution of the sample mean. The CLT says the average becomes normal, not the original data itself!

Key Takeaway: If \(n \ge 30\), you can treat the sample mean as if it belongs to a Normal Distribution, allowing you to calculate probabilities and confidence intervals easily.

3. Summary and Tips for Success

In Paper 2, you will often be asked to justify why you can use a Normal Distribution. If the question tells you the population isn't normal, or doesn't mention the shape at all, your "saving grace" is usually the Central Limit Theorem.

Step-by-Step for Exam Questions:
1. Identify the population mean (\( \mu \)) and variance (\( \sigma^2 \)).
2. Check the sample size (\(n\)). Is it 30 or more?
3. State: "Since \(n\) is large, by the Central Limit Theorem, \( \bar{X} \approx N(\mu, \frac{\sigma^2}{n}) \)."
4. Use your calculator to find the probabilities required.

Don't worry if this seems tricky at first! The idea that "averages of wonky data become normal" is a bit mind-bending. Just remember: Big samples (\(n \ge 30\)) make life easier because they let us use the Normal Distribution tools we already know.

Key Takeaway: Parameters describe the world; Statistics describe our sample; the CLT is the bridge that connects the two when the sample is large enough!