Statistical inference

From samples to populations

We rarely know population values (parameters). Statistical inference uses sample statistics to estimate them, and tells us how confident to be.

Two main tools:

1. Estimation with confidence intervals
2. Hypothesis testing

Point estimates

A point estimate is a single best guess:

• sample mean x̄ estimates the population mean μ
• sample proportion p̂ estimates the population proportion p

Example: In a random sample of 400 voters, 228 support a candidate. p̂ = 228/400 = 0.57.

Sampling variability and sampling distributions

Different samples give different estimates. If you took many samples and plotted the statistics, you’d get the sampling distribution.

Key facts:

• It’s centred on the true parameter (if sampling is random).
• Its spread is the standard error: smaller for larger samples.
• For a mean: SE = σ/√n.
• For a proportion: SE = √[p(1 − p)/n].

The central limit theorem

The central limit theorem says: for large enough samples (often n ≥ 30), the sampling distribution of the sample mean is approximately normal, even if the population isn’t.

This is why the normal distribution is so useful for inference.

Margin of error

A poll might report "57%, accurate to within ±4.9 percentage points, 19 times out of 20."

• The margin of error (≈ 2 × SE for 95% confidence) shows the likely range.
• "19 times out of 20" means 95% confidence.

Example: p̂ = 0.57, n = 400

• SE ≈ √(0.57 × 0.43/400) ≈ 0.0248
• Margin ≈ 1.96 × 0.0248 ≈ 0.049
• So 52.1% to 61.9%

Sample size matters

To halve the margin of error, you need four times the sample size (because of √n).

Conditions for valid inference

• The sample must be random.
• Observations should be independent.
• The sample must be large enough for the normal approximation.

Bias isn’t fixed by inference methods: a biased sample gives a confident wrong answer.

Worked examples

Example 1: A sample of 50 batteries has mean life 38 h. Point estimate of μ? 38 h.

Example 2: σ = 12, n = 36. Standard error of the mean? 12/6 = 2.

Example 3: A poll of 1,000 has a margin of about ±3%. To get ±1.5%, about how many people? 4,000.

Common mistakes

• Treating a point estimate as exact.
• Thinking a bigger population needs a much bigger sample. The sample size matters far more than the population size.
• Ignoring bias.

Try it yourself

1. 135 of 300 students prefer online learning. Point estimate p̂?
2. σ = 20, n = 100. Standard error?
3. What does "19 times out of 20" mean in a poll?
4. Why does the central limit theorem matter?

Answers: 1) 0.45 2) 2 3) 95% confidence 4) It lets us use the normal distribution for sample means even when the population isn’t normal.

Quick recap

• Statistical inference generalizes from samples to populations.
• Point estimates have uncertainty measured by the standard error and margin of error.
• The central limit theorem makes normal methods work for large samples.