Estimation, Confidence Intervals and Tests using the Normal Distribution

• Statistics & estimators: using sample data to estimate unknown population parameters (mean, variance), and what makes an estimator "unbiased."
• Combining samples: merging two samples' data to get a single, better estimate.
• Standard error of the mean: how "wobbly" your sample mean is as an estimate — and why bigger samples wobble less.
• Confidence intervals: building a range of plausible values for the true population mean, at a chosen confidence level (e.g. 95%, 99%).
• Central Limit Theorem (CLT): the reason all of this works — sample means become normally distributed even if the original population isn't.
• Hypothesis testing for the mean: using a sample to test whether a claim about the population mean is credible.

The apple problem

Imagine you want to know the true average weight of every apple in a massive orchard — millions of apples. You can't weigh them all. So instead, you grab a random sample of, say, 10 apples, weigh them, and use that to estimate the true average.

This is the entire chapter in one sentence: we use small, manageable samples to make honest, mathematically justified guesses about huge, unknowable populations.

Key vocabulary — read this slowly, it matters

• Population parameter — the real, true value for the whole population (e.g. the true mean weight μ, or true variance σ² of all apples). We almost never know this. It's the thing we're trying to estimate.
• Random sample — a set of observations X₁, X₂, X₃, ... Xₙ taken independently from the population. Each Xᵢ is itself a random variable and follows the same distribution as the population variable X.
• Statistic — any function of the sample data that does NOT contain an unknown parameter. For example, the sample mean x̄ is a statistic — you can calculate it purely from your data. But something like "x̄ − μ" is NOT a statistic, because μ is unknown.
• Estimator — a statistic that is being used specifically to estimate a population parameter. The sample mean x̄ is an estimator for μ. Since a statistic is a random variable (it changes every time you take a new sample), it has its own probability distribution — this is called the sampling distribution.
• Unbiased estimator — an estimator T is unbiased for parameter θ if E(T) = θ. In plain English: if you repeated the sampling process infinitely many times and averaged all your estimates together, you'd land exactly on the true value. No systematic over- or under-estimating.
• Bias — when E(T) ≠ θ, the estimator is biased, and the bias is the difference E(T) − θ.

The formulas you need