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

Combining two samples

Sometimes you're given summary statistics (mean & variance) for two separate samples and asked to combine them into one bigger, more reliable estimate. The trick: convert everything back into Σx and Σx² first, then merge, then recalculate.

Every time you take a new sample, you'll get a slightly different sample mean. The standard error of the mean measures how much these sample means typically vary — i.e., how much "wobble" there is in your estimate of μ.

The key intuition: a bigger sample gives you a more stable, trustworthy estimate. Averaging 1000 apples is far less likely to be thrown off by one weirdly heavy apple than averaging just 3. So as n increases, the standard error decreases — your estimator becomes more precise.

What a confidence interval actually means

A confidence interval is a range of values that we believe, with a stated level of confidence (e.g. 95%), contains the true population mean μ. It's built by taking your sample mean and adding/subtracting a "margin of error" on either side.

The formula

The value of z depends on how confident you want to be:

• 90% confidence → z = 1.6449
• 95% confidence → z = 1.96
• 99% confidence → z = 2.5758

The width of the interval (upper limit minus lower limit) is simply:

This is the quiet hero that makes everything else in this chapter legal. It says:

Why does this matter so much? Every confidence interval and every hypothesis test in this chapter relies on X̄ being normally distributed, because that's the only way we're allowed to use the standard normal z-tables. The CLT is the theoretical justification that lets us do that — even when we don't know or can't assume the original population itself is normal.

The core idea

Hypothesis testing answers the question: "is my sample result surprising enough, given what's claimed to be true, that I should doubt the claim?" You compare a claimed value of μ against what your sample actually shows.

The 4-step process, every time

1. State your hypotheses.
   • H₀ (null hypothesis): μ = [claimed value] — this is what we assume true unless the evidence says otherwise.
   • H₁ (alternative hypothesis): depends on the wording —
     • μ < value → one-tailed test (suspect the value is LOWER)
     • μ > value → one-tailed test (suspect the value is HIGHER)
     • μ ≠ value → two-tailed test (suspect it's just DIFFERENT, direction unknown)
2. Calculate the test statistic Z using the formula below.
3. Find the critical value(s) from the z-table for your chosen significance level, and identify the critical region.
4. Compare and conclude — if Z falls inside the critical region, reject H₀ (there IS evidence for the claim). If not, you don't have enough evidence to reject H₀.