Hypothesis Tests: z-test, t-test & chi-squared

The last lesson estimated a mean; now we want a verdict. Think of a hypothesis test as a small courtroom. You begin by presuming innocence — the null hypothesis, the boring default that nothing has changed — and then ask: if that were true, how strange would the data we actually saw be? Strange enough, and you reject the presumption. Not strange enough, and you walk away unconvinced. You never prove the null true; you only decide whether the evidence is strong enough to overturn it.

One phrase in that paragraph does more damage than any other, so read it again slowly: “do not reject H0” is not the same as “H0 is true”. The test only ever asks whether the evidence is strong enough to overturn the default. A verdict of “not strong enough” may mean the default really is right — or it may mean you gathered too little data to tell the difference. An acquittal is not a certificate of innocence.

The same misreading has a numeric twin worth heading off now. A p-value of 0.04 does not say there is a 4% chance the null is true, and α = 0.05 is not a 5% chance of the null being right. Both numbers are computed while assuming the null is true; they measure how odd the data looks under that assumption, never how likely the assumption itself is.

For GATE the bar is mostly recognition: read a scenario, name the right test, write the right statistic. Full derivations are rare, so we keep the depth there.

The framework

• Null hypothesis H0 — the default claim, usually “no effect” or a specific value, like μ = 50.
• Alternative H1 — what you suspect instead, like μ ≠ 50 (two-tailed) or μ > 50 (one-tailed).
• Test statistic — one number measuring how far the data sits from H0, in standard-error units. The standard error is how much a sample mean typically wobbles away from the truth by sheer luck of the draw. The statistic is really asking: “is this gap bigger than ordinary sampling noise?”
• Significance level α — the risk you accept of rejecting a true H0 (commonly 0.05). It fixes the critical value (e.g. 1.96 for a two-tailed z at 5%).
• Decision — reject H0 if the statistic is more extreme than the critical value, or equivalently if the p-value (the chance of data this extreme under H0) falls below α.

p-value < α     →  reject H0          |statistic| > critical value  →  reject H0
p-value ≥ α     →  do not reject       |statistic| ≤ critical value  →  do not reject

Two ways to be wrong

A verdict can err in two directions, and the courtroom names them well.

A Type I error is rejecting a true null (convicting the innocent), and its rate is exactly α. A Type II error is failing to reject a false null (acquitting the guilty), with rate β; the power 1 − β is the chance of correctly catching a real effect.

Drag the threshold below: slide it to shrink the red Type-I area and the orange Type-II area grows — you trade one error for the other. But push the effect size or the sample size up and the two bells separate, so power climbs without spending any more α. That is why bigger samples make tests more decisive.

Which test? z, t, or chi-squared

This is the choice GATE most wants you to make. It hinges on what you know and what you are testing.

Use a z-test for a mean when the population σ is known (or n is large enough that the sample SD is reliable) — the CLT from the last lesson is what justifies it.

Use a t-test when σ is unknown and estimated from the sample as s, typically with small n. It runs on Student’s t-distribution, whose heavier tails simply admit that you had to guess σ too. Extra uncertainty means extremes are less surprising, so the bar for “strange” sits further out. As n grows the guess gets good and the t-distribution settles onto the Normal.

Use a chi-squared (χ²) test for counts and categories. It covers:

• testing a variance;
• a goodness-of-fit; or
• independence in a contingency table (a grid of counts cross-classified by two categories).

The test works by summing (observed − expected)² / expected.

A worked example — a two-tailed z-test

A machine should fill bottles to μ0 = 50 ml; the fill SD is known, σ = 8 ml. A sample of n = 64 bottles averages x̄ = 52 ml. At the 5% level (two-tailed, critical value 1.96), is the machine off-target?

H0: μ = 50   vs   H1: μ ≠ 50          σ known, large n → z-test

standard error = σ/√n   = 8/√64  = 8/8 = 1
z = (x̄ − μ0)/(σ/√n)     = (52 − 50)/1 = 2.0

compare |z| = 2.0  with the two-tailed 5% critical value 1.96:
2.0 > 1.96   →   reject H0

There is significant evidence at the 5% level that the machine is off-target. Note how close it is, though — at the 1% level the critical value is 2.576, and 2.0 < 2.576, so you would not reject. The threshold matters.

A question to carry forward

You have now turned data into chances, into summaries, and into verdicts — the whole of the Probability roadmap. The next roadmap leaves randomness behind for the exact world of vectors and matrices. But one word will follow you across the border: independence. Here it meant events that tell you nothing about each other; there it will mean directions that cannot be built from one another. What might independence mean for arrows rather than events?

In one breath

• A test is a courtroom: presume H0 (the boring default), measure how surprising the data is, and reject only if it is extreme enough — you never prove H0.
• Decide by p-value < α ⇔ |statistic| > critical value (two-tailed 5% → 1.96; one-tailed 5% → 1.645).
• Errors: Type I = reject a true H0 (rate α); Type II = miss a false H0 (rate β); power = 1 − β (grows with effect size and n).
• Which test: σ known / large n → z; σ unknown / small n → t (heavier tails); counts, variance, goodness-of-fit, independence → χ² Σ(O−E)²/E.
• GATE is recognition: name the test and the right one/two-tailed critical value. z-statistic = (x̄−μ₀)/(σ/√n).

Practice