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

Think of a hypothesis test as a small courtroom. You start by assuming the defendant is innocent — the null hypothesis, the boring default — and then ask: if that were true, how strange would the data we just saw be? If the data is strange enough, you reject the null. If not, you walk away unconvinced. You never prove the null true; you only decide whether the evidence is strong enough to overturn it.

For GATE DA the bar is mostly recognition: read a scenario, name the right test, write the right statistic. Full multi-step derivations are rare, so this lesson keeps the depth at exactly that level.

The framework

• Null hypothesis H0 — the default claim, usually “no effect” or a specific value, e.g. μ = 50.
• Alternative H1 — what you suspect instead, e.g. μ ≠ 50 (two-tailed) or μ > 50 (one-tailed).
• Test statistic — a single number measuring how far the data sits from H0, in standard-error units.
• Significance level α — the risk you accept of rejecting a true H0 (commonly α = 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 probability of data this extreme under H0) is below α.

p-value  <  α     →  reject H0          |statistic|  >  critical value  →  reject H0
p-value  ≥  α     →  fail to reject H0   |statistic|  ≤  critical value  →  fail to reject

Two ways to be wrong

Drag the decision threshold below and watch the trade. Slide it right and the red area (Type I, α) shrinks but the orange area (Type II, β) grows. Bump the effect size or the sample size and the two bells pull apart, so the power (1 − β) climbs without you giving up any α. That is exactly why bigger samples make tests more decisive.

Which test? z vs t vs chi-squared

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

• z-test — for a mean when the population standard deviation σ is known (or n is large so the sample SD is reliable). The statistic uses the Normal, and the CLT from the last lesson is what justifies it.
• t-test — for a mean when σ is unknown and you estimate it from the sample (s), typically with small n. It uses Student’s t-distribution, which has heavier tails than the Normal (extra uncertainty from estimating σ) and approaches the Normal as n grows. The statistic swaps σ for s.
• chi-squared (χ²) test — for counts and categories: testing a variance, goodness-of-fit (do observed frequencies match expected?), or independence in a contingency table (a cross-tab counting how often each combination of two categories occurs). It sums (observed − expected)² / expected.

How GATE asks this

Usually an MCQ at recognition level: a scenario describing the data (“σ unknown, n = 12”) and four candidate tests — pick the right one and the right one- vs two-tailed critical value. Occasionally a short NAT asks you to plug numbers into a z- or t-statistic. A full p-value derivation is rare; what GATE reliably rewards is knowing the framework and the z/t/χ² decision cold.

Worked example — a two-tailed z-test

A machine is supposed to fill bottles to μ0 = 50 ml. The fill standard deviation is known to be σ = 8 ml. A sample of n = 64 bottles has mean x̄ = 52 ml. At the 5% level (two-tailed, critical value 1.96), is the machine off-target?

Step 1 — hypotheses. H0: μ = 50 versus H1: μ ≠ 50 (two-tailed: “off-target” in either direction).

Step 2 — choose the test. σ is known and n is large, so this is a z-test.

Step 3 — standard error and statistic.

standard error  =  σ / √n      =  8 / √64   =  8 / 8   =  1

z  =  (x̄ − μ0) / (σ/√n)  =  (52 − 50) / 1  =  2 / 1  =  2.0

Step 4 — decide. Compare |z| = 2.0 with the two-tailed 5% critical value 1.96. Since 2.0 > 1.96, the result falls in the rejection region:

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 — had the critical value been the 1% value 2.576, we would not reject. The threshold matters.)

Quick check