Library › Further Hypothesis Tests

Further Mathematics

Further Hypothesis Tests

Revise Further Hypothesis Tests for Further Mathematics — revision notes and instant AI marking. Free to start.

📖 Revision notes · preview

Edexcel FS2 · Chapter 6

Further Hypothesis Tests

Everything you need to go from "I've seen this formula before" to "I can defend every step of this test in an exam." Confidence intervals for variance, hypothesis tests for variance, and the F-test — explained properly, not just listed.

Variance & χ² F-distribution F-test

Start hereThe whole chapter in one breath

This chapter is about testing spread, not average. Earlier in your stats course you tested whether a mean had changed. Here, you test whether a variance has changed — either on its own, or by comparing two variances against each other.

There are really only three tools in this chapter, and they build on each other in a straight line:

The chi-squared (χ²) distribution — describes how the sample variance S² of a single normal sample behaves. Used for confidence intervals and hypothesis tests on one variance.
The F-distribution — describes what happens when you take the ratio of two sample variances. It has two "degrees of freedom" instead of one, because two samples are involved.
The F-test — the actual hypothesis test that uses the F-distribution to decide whether two populations really do have equal variance.

Big idea

When you want to know if things have become more spread out — one sample compared to a claimed value, or two samples compared to each other — you divide variances instead of subtracting means, and you check the result against the χ² or F tables instead of the z or t tables.

Topic 1Confidence Intervals for the Variance

Why we need a new distribution

You already know that the sample mean X̄ is an unbiased estimator of the population mean μ. In exactly the same spirit, the sample variance

Unbiased estimator of population variance

S² = Σ(Xᵢ − X̄)² / (n − 1)

Notice the n − 1 on the bottom, not n. Dividing by n − 1 instead of n is what makes S² unbiased — it corrects for the fact that X̄ is estimated from the same data, which would otherwise make the spread look artificially small.

Different samples from the same population give different values of s² — it's a particular observed value of the random variable S². To build a confidence interval for the true population variance σ², we need to know how S² is distributed. That's where chi-squared comes in.

The key result

If X₁, X₂, …, Xₙ ~ N(μ, σ²), then (n − 1)S² / σ² ~ χ²ₙ₋₁

Take your sample variance, scale it by (n − 1) and divide by the true variance — the result follows a chi-squared distribution with n − 1 degrees of freedom. This single fact is the engine behind everything in this section: confidence intervals AND hypothesis tests for variance both come straight from it.

Think of it like this χ² only takes positive values and is skewed — which makes sense, because variance can never be negative, and small variances are far more common than huge ones by chance. That's why we can't just use a symmetric z-table here the way we do for means.

Building the confidence interval

Because (n−1)S²/σ² follows a χ² distribution, we can trap it between two critical values from the χ² table with a chosen probability. Rearranging for σ² flips which critical value goes on top and bottom — this trips people up constantly, so watch closely in the example.

100(1 − α)% confidence interval for σ²

( (n−1)s² / χ²ₙ₋₁(α/2) , (n−1)s² / χ²ₙ₋₁(1 − α/2) )

The upper confidence limit uses the smaller χ² value (α/2), and the lower confidence limit uses the larger χ² value (1 − α/2). It feels backwards the first time — dividing by a small number gives a big result, and vice versa.

Worked Example — Salmon lengths

A fisherman catches 10 fish from a pink salmon pond. Lengths (cm): 67, 71, 62, 69, 75, 77, 63, 75, 61, 72. Find a 90% confidence interval for the population variance, assuming lengths are normally distributed.

1. Find x̄ and s²	x̄ = 69.2, s² = Σ(xᵢ − 69.2)² / 9 = 33.5111
2. Degrees of freedom	n = 10, so use χ²₉ (that's n − 1, not n — easy slip)
3. Read the χ² table	χ²₉(0.95) = 3.325 χ²₉(0.05) = 16.919
4. Lower limit	(n−1)s² / χ²₉(0.05) = (9 × 33.5111) / 16.919 = 17.826
5. Upper limit	(n−1)s² / χ²₉(0.95) = (9 × 33.5111) / 3.325 = 90.706
6. Conclusion	90% CI for σ²: (17.826, 90.706)

Practice Question 1

A random sample of 15 observations from a normal distribution gives s² = 12.4. Find a 95% confidence interval for σ². (You'll need χ²₁₄(0.975) = 5.629 and χ²₁₄(0.025) = 26.119.)

Practice Question 2

Explain, in your own words, why the upper confidence limit for σ² uses the lower percentage-point value of χ² (e.g. χ²(0.95)) rather than the upper one.

Topic 2Hypothesis Testing for a Single Variance

Same underlying result — (n−1)S²/σ² ~ χ²ₙ₋₁ — but now we're using it to test a claim rather than build an interval. The logic is identical to every hypothesis test you've ever done: state hypotheses, find a critical region, calculate a test statistic, compare, conclude.

The 5-step method

1. State H₀: σ² = (claimed value) and H₁ (>, <, or ≠ depending on the wording)
2. Specify α, the significance level, and ν = n − 1, the degrees of freedom
3. Find the critical region using the χ² tables
4. Calculate the test statistic: (n − 1)s² / σ² (using the claimed σ² from H₀, not the sample estimate)
5. Compare, state your conclusion, and link it back to the actual context of the question

Where students lose marks In step 4, you divide by σ² — the value stated in H₀ — not by your calculated s². It's easy to instinctively plug in s² on the bottom because that's what you just calculated, but the test statistic formula uses the population value being tested against.

Worked Example — Chicken weights

A farmer suspects some chickens are eating more than their fair share, so the variance of their weights has increased above the usual 0.9kg. A sample of 8 chickens gives weights (kg): 2.5, 4.3, 2.1, 4.5, 3, 3.2, 2.7, 4.4. Test at the 2.5% significance level whether the variance has increased.

1. Hypotheses	H₀: σ² = 0.9 H₁: σ² > 0.9 (one-tailed — "increased")
2. α and ν	α = 2.5%, ν = 8 − 1 = 7
3. Critical value	χ²₇(0.025) = 16.013 → critical region: (n−1)s²/σ² ≥ 16.013 (Note: the cheat sheet's worked source shows χ²₇(0.05)=14.067 for a 5% test — always match the tail probability to the α given in your question.)
4. Calculate s²	s² = 0.8827 (from the 8 data values)
5. Test statistic	(n−1)s²/σ² = (7 × 0.8827) / 0.9 = 6.865
6. Conclusion	6.865 is not in the critical region → insufficient evidence to reject H₀. The farmer cannot conclude the variance in chicken weights has increased.

Practice Question 3

A machine is set to fill bottles with variance 4 ml². A quality check on 12 bottles gives s² = 7.6 ml². Test at the 5% level whether the variance has changed (two-tailed). Use χ²₁₁(0.975) = 3.816 and χ²₁₁(0.025) = 21.920.

Topic 3The F-Distribution

Real problems often ask you to compare two variances, not just check one against a claimed number. Think of a retailer choosing between two suppliers of the same product — both might have similar average quality, but the retailer wants the supplier whose product is more consistent (smaller variance). The F-distribution is what lets you compare two variances directly, by looking at their ratio.

The F-distribution, defined

Sₓ² / Sᵧ² ~ F₍ₙₓ₋₁, ₙᵧ₋₁₎ (when σₓ² = σᵧ²)

If two independent samples come from normal distributions with the same (unknown) variance, the ratio of their sample variances follows an F-distribution. It has two degrees-of-freedom parameters, ν₁ and ν₂ — one for each sample — because two separate pieces of information went into building the ratio.

Order matters — a lot F₅,₈ and F₈,₅ are completely different distributions. Always write the degrees of freedom for the numerator's sample first, then the denominator's sample. Mixing up the order is the single most common F-distribution mistake.

Upper and lower critical values

F-tables (like χ² tables) only give you the upper critical values directly — the tables would be impossibly long otherwise. To find a lower critical value, you use a clever trick: flip the fraction and swap the degrees of freedom.

The reciprocal rule

F_{ν₁,ν₂}(p) = 1 / F_{ν₂,ν₁}(1 − p)

To find a lower critical value for F_{ν₁,ν₂}, look up the upper critical value for F_{ν₂,ν₁} (swap the degrees of freedom order) at the complementary probability, then take the reciprocal.

Worked Example — Reciprocal rule

Find the upper critical value F₅,₈(0.95).

1. Table only gives upper (small p) values	Read off F₈,₅(0.05) = 4.82 (swap order, complement the probability)
2. Apply the reciprocal rule	F₅,₈(0.95) = 1 / F₈,₅(0.05) = 1 / 4.82 = 0.21

Compare this to F₈,₅(0.95): reading F₅,₈(0.05) = 3.69 gives F₈,₅(0.95) = 1/3.69 = 0.27 — a different answer. This is exactly why order can't be swapped carelessly.

Worked Example — Probability between two values

X ~ F₁₂,₁₀. Find P(1/4.30 < X < 2.91).

1. Upper tail	F₁₂,₁₀(5%) = 2.91, so P(X < 2.91) = 1 − 0.05 = 0.95
2. Lower tail — use the rule P(F_{ν₁,ν₂} < f) = P(1/f < F_{ν₂,ν₁})	P(X < 1/4.3) = P(F₁₀,₁₂ > 4.3)
3. Look up F₁₀,₁₂(1%) = 4.3	P(X < 1/4.3) = 0.01
4. Subtract	P(1/4.30 < X < 2.91) = 0.95 − 0.01 = 0.94

Practice Question 4

Given F₆,₉(0.05) = 3.37, find the lower critical value F₉,₆(0.95).

Topic 4The F-Test

This is where everything comes together. The F-test uses the F-distribution to formally test whether two populations have equal variance — exactly the retailer-and-suppliers situation from earlier.

The 7-step method

1. Identify sₓ² as the larger sample variance and sᵧ² as the smaller
2. H₀: σₓ² = σᵧ²
3. H₁: σₓ² > σᵧ² (one-tailed) or σₓ² ≠ σᵧ² (two-tailed)
4. Find the critical value F_{νₓ,νᵧ} — νₓ (larger-variance sample) goes first
5. Write down the critical region
6. Test statistic: F = sₓ² / sᵧ²
7. Compare, conclude, and relate back to the context

Two-tailed trap For a two-tailed test, you still only look up one critical value from the table — but you halve the significance level first (e.g. a 10% two-tailed test uses the 5% critical value). Because you've already put the larger variance on top, the test statistic is automatically pushed into the upper tail, so you never need the lower critical value separately.

Worked Example — Two suppliers, one-tailed

Samples of size 6 (X) and 13 (Y) give sₓ² = 7 and sᵧ² = 26. Test at the 5% level whether the variance of Y is greater than the variance of X.

1. Hypotheses	H₀: σₓ² = σᵧ² H₁: σₓ² < σᵧ²
2. Degrees of freedom	νₓ = 5, νᵧ = 12
3. Critical value	Since Y's variance is the one claimed larger, put it on top: F₁₂,₅(0.05) = 4.68
4. Test statistic	The larger sample variance divides the smaller: F = sᵧ²/sₓ² = 26/7 = 3.714
5. Conclusion	3.714 < 4.68 → not in the critical region → insufficient evidence that Y's variance is greater than X's.

Worked Example — Two-tailed test

Samples of size 25 (X) and 17 (Y) give sₓ² = 8.3 and sᵧ² = 3.1. Test at the 10% level whether the variances are different.

1. Hypotheses	H₀: σₓ² = σᵧ² H₁: σₓ² ≠ σᵧ²
2. Halve α for two tails	10% two-tailed → look up the 5% critical value
3. Critical value	νₓ = 24, νᵧ = 16 → F₂₄,₁₆(0.05) = 2.24
4. Test statistic	F = sₓ²/sᵧ² = 8.3/3.1 = 2.677
5. Conclusion	2.677 > 2.24 → in the critical region → reject H₀. There is sufficient evidence the variances differ.

Practice Question 5

Two factories, P and Q, produce the same component. A sample of 9 from P gives sₚ² = 12.4; a sample of 7 from Q gives s_Q² = 3.8. Test at the 5% level whether the variances differ. (You'll need F₈,₆(0.025) = 5.60.)

Practice Question 6

Why does the F-test procedure insist you always divide the larger sample variance by the smaller one, rather than just following whatever order the question states the samples in?

Quick referenceWhat to Memorise

Unbiased variance estimator

S² = Σ(Xᵢ − X̄)² / (n − 1)

Divide by n − 1, not n.

χ² result for one sample

(n − 1)S² / σ² ~ χ²ₙ₋₁

Foundation for CI and hypothesis tests on a single variance.

CI for σ²

( (n−1)s²/χ²(α/2) , (n−1)s²/χ²(1−α/2) )

Small χ² value → upper limit. Large χ² value → lower limit.

F-distribution

Sₓ²/Sᵧ² ~ F_{nₓ−1, nᵧ−1}

Order of degrees of freedom must match order of variances.

Reciprocal rule

F_{ν₁,ν₂}(p) = 1 / F_{ν₂,ν₁}(1−p)

Swap df order + complement p + take reciprocal for lower critical values.

F-test statistic

F = larger s² / smaller s²

Always ≥ 1. Compare against the upper critical value only.

Self-checkConcepts Checklist

I can state and use S² = Σ(Xᵢ − X̄)²/(n−1) as the unbiased estimator of σ²

I know (n−1)S²/σ² follows a χ²ₙ₋₁ distribution and can explain what n−1 represents

I can construct a confidence interval for σ² and correctly match small/large χ² values to upper/lower limits

I can carry out a full 5-step hypothesis test for a single population variance, one- or two-tailed

I understand why the test statistic uses the claimed σ² from H₀, not the calculated s²

I can define the F-distribution and explain why it needs two degrees-of-freedom parameters

I always write F-distribution subscripts in the correct order (numerator sample first)

I can use the reciprocal rule to find lower critical values from F-tables

I can carry out a full 7-step F-test comparing two population variances

I remember to halve α for a two-tailed F-test rather than looking up two separate critical values

I always put the larger sample variance on top of the F-test ratio

I can link every conclusion back to the real-world context of the question, not just "reject/don't reject H₀"

Before the examExam Tips & Common Mistakes

n vs n − 1

Using n instead of n − 1 anywhere in this chapter — in S², in degrees of freedom, or in the test statistic — is the single most common way to lose easy marks. Every degrees-of-freedom value here comes from sample size minus 1.

Wrong table value

Mixing up χ²(α) with χ²(1−α), or F_{ν₁,ν₂} with F_{ν₂,ν₁}, gives an answer that looks plausible but is wrong. Always write out exactly which value you need before opening the table.

Forgetting to halve α

Two-tailed tests need α/2 in each tail. For the F-test specifically, this means looking up the critical value at half the stated significance level — not looking up two separate values.

No context in the conclusion

"Reject H₀" alone rarely gets full marks. Examiners want the conclusion tied back to the scenario — e.g. "there is sufficient evidence that Company B's lightbulbs have greater variance and are therefore less reliable."

Always identify which variance is larger first

Before touching the F-test, label sₓ² as the larger variance. This single habit prevents almost every ordering mistake that follows in the test.

Write the critical region before calculating

State the critical region as an inequality (e.g. "reject if F ≥ 2.24") before you compute the test statistic. It stops you from unconsciously working backwards from the answer you expect.

What's inside

📖 Revision notes ✦ AI flashcards ✓ Instant AI marking

Read the full Further Hypothesis Tests notes free

That's the preview — create a free account to read the rest, plus flashcards and practice questions with instant AI marking. No credit card.

Unlock the full notes free →