Pearson Edexcel International Advanced Level Mathematics: Samples and hypothesis tests (YMA01) – Revision Notes

These are condensed notes for section 4 (topics 4.1 to 4.6, “Hypothesis tests”) of Unit S2: Statistics 2 in the Pearson Edexcel International Advanced Subsidiary/Advanced Level Mathematics, Further Mathematics and Pure Mathematics specification (YMA01), Issue 3, April 2019. The specification lists S2 as an optional unit for IAS Further Mathematics and for IAL Mathematics and Further Mathematics. The study guide explains each step in depth, and the practice questions let you check what you recall.

Course links: hub, checklist, free 10-minute diagnostics.

Definitions to recall (4.1 to 4.3)

Term	Meaning
Population	Every item or person of interest
Census	Every member of the population is observed
Sample	A subset of the population
Sampling unit	One individual member that can be selected
Sampling frame	The list of all sampling units the sample is chosen from
Statistic	A quantity calculated from the sample only, with no unknown parameter in it
Sampling distribution	The distribution of a statistic over all possible samples of a given size
Null hypothesis H₀	The assumed value of the parameter, written with an equals sign
Alternative hypothesis H₁	How the parameter differs: >, < or ≠
Test statistic	The statistic used to decide, for example the count X
Critical region	Outcomes of the test statistic for which H₀ is rejected
Critical value	The boundary of the critical region
Actual significance level	P(test statistic in the critical region

Census versus sample. Census: exact, but slow, costly, and impossible if testing destroys the item. Sample: cheaper and faster, but results vary and a poor frame causes bias.

Statistic or not. For a sample X₁, X₂, X₃ from a population with unknown mean μ: the sample mean and the maximum are statistics. X₁ − μ is not.

Small sampling distribution. Population {0, 2} equally likely, samples of size 2 with replacement, statistic = sample mean.

Pairs: 00, 02, 20, 22
Mean:   0,  1,  1,  2
x            0     1     2
P(mean = x) 1/4   1/2   1/4

Method box: any hypothesis test

1. Define the parameter in words (p, or λ per stated interval).
2. Write H₀ and H₁. Pick one-tailed or two-tailed from the wording before using data.
3. Give the distribution of the test statistic when H₀ holds.
4. Find the critical region, or the probability of a result as extreme as, or more extreme than, the one seen.
5. Compare with the significance level (halve it for each tail of a two-tailed test).
6. Say whether H₀ is rejected, then write a conclusion in context. Do not say “proved”.

Choosing the tails (4.5)

Wording	H₁	Tails
increased, higher, more than	p > p₀	upper tail
decreased, lower, fewer than	p < p₀	lower tail
changed, different, not equal	p ≠ p₀	both tails, half the level each

Binomial test (4.6)

Under H₀, X ~ B(n, p₀). Choose the critical region so its probability is as large as possible without exceeding the level (or the half-level in each tail).

Reminder. X ~ B(14, 0.5), H₁: p < 0.5, 5% level.

P(X <= 3) = 0.0287  (< 0.05)
P(X <= 4) = 0.0898  (> 0.05)
Critical region: X <= 3.  Actual significance level 2.87%.

Poisson test (4.6)

Under H₀, X ~ Po(λ₀ × length of interval counted). Scale the mean first, always.

Reminder. H₀: λ = 6 per week, H₁: λ ≠ 6, 5% level, counted over one week. X ~ Po(6), 2.5% in each tail.

Lower: P(X <= 1) = 0.0174 (< 0.025),  P(X <= 2) = 0.0620 (> 0.025)  -> X <= 1
Upper: P(X >= 12) = 0.0201 (< 0.025), P(X >= 11) = 0.0426 (> 0.025) -> X >= 12
Critical region: X <= 1 or X >= 12.  Actual level 0.0174 + 0.0201 = 0.0374 (3.74%)

Normal approximation to a binomial test (4.6)

For large n, X ~ B(n, p₀) is approximated by N(np₀, np₀(1 − p₀)). Apply a continuity correction.

Required	Use
P(X ≥ k)	P(Y > k − 0.5)
P(X ≤ k)	P(Y < k + 0.5)
P(X > k)	P(Y > k + 0.5)
P(X < k)	P(Y < k − 0.5)

Reminder. H₀: p = 0.5, H₁: p < 0.5, n = 80, observed X = 32. Under H₀, X ≈ N(40, 20).

P(X <= 32) = P(Y < 32.5) = P(Z < (32.5 - 40)/sqrt(20)) = P(Z < -1.677) = 0.0468
0.0468 < 0.05: reject H0

The exact binomial probability P(X ≤ 32) is 0.0465, so the approximation is close here.

One-tailed Poisson reminder using the p-value

H₀: λ = 3 per day, H₁: λ > 3, 5% level. A 4-day record gives 19 events. Under H₀ the 4-day count is X ~ Po(12).

P(X >= 19) = 0.0374  < 0.05

Reject H₀. At the 5% level the sample gives enough evidence that the daily rate has increased. If the record had shown 18 events, P(X ≥ 18) = 0.0630 and H₀ would not be rejected.

Binomial reminder using a p-value

H₀: p = 0.15, H₁: p > 0.15, n = 25, 5% level. Suppose X = 7 is observed.

P(X >= 7) = 0.0695  > 0.05   -> do not reject H0
P(X >= 8) = 0.0255           -> the critical region would be X >= 8

Observing 7 gives insufficient evidence. The same data would reject H₀ at the 10% level, so always read the level from the question.

Writing the conclusion

Use this shape every time.

1. Decision: “Reject H₀” or “Do not reject H₀”, with the comparison that led to it (for example 0.0374 < 0.05).
2. Evidence statement: “There is sufficient (or insufficient) evidence at the 5% level to support the claim that …”
3. Context: name the quantity, such as “the daily rate of events” or “the proportion of parcels arriving on time”.

Spotting which test to use

The question gives	Likely test
A fixed number of trials, each success or failure, and a proportion	Binomial test for p
A count of random events over time, area or length, and an average rate	Poisson test for λ
A binomial test with a large number of trials	Binomial test with a normal approximation

Must-know distinctions

• Parameter and statistic. Hypotheses are about the parameter (p or λ). The test statistic is the sample value.
• Critical region and p-value. Both give the same decision. A critical region is set before looking at the data.
• Nominal and actual level. Nominal is what the question asks for. Actual is what the critical region really has.
• “Not reject” and “accept”. A result outside the critical region means insufficient evidence against H₀, not proof of it.
• Binomial and Poisson. A binomial variable counts successes in a set number of trials. A Poisson variable counts random events in an interval, and its mean scales with that interval.

Quick self-test

1. In one phrase, what is a sampling frame?
2. Is X₁ + X₂ − 2μ a statistic when μ is unknown?
3. A company’s mean number of faults per batch was 3. It now thinks the mean has changed. Write H₀ and H₁.
4. X ~ B(14, 0.5), H₁: p < 0.5, 5% level. State the critical region and the actual significance level.
5. For a Poisson test over 10 minutes, the rate is 0.7 per minute under H₀. What is the distribution of X?
6. X ~ Po(6), H₁: λ ≠ 6, total level 5%. Give the critical region.
7. A one-tailed binomial test gives P(X ≥ 11) = 0.0679 against a 5% level. Decision?
8. X ~ B(80, 0.5) is approximated by a normal variable Y. Write P(X ≤ 32) in terms of Y.
9. What is the mean and variance of the normal approximation in question 8?
10. Give one disadvantage of a census and one of a sample.

Answers

1. The list of all sampling units from which the sample is taken.
2. No. It contains the unknown μ.
3. H₀: λ = 3, H₁: λ ≠ 3 (λ is the mean faults per batch).
4. X ≤ 3. Actual level 0.0287.
5. X ~ Po(7).
6. X ≤ 1 or X ≥ 12 (actual level 0.0374).
7. Do not reject H₀, because 0.0679 is greater than 0.05.
8. P(Y < 32.5).
9. Mean 40, variance 20.
10. Census: costly or time-consuming, and impossible if testing destroys the item. Sample: results vary, and a poor frame gives bias.

Where marks are usually lost

• Hypotheses written about X or x̄ instead of p or λ.
• A Poisson mean not rescaled to the interval in the question.
• The wrong tail, because the wording was not read before the data.
• Setting a single tail against the whole level in a two-tailed test.
• A critical region that overshoots the level, such as taking X ≥ 10 when P(X ≥ 10) is above the level.
• A missing continuity correction, or one shifted the wrong way.
• Not quoting the actual significance level when the question asks for it.
• A conclusion that says “H₀ is accepted” or “the claim is proved”, or has no context.
• Advantage and disadvantage answers that are generic and not tied to the scenario.

Official syllabus

Pearson Edexcel International Advanced Subsidiary/Advanced Level in Mathematics, Further Mathematics and Pure Mathematics, Specification, Issue 3, April 2019, Unit S2: Statistics 2, section 4 (Hypothesis tests), topics 4.1 to 4.6. Published by Pearson Education Limited.