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Edexcel A-level Further Maths revision notes

Hypothesis testing

Section FS1-4
Both years
Both years: this holds AS subject content and content the exam board adds beyond it for the full A-level.
2 specification points

Notes and three levels of exam-style practice for each registered specification point in this section.

Checked against Edexcel 9FM0 section FS1-4

Checked against Edexcel 9FM0 section FS1-4. Review basis: the qualification registry sourced from the Pearson Edexcel Level 3 Advanced GCE in Further Mathematics (9FM0) specification; registry verification recorded 17 July 2026.

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In the exam: Formulae booklet provided · calculator allowed in every paper

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FS1-4.1

Extend ideas of hypothesis tests to test for the mean of a Poisson distribution.

Notes
Evidence from your answers: none yet
Your confidence:

A self-report of how sure you feel. It does not measure mastery. Evidence from your answers reaches secure after the latest Tier 2/3 attempt is correct, with three correct distinct drills across at least two dates and two practice sources.

Explanation

  • A Poisson-mean test states H0H_0 and H1H_1 in terms of the population parameter λ\lambda or μ\mu, not the observed count. Under H0H_0, scale the mean to the complete observation period; independent periods may be combined into one Poisson total.
  • The direction of H1H_1 selects the tail.
  • A critical region is the most extreme attainable tail whose probability under H0H_0 does not exceed the significance level, and that probability is the size of the test.
  • A pp-value is the null probability of an outcome at least as extreme as observed.
  • Examiners expect a contextual conclusion: reject H0H_0 for evidence supporting H1H_1, or do not reject it for insufficient evidence.
Worked example

A process has claimed mean 33 faults per day. Over 55 independent days, 2424 faults occur. At the 5%5\% level, test whether the daily mean has increased, given that P(Po⁡(15)≥23)=0.0327P(\operatorname{Po}(15)\geq23)=0.0327 and P(Po⁡(15)≥22)=0.0531P(\operatorname{Po}(15)\geq22)=0.0531.

  1. 1.Let λ\lambda be the daily population mean: H0:λ=3H_0:\lambda=3, H1:λ>3H_1:\lambda>3.
  2. 2.Under H0H_0, the total X∼Po⁡(15)X\sim\operatorname{Po}(15).
  3. 3.The critical region is X≥23X\geq23, with size 0.03270.0327.
  4. 4.Since 2424 is critical, reject H0H_0.

Answer: There is sufficient evidence at the 5%5\% level that the daily mean fault rate has increased.

Common mistakes

  • Don't state hypotheses using the sample total rather than the population mean parameter.
  • Don't use the daily mean as the Poisson parameter for a multi-day total.
  • Don't say that failing to reject H0H_0 proves the null hypothesis.

Exam tip

A full test conclusion must name the significance level, evidence and contextual parameter direction.

Tier 1 · Easy

ORIGINAL

1.

A manager claims that the mean number of alerts per shift is 4.24.2. State hypotheses to test whether the mean has increased, defining your parameter.

(2)

(Total for Question 1 is 2 marks)

Tier 2 · Standard

ORIGINAL

1.

Under H0H_0, defects occur at a mean rate of 2.52.5 per hour. Eight independent hours give a total of 2828 defects. Test at a 5%5\% significance level whether the mean rate has increased. State the critical region and the size of the test.

(7)

(Total for Question 1 is 7 marks)

Tier 3 · Hard

ORIGINAL

1.

A Poisson model has mean 1.81.8 events per batch. Twelve independent batches produce 1414 events in total. Carry out a test at the 5%5\% significance level of whether the mean per batch has decreased, giving the critical region, its size and your conclusion.

(8)

(Total for Question 1 is 8 marks)

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FS1-4.2

Extend hypothesis testing to test for the parameter p of a geometric distribution.

Notes
Evidence from your answers: none yet
Your confidence:

A self-report of how sure you feel. It does not measure mastery. Evidence from your answers reaches secure after the latest Tier 2/3 attempt is correct, with three correct distinct drills across at least two dates and two practice sources.

Explanation

  • A geometric-parameter test states hypotheses in terms of the success probability pp. A larger pp tends to produce earlier success, so small waiting times support H1:p>p0H_1:p>p_0 and large waiting times support H1:p<p0H_1:p<p_0.
  • The sum of rr independent geometric variables with common pp has a negative binomial distribution counting the trial of the rrth success.
  • Use this distribution to find a discrete critical region whose null probability does not exceed the significance level.
  • The size of the test may be below the nominal level.
  • Examiners expect the tail to be justified from the effect of changing pp and conclusions to refer to evidence about the population success probability.
Worked example

Four independent geometric waiting times have common parameter pp. Their sum is SS. Test H0:p=0.5H_0:p=0.5 against H1:p<0.5H_1:p<0.5. Explain which tail is critical and interpret a result in that tail.

  1. 1.A smaller pp makes successes less frequent and waiting times larger.
  2. 2.Under H0H_0, SS is negative binomial with r=4r=4, p=0.5p=0.5.
  3. 3.Therefore the critical region has form S≥cS\geq c, where cc is the least integer making P(S≥c)≤αP(S\geq c)\leq\alpha.

Answer: Use the upper tail; a critical result gives evidence that the success probability is below 0.50.5.

Common mistakes

  • Don't use the lower waiting-time tail for the alternative p<p0p<p_0.
  • Don't model a sum of geometric waiting times as another geometric variable.
  • Don't state hypotheses using the observed sum SS rather than the parameter pp.

Exam tip

Before calculating a critical value, write one sentence linking larger or smaller pp to waiting-time direction.

Tier 1 · Easy

ORIGINAL

1.

A geometric model uses p=0.35p=0.35. State hypotheses for testing whether the probability of success has fallen, and state which tail of the waiting time is critical.

(3)

(Total for Question 1 is 3 marks)

Tier 2 · Standard

ORIGINAL

1.

Six independent geometric observations count trials to first success. Test H0:p=0.3H_0:p=0.3 against H1:p>0.3H_1:p>0.3 at the 5%5\% level of significance using their sum SS. Find the critical region and its size, then state the conclusion when S=9S=9.

(7)

(Total for Question 1 is 7 marks)

Tier 3 · Hard

ORIGINAL

1.

Five independent geometric observations have common parameter pp. Use their sum SS to test H0:p=0.4H_0:p=0.4 against H1:p<0.4H_1:p<0.4 at the 5%5\% level. Determine the critical region and its size, then carry out the test and state your conclusion when S=19S=19.

(8)

(Total for Question 1 is 8 marks)

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