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Edexcel A-level Further Maths revision notes

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Section FS1-8
Year 2
Year 2: this is content the exam board adds beyond the AS subject content, for the full A-level.
1 specification point

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

Checked against Edexcel 9FM0 section FS1-8

Checked against Edexcel 9FM0 section FS1-8. 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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FS1-8.1

Type I and Type II errors. Size and Power of Test. The power function.

Notes
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Explanation

  • A Type I error rejects H0H_0 when it is true; a Type II error does not reject H0H_0 when a specified alternative is true. The size is P(reject H0∣H0 true)P(\text{reject }H_0\mid H_0\text{ true}), the attainable null probability of the critical region, which may differ from a nominal significance level in a discrete test.
  • The power function is π(θ)=Pθ(reject H0)\pi(\theta)=P_\theta(\text{reject }H_0).
  • At a specified alternative, power equals one minus the Type II error probability.
  • A more effective test has higher power against relevant alternatives while controlling size.
  • Examiners expect errors to be described in context and power to be calculated from the critical region at the alternative parameter.
A typical upper-tailed power function, with the test size shown at the null parameter.
Worked example

For X∼Bin⁡(10,p)X\sim\operatorname{Bin}(10,p), a test rejects H0:p=0.3H_0:p=0.3 when X≥6X\geq6. Find the size, write the power function and express the Type II error probability at p=0.6p=0.6.

  1. 1.Size =P0.3(X≥6)=∑x=610(10x)0.3x0.710−x=P_{0.3}(X\geq6)=\sum_{x=6}^{10}\binom{10}{x}0.3^x0.7^{10-x}.
  2. 2.π(p)=∑x=610(10x)px(1−p)10−x\pi(p)=\sum_{x=6}^{10}\binom{10}{x}p^x(1-p)^{10-x}.
  3. 3.At p=0.6p=0.6, Type II probability =1−π(0.6)=P0.6(X≤5)=1-\pi(0.6)=P_{0.6}(X\leq5).

Answer: Size =0.04735=0.04735 to 55 significant figures; power is the stated upper-tail sum, and Type II probability at p=0.6p=0.6 is 1−π(0.6)1-\pi(0.6).

Common mistakes

  • Don't call the nominal 5%5\% level the size without calculating the discrete critical-region probability.
  • Don't calculate power at an alternative parameter using the null distribution instead.
  • Don't add power and Type I error probability rather than power and Type II error probability.

Exam tip

Use the same critical-region event for size and power; only the parameter value in its probability changes.

Tier 1 · Easy

ORIGINAL

1.

In a test of H0:p=0.4H_0:p=0.4 against H1:p>0.4H_1:p>0.4, describe Type I and Type II errors in terms of pp, and define the size of the test.

(4)

(Total for Question 1 is 4 marks)

Tier 2 · Standard

ORIGINAL

1.

Let X∼Bin⁡(12,p)X\sim\operatorname{Bin}(12,p). A test of H0:p=0.3H_0:p=0.3 against H1:p>0.3H_1:p>0.3 rejects H0H_0 when X≥7X\geq7. Find the size, the power at p=0.5p=0.5, and the probability of a Type II error at p=0.5p=0.5.

(6)

(Total for Question 1 is 6 marks)

Tier 3 · Hard

ORIGINAL

1.

For X∼Bin⁡(12,p)X\sim\operatorname{Bin}(12,p), a test of H0:p=0.4H_0:p=0.4 against H1:p>0.4H_1:p>0.4 uses critical region X≥8X\geq8. Write its power function. Calculate its size, its power and Type II error probability when p=0.65p=0.65, and the expected number of rejections in 4040 independent repetitions at p=0.65p=0.65. Comment on whether it is a 5%5\% test.

(9)

(Total for Question 1 is 9 marks)

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