Edexcel A-level Further Maths coverage

Quality of tests

Section FS1-8
1 spec leaf

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

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FS1-8.1

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

  • 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}), while the power function is π(θ)=Pθ(reject H0)\pi(\theta)=P_{\theta}(\text{reject }H_0).
  • At a specified alternative parameter, power equals 1−P(Type II error)1-P(\text{Type II error}); higher power means the test is more likely to detect that departure from H0H_0.
  • A common error is to call the nominal significance level the size without calculating the attainable probability of the discrete critical region.

Tier 1 · Easy

4 marks
ORIGINAL

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.

Tier 2 · Standard

6 marks
ORIGINAL

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.

Tier 3 · Hard

9 marks
ORIGINAL

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.