Quality of tests questions

Questions to attempt

3 questions on Quality of tests, each with its mark scheme and worked method. Reveal the answer when you have had a go.

1. FS1-8.1 · Type I and Type II errors. Size and Power of Test. The power function.

   Q1

   ORIGINAL

   Year 2

   Year 2: this is content the exam board adds beyond the AS subject content, for the full A-level.

   1.

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

   (6)

   (Total for Question 1 is 6 marks)

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2. FS1-8.1 · Type I and Type II errors. Size and Power of Test. The power function.

   Q2

   ORIGINAL

   Year 2

   Year 2: this is content the exam board adds beyond the AS subject content, for the full A-level.

   2.

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

   (4)

   (Total for Question 2 is 4 marks)

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3. FS1-8.1 · Type I and Type II errors. Size and Power of Test. The power function.

   Q3

   ORIGINAL

   Year 2

   Year 2: this is content the exam board adds beyond the AS subject content, for the full A-level.

   3.

   For $X\sim\operatorname{Bin}(12,p)$, a test of $H_0:p=0.4$ against $H_1:p>0.4$ uses critical region $X\geq8$. Write its power function. Calculate its size, its power and Type II error probability when $p=0.65$, and the expected number of rejections in $40$ independent repetitions at $p=0.65$. Comment on whether it is a $5\%$ test.

   (9)

   (Total for Question 3 is 9 marks)

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FS1-8 Quality of tests revision notes