CAIE A-Level Mathematics A2 6.5.5 Normal Distribution Questions
Practise calculating Type II error probabilities under specified alternative parameter values.
- Syllabus
- 2028–2030
- Course
- Mathematics 9709
- Level
- A2
Exam points
- identify the acceptance region for a hypothesis test
- calculate Type II error probabilities under an alternative hypothesis
- interpret the power and error probabilities of a test
CAIE A-Level Mathematics A2 6.5.5 Normal Distribution Questions question 1
[Maximum number: 5]
Anton believes that 10% of students at his college are left-handed. Aliya believes that this is an under
estimate. She plans to carry out a hypothesis test of the null hypothesis p=0.1 against the alternative hypothesis p>0.1, where p is the actual proportion of students at the college that are left-handed. She chooses a random sample of 20 students from the college. She will reject the null hypothesis if at least 5 of these students are left-handed.
Question (a)
(a)
Find the probability of a Type I error in the test.
[ 3 ]
1-(0.9^20+20 x 0.9^19 x 0.1+ ^20 C_2 x 0.9^18 x 0.1^2+ ^20 C_3 x 0.9^17 x 0.1^3+ ^20 C_4 x. 0.9^16 x 0.1^4 )
1-(0.9^20+20 x 0.9^19 x 0.1+ ^20 C_2 x 0.9^18 x 0.1^2+ ^20 C_3 x 0.9^17 x 0.1^3+ ^20 C_4 x.0.9^16 x 0.1^4 )
M2: fully correct
M1: attempt 1-P(X=0,1,2,3,4);
allow 1-P(X=0,1,2,3,4,5) or 1-P(X=0,1,2,3) need 1 -... the method mark cannot be implied
0.0432 (3 s.f.)
If M0 awarded allow SC B2 for 0.0432
Question (b)
(b)
Given that the true value of p is 0.3, find the probability of a Type II error in the test.
[ 2 ]
0.7^20+20 x 0.7^19 x 0.3+ ^20 C_2 x 0.7^18 x 0.3^2+ ^20 C_3 x 0.7^17 x 0.3^3+ ^20 C_4 x 0.7^16x 0.3^4
Attempt to find P(<=slant 4) using B(20,0.3)
Allow one end error The method mark cannot be implied
0.238 or 0.237 (3 s.f.)
If M0 awarded allow SC B1for 0.238 or 0.237
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