Practise calculating Type I and Type II error probabilities from a stated rejection region under the null or a specified alternative binomial, Poisson or normal model.
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.
Find the probability of a Type I error in the test.
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
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.)
A1
If M0 awarded allow SC B2 for 0.0432
Given that the true value of p is 0.3, find the probability of a Type II error in the test.
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
M1
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.)
A1
If M0 awarded allow SC B1for 0.238 or 0.237