Date | May 2018 | Marks available | 3 | Reference code | 18M.1.hl.TZ0.7 |
Level | HL only | Paper | 1 | Time zone | TZ0 |
Command term | Find | Question number | 7 | Adapted from | N/A |
Question
A sample of size 100 is taken from a normal population with unknown mean μ and known variance 36.
Another investigator decides to use the same data to test the hypotheses H0 : μ = 65 , H1 : μ = 67.9.
An investigator wishes to test the hypotheses H0 : μ = 65, H1 : μ > 65.
He decides on the following acceptance criteria:
Accept H0 if the sample mean ˉx ≤ 66.5
Accept H1 if ˉx > 66.5
Find the probability of a Type I error.
She decides to use the same acceptance criteria as the previous investigator. Find the probability of a Type II error.
Find the critical value for ˉx if she wants the probabilities of a Type I error and a Type II error to be equal.
Markscheme
ˉX∼N(μ,σ2n)
ˉX∼N(65,36100) (A1)
P(Type I Error) =P(ˉX>66.5) (M1)
= 0.00621 A1
[3 marks]
P(Type II Error) = P(accept H0 | H1 is true)
=P(ˉX⩽ (M1)
= {\text{P}}\left( {\bar X \leqslant 66.5} \right) when \bar X \sim {\text{N}}\left( {67.9,\,\frac{{36}}{{100}}} \right) (M1)
= 0.00982 A1
[3 marks]
the variances of the distributions given by H0 and H1 are equal, (R1)
by symmetry the value of {\bar x} lies midway between 65 and 67.9 (M1)
\Rightarrow \bar x = \frac{1}{2}\left( {65 + 67.9} \right) = 66.45 A1
[3 marks]