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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.

[3]
a.

She decides to use the same acceptance criteria as the previous investigator. Find the probability of a Type II error.

[3]
b.i.

Find the critical value for ˉx if she wants the probabilities of a Type I error and a Type II error to be equal.

[3]
b.ii.

Markscheme

ˉXN(μ,σ2n)

ˉXN(65,36100)     (A1)

P(Type I Error) =P(ˉX>66.5)      (M1)

= 0.00621       A1

[3 marks]

a.

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]

b.i.

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]

b.ii.

Examiners report

[N/A]
a.
[N/A]
b.i.
[N/A]
b.ii.

Syllabus sections

Topic 4 - Sets, relations and groups » 4.1 » Finite and infinite sets. Subsets.

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