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 \(\bar x\) ≤ 66.5

Accept H₁ if \(\bar x\) > 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 \({\bar x}\) if she wants the probabilities of a Type I error and a Type II error to be equal.

\(\bar X \sim {\text{N}}\left( {\mu ,\,\frac{{{\sigma ^2}}}{n}} \right)\)

\(\bar X \sim {\text{N}}\left( {65,\,\frac{{36}}{{100}}} \right)\)     (A1)

P(Type I Error) \( = {\text{P}}\left( {\bar X > 66.5} \right)\)      (M1)

= 0.00621       A1

[3 marks]

P(Type II Error) = P(accept H₀| H₁ is true)

\( = {\text{P}}\left( {\bar X \leqslant 66.5\left| {\mu  = 67.9} \right.} \right)\)        (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]