Sarah is the quality control manager for the Stronger Steel Corporation which makes steel sheets. The steel sheets should have a mean tensile strength of 430 MegaPascals (MPa). If the mean tensile strength drops to 400 MPa, then Sarah must recommend a change in composition. The tensile strength of these steel sheets follows a normal distribution with a standard deviation of 35 MPa. Sarah defines the following hypotheses

\[{H_0}:\mu  = 430\]

\[{H_1}:\mu  = 400\]

where \(\mu \) denotes the mean tensile strength in MPa. She takes a random sample of \(n\) steel sheets and defines the critical region as \(\bar x \leqslant k\), where \(\bar x\) notes the mean tensile strength of the sample in MPa and \(k\) is a constant.

Given that the \(P{\text{(Type I Error)}} = 0.0851\) and \(P{\text{(Type II Error)}} = 0.115\), both correct to three significant figures, find the value of \(k\) and the value of \(n\).

\(\bar X \sim N\left( {430,{\text{ }}\frac{{{{35}^2}}}{n}} \right)\)     (M1)(A1)

Note: The M1 is for considering the distribution of \(\bar X\)

type I error gives \({\text{P}}(\bar X \leqslant k/\mu  = 430) = 0.0851\)

\(\frac{{k - 430}}{{\frac{{35}}{{\sqrt n }}}} =  - 1.37156 \ldots \)     M1A1

type II error gives \({\text{P}}(\bar X > k/\mu  = 400) = 0.115\)

\(\frac{{k - 400}}{{\frac{{35}}{{\sqrt n }}}} = 1,20035 \ldots \)     M1A1

Note: The two M1 marks above are for attempting to standardize \({\bar X}\) and obtain the corresponding equations with inverse normal values

solving simultaneously     (M1)

\(k = 414\)     A1

\(n = 9\)     A1

This proved to be a difficult question for most candidates with only a minority giving a correct solution. Most candidates either made no attempt at the question or just wrote several lines of irrelevant mathematics.