The independent random variables \(S\) and \(L\) have means \(\mu _ { S }\) and \(\mu _ { L }\) respectively, and a common variance of \(\sigma ^ { 2 }\).
The variable \(\bar { S }\) denotes the mean of a random sample of \(n\) observations on \(S\) and the variable \(\bar { L }\) denotes the mean of a random sample of \(n\) observations on \(L\).
Find a simplified expression, in terms of \(\sigma ^ { 2 }\), for the variance of \(\bar { L } - 2 \bar { S }\).
A machine fills both small bottles and large bottles with shower gel. It is known that the volume of shower gel delivered by the machine is normally distributed with a standard deviation of 8 ml .
A random sample of 25 small bottles filled by the machine contained a mean volume of \(\bar { s } = 258 \mathrm { ml }\) of shower gel.
An independent random sample of 25 large bottles filled by the machine contained a mean volume of \(\bar { l } = 522 \mathrm { ml }\) of shower gel.
Investigate, at the \(10 \%\) level of significance, the hypothesis that the mean volume of shower gel in a large bottle is more than twice that in a small bottle. [0pt]
[7 marks]
Deduce that, for the test of the hypothesis in part (b)(i), the critical value of \(\bar { L } - 2 \bar { S }\) is 4.585 , correct to three decimal places. [0pt]
[2 marks]
In fact, the mean volume of shower gel in a large bottle exceeds twice that in a small bottle by 10 ml .
Determine the probability of a Type II error for a test of the hypothesis in part (b)(i) at the 10\% level of significance, based upon random samples of 25 small bottles and 25 large bottles. [0pt]
[4 marks]