N2012 P2 Q6

On a remote island a zoologist measures the tail lengths of a random sample of 20 squirrels. In a species of squirrel known to her, the tail lengths have mean \(14.0\text{ cm}\). She carries out a test, at the \(5\%\) significance level, of whether squirrels on the island have the same mean tail length as the species known to her. She assumes that the tail lengths of squirrels on the island are normally distributed with standard deviation \(3.8\text{ cm}\).

1. State appropriate hypotheses for the test.[1]

The sample mean tail length is denoted by \(\bar x\text{ cm}\).

2. Use an algebraic method to calculate the set of values of \(\bar x\) for which the null hypothesis would not be rejected.
   (Answers obtained by trial and improvement from a calculator will obtain no marks.)[3]
3. State the conclusion of the test in the case where \(\bar x=15.8\).[2]