3 It is given that $X _ { 1 }$ and $X _ { 2 }$ are independent random variables with $X _ { 1 } \sim \mathrm {~N} \left( \mu _ { 1 } , 2.47 \right)$ and $X _ { 2 } \sim \mathrm {~N} \left( \mu _ { 2 } , 4.23 \right)$. Random samples of $n _ { 1 }$ observations of $X _ { 1 }$ and $n _ { 2 }$ observations of $X _ { 2 }$ are taken. The sample means are denoted by $\bar { X } _ { 1 }$ and $\bar { X } _ { 2 }$.\\
(i) State the distribution of $\bar { X } _ { 1 } - \bar { X } _ { 2 }$, giving its parameters.

For two particular samples, $n _ { 1 } = 5 , \Sigma x _ { 1 } = 48.25 , n _ { 2 } = 10$ and $\Sigma x _ { 2 } = 72.30$.\\
(ii) Test at the $2 \%$ significance level whether $\mu _ { 1 }$ differs from $\mu _ { 2 }$.

A student stated that because of the Central Limit Theorem the sample means will have normal distributions so it is unnecessary for $X _ { 1 }$ and $X _ { 2 }$ to have normal distributions.\\
(iii) Comment on the student's statement.

\hfill \mbox{\textit{OCR S3 2010 Q3 [10]}}