3 The fuel economy of two similar cars produced by manufacturers $A$ and $B$ was compared. A random sample of 15 cars was selected from manufacturer $A$ and a random sample of 10 cars was selected from manufacturer $B$. All the selected cars were driven over the same distance and the petrol consumption in miles per gallon (mpg) was calculated for each car. The results, $x _ { A } \operatorname { mpg }$ and $x _ { B } \operatorname { mpg }$ for cars from manufacturers $A$ and $B$ respectively, are summarised below, where $\bar { x }$ denotes the sample mean and $n$ the sample size.

$$\begin{array} { l l l } 
\Sigma x _ { A } = 460.5 & \Sigma \left( x _ { A } - \bar { x } _ { A } \right) ^ { 2 } = 156.88 & n _ { A } = 15 \\
\Sigma x _ { B } = 334 & \Sigma \left( x _ { B } - \bar { x } _ { B } \right) ^ { 2 } = 123.97 & n _ { B } = 10
\end{array}$$

(i) (a) Assuming that the populations are normally distributed with a common variance, show that the pooled estimate of this common variance is 12.21 , correct to 4 significant figures. [2]\\
(b) Construct a 95\% confidence interval for $\mu _ { B } - \mu _ { A }$, the difference in the population means for manufacturers $A$ and $B$.\\
(ii) Comment on a claim that the fuel economy for manufacturer $B$ 's cars is better than that for manufacturer $A$ 's cars.\\
(i) A random variable $X$ has probability density function given by

$$\mathrm { f } ( x ) = \begin{cases} \frac { 1 } { \theta } \mathrm { e } ^ { - \frac { x } { \theta } } & x \geqslant 0 \\ 0 & x < 0 \end{cases}$$

where $\theta$ is a positive constant. Find $\mathrm { E } \left( X ^ { 2 } \right)$.\\
(ii) A random sample $X _ { 1 } , X _ { 2 } , \ldots , X _ { n }$ is taken from a population with the distribution in part (i). The estimator $T$ is defined by $T = k \sum _ { i = 1 } ^ { n } X _ { i } ^ { 2 }$, where $k$ is a constant. Find the value of $k$ such that $T$ is an unbiased estimator of $\theta ^ { 2 }$.\\
(i) The discrete random variable $X$ has distribution $\operatorname { Geo } ( p )$. Show that the moment generating function of $X$ is given by $\mathrm { M } _ { X } ( t ) = \frac { p \mathrm { e } ^ { t } } { 1 - q \mathrm { e } ^ { t } }$, where $q = 1 - p$.\\
(ii) Use the moment generating function to find\\
(a) $\mathrm { E } ( X )$,\\
(b) $\operatorname { Var } ( X )$.\\
(iii) An unbiased six-sided die is thrown repeatedly until a five is obtained, and $Y$ denotes the number of throws up to and including the throw on which the five is obtained. Find $\mathrm { P } ( | Y - \mu | < \sigma )$, where $\mu$ and $\sigma$ are the mean and standard deviation, respectively, of the distribution of $Y$.\\
(i) The continuous random variable $X$ has a uniform distribution over the interval $0 < x < \frac { 1 } { 2 } \pi$. Show that the probability density function of $Y$, where $Y = \sin X$, is given by

$$\mathrm { f } ( y ) = \begin{cases} \frac { 2 } { \pi \sqrt { 1 - y ^ { 2 } } } & 0 < y < 1 \\ 0 & \text { otherwise. } \end{cases}$$

(ii) Deduce, using the probability density function, the exact values of\\
(a) the median value of $Y$,\\
(b) $\mathrm { E } ( Y )$.

\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2011 Q3 [10]}}