Moderate -0.3 This is a multi-part S2 question covering standard topics (rectangular distribution, confidence intervals, chi-squared test). Part (a) involves routine application of formulas for rectangular distributions and linear transformations. Part (b) is a textbook confidence interval calculation. The chi-squared component appears incomplete but would involve standard test procedures. All parts require straightforward application of learned techniques with no novel problem-solving, making it slightly easier than average.
The continuous random variable \(X\) has a rectangular distribution defined by the probability density function
$$f ( x ) = \begin{cases} 0.01 \pi & u \leqslant x \leqslant 11 u \\ 0 & \text { otherwise } \end{cases}$$
where \(u\) is a constant.
Show that \(u = \frac { 10 } { \pi }\).
Using the formulae for the mean and the variance of a rectangular distribution, find, in terms of \(\pi\), values for \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
Calculate exact values for the mean and the variance of the circumferences of circles having diameters of length \(\left( X + \frac { 10 } { \pi } \right)\).
A machine produces circular discs which have an area of \(Y \mathrm {~cm} ^ { 2 }\). The distribution of \(Y\) has mean \(\mu\) and variance 25 .
A random sample of 100 such discs is selected. The mean area of the discs in this sample is calculated to be \(40.5 \mathrm {~cm} ^ { 2 }\).
Calculate a 95\% confidence interval for \(\mu\).
Emily believed that the performances of 16-year-old students in their GCSEs are associated with the schools that they attend. To investigate her belief, Emily collected data on the GCSE results for 2010 from four schools in her area.
The table shows Emily's collected data, denoted by \(O _ { i }\), together with the corresponding expected frequencies, \(E _ { i }\), necessary for a \(\chi ^ { 2 }\) test.
2
\begin{enumerate}[label=(\alph*)]
\item The continuous random variable $X$ has a rectangular distribution defined by the probability density function
$$f ( x ) = \begin{cases} 0.01 \pi & u \leqslant x \leqslant 11 u \\ 0 & \text { otherwise } \end{cases}$$
where $u$ is a constant.
\begin{enumerate}[label=(\roman*)]
\item Show that $u = \frac { 10 } { \pi }$.
\item Using the formulae for the mean and the variance of a rectangular distribution, find, in terms of $\pi$, values for $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$.
\item Calculate exact values for the mean and the variance of the circumferences of circles having diameters of length $\left( X + \frac { 10 } { \pi } \right)$.
\end{enumerate}\item A machine produces circular discs which have an area of $Y \mathrm {~cm} ^ { 2 }$. The distribution of $Y$ has mean $\mu$ and variance 25 .
A random sample of 100 such discs is selected. The mean area of the discs in this sample is calculated to be $40.5 \mathrm {~cm} ^ { 2 }$.
Calculate a 95\% confidence interval for $\mu$.
Emily believed that the performances of 16-year-old students in their GCSEs are associated with the schools that they attend. To investigate her belief, Emily collected data on the GCSE results for 2010 from four schools in her area.
The table shows Emily's collected data, denoted by $O _ { i }$, together with the corresponding expected frequencies, $E _ { i }$, necessary for a $\chi ^ { 2 }$ test.
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|}
\hline
\multirow{2}{*}{} & \multicolumn{2}{|c|}{$\boldsymbol { \geq } \mathbf { 5 }$ GCSEs} & \multicolumn{2}{|c|}{$\mathbf { 1 } \boldsymbol { \leqslant }$ GCSEs < $\mathbf { 5 }$} & \multicolumn{2}{|c|}{No GCSEs} \\
\hline
& $O _ { i }$ & $E _ { i }$ & $O _ { i }$ & $E _ { i }$ & $O _ { i }$ & $E _ { i }$ \\
\hline
Jolliffe College for the Arts & 187 & 193.15 & 93 & 90.62 & 30 & 26.23 \\
\hline
Volpe Science Academy & 175 & 184.43 & 97 & 86.52 & 24 & 25.05 \\
\hline
Radok Music School & 183 & 183.81 & 78 & 86.23 & 34 & 24.96 \\
\hline
Bailey Language School & 265 & 248.61 & 112 & 116.63 & 22 & 33.76 \\
\hline
\end{tabular}
\end{center}
Emily used these values to correctly conduct a $\chi ^ { 2 }$ test at the $1 \%$ level of significance.
\end{enumerate}
\hfill \mbox{\textit{AQA S2 2011 Q2 [11]}}