Moderate -0.8 This is a straightforward S3 question testing standard techniques. Part (a) involves routine integration to find k, sketching a linear pdf, and basic interpretation. The CDF calculation and probability are mechanical applications of integration. The median requires solving a quadratic but is standard. Part (b) tests textbook definitions and explanations of sampling methods with no mathematical content. All parts are direct recall and application of standard methods with no problem-solving or novel insight required.
Sarah travels home from work each evening by bus; there is a bus every 20 minutes. The time at which Sarah arrives at the bus stop varies randomly in such a way that the probability density function of \(X\), the length of time in minutes she has to wait for the next bus, is given by
$$f(x) = k(20-x) \text{ for } 0 \leq x \leq 20, \text{ where } k \text{ is a constant.}$$
Find \(k\). Sketch the graph of \(f(x)\) and use its shape to explain what can be deduced about how long Sarah has to wait. [5]
Find the cumulative distribution function of \(X\) and hence, or otherwise, find the probability that Sarah has to wait more than 10 minutes for the bus. [4]
Find the median length of time that Sarah has to wait. [3]
Define the term 'simple random sample'. [2]
Explain briefly how to carry out cluster sampling. [3]
A researcher wishes to investigate the attitudes of secondary school pupils to pollution. Explain why he might prefer to collect his data using a cluster sample rather than a simple random sample. [2]
\begin{enumerate}[label=(\alph*)]
\item Sarah travels home from work each evening by bus; there is a bus every 20 minutes. The time at which Sarah arrives at the bus stop varies randomly in such a way that the probability density function of $X$, the length of time in minutes she has to wait for the next bus, is given by
$$f(x) = k(20-x) \text{ for } 0 \leq x \leq 20, \text{ where } k \text{ is a constant.}$$
\begin{enumerate}[label=(\roman*)]
\item Find $k$. Sketch the graph of $f(x)$ and use its shape to explain what can be deduced about how long Sarah has to wait. [5]
\item Find the cumulative distribution function of $X$ and hence, or otherwise, find the probability that Sarah has to wait more than 10 minutes for the bus. [4]
\item Find the median length of time that Sarah has to wait. [3]
\end{enumerate}
\item \begin{enumerate}[label=(\roman*)]
\item Define the term 'simple random sample'. [2]
\item Explain briefly how to carry out cluster sampling. [3]
\item A researcher wishes to investigate the attitudes of secondary school pupils to pollution. Explain why he might prefer to collect his data using a cluster sample rather than a simple random sample. [2]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{OCR MEI S3 2008 Q1 [19]}}