| Exam Board | OCR |
|---|---|
| Module | H240/02 (Pure Mathematics and Statistics) |
| Year | 2020 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Mixed calculations with boundaries |
| Difficulty | Moderate -0.3 This is a straightforward statistics question testing standard A-level techniques: calculating mean/SD from grouped data, using normal distribution tables/calculator for probabilities and inverse problems, and commenting on model appropriateness. All parts are routine applications of learned procedures with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.02f Measures of average and spread2.02g Calculate mean and standard deviation2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
| Mass (g) | \(50 \leq m < 150\) | \(150 \leq m < 200\) | \(200 \leq m < 250\) | \(250 \leq m < 350\) |
| Frequency | 162 | 318 | 355 | 165 |
As part of a research project, the masses, $m$ grams, of a random sample of 1000 pebbles from a certain beach were recorded. The results are summarised in the table.
\begin{tabular}{|c|c|c|c|c|}
\hline
Mass (g) & $50 \leq m < 150$ & $150 \leq m < 200$ & $200 \leq m < 250$ & $250 \leq m < 350$ \\
\hline
Frequency & 162 & 318 & 355 & 165 \\
\hline
\end{tabular}
\begin{enumerate}[label=(\alph*)]
\item Calculate estimates of the mean and standard deviation of these masses. [2]
\end{enumerate}
The masses, $x$ grams, of a random sample of 1000 pebbles on a different beach were also found. It was proposed that the distribution of these masses should be modelled by the random variable $X \sim N(200, 3600)$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Use the model to find $P(150 < X < 210)$. [1]
\item Use the model to determine $x_1$ such that $P(160 < X < x_1) = 0.6$, giving your answer correct to five significant figures. [3]
\end{enumerate}
It was found that the smallest and largest masses of the pebbles in this second sample were 112 g and 288 g respectively.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item Use these results to show that the model may not be appropriate. [1]
\item Suggest a different value of a parameter of the model in the light of these results. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR H240/02 2020 Q11 [9]}}