| Exam Board | OCR |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2007 |
| Session | January |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Modal value or most probable value |
| Difficulty | Moderate -0.3 This question tests basic understanding of distribution properties through visual interpretation. Part (i) requires recognizing symmetry and skewness from diagrams (standard S1 skill), part (ii) needs recall that geometric distributions are monotonically decreasing, and part (iii) requires knowing binomial distributions are unimodal. All parts are conceptual recognition rather than calculation, making it slightly easier than average but still requiring proper understanding of distribution characteristics. |
| Spec | 2.02f Measures of average and spread5.02a Discrete probability distributions: general |
| Answer | Marks | Guidance |
|---|---|---|
| i a) W & Y oe | B1 | 1 mark |
| i b) X oe | B1 | 1 mark |
| ii) Geo probs always decrease or Geo has no upper limit to x or \(x \neq 0\) | B1 | 1 mark; Geo not fixed no. of values; diags have fixed no of trials; not Geo has +ve skew |
| iii) W | B1 | indep; allow Bin probs rise then fall |
| Bin probs cannot fall then rise or bimodal | B1dep | 2 marks |
**i a)** W & Y oe | B1 | 1 mark
**i b)** X oe | B1 | 1 mark
**ii)** Geo probs always decrease or Geo has no upper limit to x or $x \neq 0$ | B1 | 1 mark; Geo not fixed no. of values; diags have fixed no of trials; not Geo has +ve skew
**iii)** W | B1 | indep; allow Bin probs rise then fall
Bin probs cannot fall then rise or bimodal | B1dep | 2 marks
**Total for Question 4:** 5 marks4 Each of the variables $W , X , Y$ and $Z$ takes eight integer values only. The probability distributions are illustrated in the following diagrams.\\
\includegraphics[max width=\textwidth, alt={}, center]{43f7e091-9ae7-4373-a209-e2ebdba5260f-3_437_394_397_280}\\
\includegraphics[max width=\textwidth, alt={}, center]{43f7e091-9ae7-4373-a209-e2ebdba5260f-3_433_380_397_685}\\
\includegraphics[max width=\textwidth, alt={}, center]{43f7e091-9ae7-4373-a209-e2ebdba5260f-3_428_383_402_1082}\\
\includegraphics[max width=\textwidth, alt={}, center]{43f7e091-9ae7-4373-a209-e2ebdba5260f-3_425_376_402_1482}\\
(i) For which one or more of these variables is
\begin{enumerate}[label=(\alph*)]
\item the mean equal to the median,
\item the mean greater than the median?\\
(ii) Give a reason why none of these diagrams could represent a geometric distribution.\\
(iii) Which one of these diagrams could not represent a binomial distribution? Explain your answer briefly.
\end{enumerate}
\hfill \mbox{\textit{OCR S1 2007 Q4 [5]}}