| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2012 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Measures of Location and Spread |
| Type | Calculate statistics from raw data |
| Difficulty | Moderate -0.8 This is a straightforward probability and statistics question requiring basic calculations: finding mean/SD from a discrete distribution, constructing a probability table by enumeration, and applying standard binomial distribution formulas. All parts use routine methods with no problem-solving insight needed, making it easier than average but not trivial due to the multi-part nature and careful enumeration required in part (ii). |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables |
| Answer | Marks | Guidance |
|---|---|---|
| (i) mean = \(11/6\) (\(\frac{5}{6}\), 1.83); \(sd = \sqrt{(1 + 1 + 4 + 9 + 9)/6 - (11/6)^2} = \sqrt{29/6}\) (0.898) | B1 M1 A1 [3] | correct answer; numerical use of a correct sd/variance formula; correct answer |
| (ii) \(\begin{array}{c | cccccc} x & 2 & 3 & 4 & 5 & 6 \\ \hline Pr & 9/36 & 6/36 & 13/36 & 4/36 & 4/36 \end{array}\) | B1 B1 M1 A1 [4] |
| (iii) \(p = 1/3\); \(np = 8\); \(n = 24\); \(Var = 24 \times 1/3 \times 2/3 = 16/3\) (5.33) | B1 M1 A1ft [3] | correct \(p\); using \(np = 8\) to find \(n\) or \(8(1 - p)\) to find var, \(0 |
(i) mean = $11/6$ ($\frac{5}{6}$, 1.83); $sd = \sqrt{(1 + 1 + 4 + 9 + 9)/6 - (11/6)^2} = \sqrt{29/6}$ (0.898) | B1 M1 A1 [3] | correct answer; numerical use of a correct sd/variance formula; correct answer
(ii) $\begin{array}{c|cccccc} x & 2 & 3 & 4 & 5 & 6 \\ \hline Pr & 9/36 & 6/36 & 13/36 & 4/36 & 4/36 \end{array}$ | B1 B1 M1 A1 [4] | all correct $x$ values; P(2) and P(6) correct; considering more than 1 case for a sum of 3 or 4 or 5; P(3), P(4) and P(5) correct
(iii) $p = 1/3$; $np = 8$; $n = 24$; $Var = 24 \times 1/3 \times 2/3 = 16/3$ (5.33) | B1 M1 A1ft [3] | correct $p$; using $np = 8$ to find $n$ or $8(1 - p)$ to find var, $0<p<1$; correct answer, ft their $p$4 The six faces of a fair die are numbered $1,1,1,2,3,3$. The score for a throw of the die, denoted by the random variable $W$, is the number on the top face after the die has landed.\\
(i) Find the mean and standard deviation of $W$.\\
(ii) The die is thrown twice and the random variable $X$ is the sum of the two scores. Draw up a probability distribution table for $X$.\\
(iii) The die is thrown $n$ times. The random variable $Y$ is the number of times that the score is 3 . Given that $\mathrm { E } ( Y ) = 8$, find $\operatorname { Var } ( Y )$.
\hfill \mbox{\textit{CAIE S1 2012 Q4 [10]}}