| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2013 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Modelling and Hypothesis Testing |
| Type | Variance calculation verification |
| Difficulty | Standard +0.3 Part (i) is routine calculation of mean and variance for a discrete uniform distribution using standard formulas. Parts (ii) and (iii) involve straightforward application of hypothesis testing concepts (Type I/II errors) with normal approximation to find a probability. The question requires recall of definitions and standard techniques but no novel problem-solving or insight. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(E(X) = 3.5\) | B1 | \(21/6\) |
| \((1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2) \div 6 - 3.5^2\) \(\left(= \frac{35}{12}\ \textbf{AG}\right)\) | B1 | [2] Must see correct expression and no incorrect working |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Attempt \(P(X < 3)\) or \(1 - P(X \geq 3)\) | M1 | Seen or implied |
| \(N\!\left(3.5,\ \frac{35}{12}/50\right)\) | M1 | Seen or implied |
| \(\frac{3 - 3.5}{\sqrt{\frac{35}{12}/50}}\ (= -2.070)\) | M1 | or \(\frac{2.99 - 3.5}{\sqrt{\frac{35}{12}/50}}\ (= -2.111)\) |
| \(\Phi(-2.070) = 1 - \Phi(2.070) = 0.0192\) | M1 | \(\Phi(-2.111) = 1 - \Phi(2.111) = 0.0174\) or \(0.0173\) |
| As final answer | A1 | [5] Allow with incorrect cc (e.g. 2.5) OR no \(\sqrt{\phantom{x}}\). Must have \(\div 50\) |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Die is biased (towards lower numbers) | B1 indep | Comment implying die is biased |
| Mean of 50 throws \(\geq 3\) (Allow \(> 3\)); or Equal nos of high and low scores; or More high scores | B1 indep | [2] Comment implying results of experiment do not indicate bias (or indicate bias towards higher numbers). Both must be in context. |
## Question 5(i):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $E(X) = 3.5$ | B1 | $21/6$ |
| $(1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2) \div 6 - 3.5^2$ $\left(= \frac{35}{12}\ \textbf{AG}\right)$ | B1 | **[2]** Must see correct expression and no incorrect working |
## Question 5(ii):
| Working/Answer | Mark | Guidance |
|---|---|---|
| Attempt $P(X < 3)$ or $1 - P(X \geq 3)$ | M1 | Seen or implied |
| $N\!\left(3.5,\ \frac{35}{12}/50\right)$ | M1 | Seen or implied |
| $\frac{3 - 3.5}{\sqrt{\frac{35}{12}/50}}\ (= -2.070)$ | M1 | or $\frac{2.99 - 3.5}{\sqrt{\frac{35}{12}/50}}\ (= -2.111)$ |
| $\Phi(-2.070) = 1 - \Phi(2.070) = 0.0192$ | M1 | $\Phi(-2.111) = 1 - \Phi(2.111) = 0.0174$ or $0.0173$ |
| As final answer | A1 | **[5]** Allow with incorrect cc (e.g. 2.5) OR no $\sqrt{\phantom{x}}$. Must have $\div 50$ |
## Question 5(iii):
| Working/Answer | Mark | Guidance |
|---|---|---|
| Die is biased (towards lower numbers) | B1 indep | Comment implying die is biased |
| Mean of 50 throws $\geq 3$ (Allow $> 3$); or Equal nos of high and low scores; or More high scores | B1 indep | **[2]** Comment implying results of experiment do not indicate bias (or indicate bias towards higher numbers). Both must be in context. |
---5 A fair six-sided die has faces numbered $1,2,3,4,5,6$. The score on one throw is denoted by $X$.\\
(i) Write down the value of $\mathrm { E } ( X )$ and show that $\operatorname { Var } ( X ) = \frac { 35 } { 12 }$.
Fayez has a six-sided die with faces numbered $1,2,3,4,5,6$. He suspects that it is biased so that when it is thrown it is more likely to show a low number than a high number. In order to test his suspicion, he plans to throw the die 50 times. If the mean score is less than 3 he will conclude that the die is biased.\\
(ii) Find the probability of a Type I error.\\
(iii) With reference to this context, describe circumstances in which Fayez would make a Type II error.
\hfill \mbox{\textit{CAIE S2 2013 Q5 [9]}}