| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2020 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Modelling and Hypothesis Testing |
| Type | Type I and Type II errors |
| Difficulty | Standard +0.3 This is a straightforward hypothesis testing question requiring calculation of variance (routine), finding P(Type I error) using normal approximation with given variance, and stating the definition of Type II error. All steps are standard textbook procedures with no novel insight required, making it slightly easier than average. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables5.05a Sample mean distribution: central limit theorem |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \((1^2 + 2^2 + 3^2 + 4^2 + 5^2) \div 5 - 3^2\ (= 2\ \mathbf{AG})\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(N(3,\ 2)\) | M1 | |
| \(\dfrac{2.6 - \text{"3"}}{\sqrt{\dfrac{2}{40}}}\) \((= -1.789)\) | M1 | |
| \(\Phi(\text{"}-1.789\text{"}) = 1 - \Phi(\text{"1.789"})\) | M1 | |
| \(0.0367\) to \(0.0368\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Concluding that spinner is unbiased when it is biased | B1 |
## Question 4:
**Part 4(a):**
| Answer | Mark | Guidance |
|--------|------|----------|
| $(1^2 + 2^2 + 3^2 + 4^2 + 5^2) \div 5 - 3^2\ (= 2\ \mathbf{AG})$ | B1 | |
**Part 4(b):**
| Answer | Mark | Guidance |
|--------|------|----------|
| $N(3,\ 2)$ | M1 | |
| $\dfrac{2.6 - \text{"3"}}{\sqrt{\dfrac{2}{40}}}$ $(= -1.789)$ | M1 | |
| $\Phi(\text{"}-1.789\text{"}) = 1 - \Phi(\text{"1.789"})$ | M1 | |
| $0.0367$ to $0.0368$ | A1 | |
**Part 4(c):**
| Answer | Mark | Guidance |
|--------|------|----------|
| Concluding that spinner is unbiased when it is biased | B1 | |
---4 A fair spinner has five sides numbered $1,2,3,4,5$. The score on one spin is denoted by $X$.
\begin{enumerate}[label=(\alph*)]
\item Show that $\operatorname { Var } ( X ) = 2$.\\
Fiona has another spinner, also with five sides numbered $1,2,3,4,5$. She suspects that it is biased so that the expected score is less than 3 . In order to test her suspicion, she plans to spin her spinner 40 times. If the mean score is less than 2.6 she will conclude that her spinner is biased in this way.
\item Find the probability of a Type I error.
\item State what is meant by a Type II error in this context.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2020 Q4 [6]}}