| Exam Board | AQA |
|---|---|
| Module | Further AS Paper 2 Statistics (Further AS Paper 2 Statistics) |
| Year | 2022 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Measures of Location and Spread |
| Type | Calculate statistics from discrete frequency table |
| Difficulty | Easy -1.2 This is a straightforward application of standard formulas for discrete probability distributions. Part (a) requires finding the median by accumulating probabilities (routine), part (b) uses the standard deviation formula E(X²) - [E(X)]² (standard calculation), and part (c) applies the variance transformation rule Var(aX+b) = a²Var(X) (direct recall). All three parts are textbook exercises with no problem-solving or insight required, making this easier than average. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Median \(= 1\) | B1 | Obtains correct value of the median of \(A\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(E(A) = 0 \times 0.45 + 1 \times 0.25 + 2 \times 0.3\) | M1 | Uses correct formula for \(E(A)\) or \(E(A^2)\) |
| \(E(A) = 0.85\) | A1 | Obtains correct value of \(E(A)\) or \(E(A^2)\); oe PI by correct variance or standard deviation |
| \(E(A^2) = 0^2 \times 0.45 + 1^2 \times 0.25 + 2^2 \times 0.3 = 1.45\) | M1 | Uses correct formula for \(\text{Var}(A)\) or standard deviation of \(A\) with their values for \(E(A)\) and \(E(A^2)\) |
| \(\text{Var}(A) = 1.45 - 0.85^2 = 0.7275\) | ||
| Standard deviation \(= \sqrt{0.7275} = 0.853\) | A1 | AWRT 0.853 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\text{Var}(9A-2) = 9^2\,\text{Var}(A)\) | M1 | Uses correct formula for \(\text{Var}(9A-2)\) with their variance; condone substitution of their standard deviation for \(\text{Var}(A)\) provided formula stated |
| \(\text{Var}(9A-2) = 9^2 \times 0.7275 = 58.9275\) | A1F | AWRT 58.9 oe; FT their variance or their standard deviation squared multiplied by 81 from 3(b) to at least 3 s.f. |
## Question 3(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Median $= 1$ | B1 | Obtains correct value of the median of $A$ |
## Question 3(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $E(A) = 0 \times 0.45 + 1 \times 0.25 + 2 \times 0.3$ | M1 | Uses correct formula for $E(A)$ or $E(A^2)$ |
| $E(A) = 0.85$ | A1 | Obtains correct value of $E(A)$ or $E(A^2)$; oe PI by correct variance or standard deviation |
| $E(A^2) = 0^2 \times 0.45 + 1^2 \times 0.25 + 2^2 \times 0.3 = 1.45$ | M1 | Uses correct formula for $\text{Var}(A)$ or standard deviation of $A$ with their values for $E(A)$ and $E(A^2)$ |
| $\text{Var}(A) = 1.45 - 0.85^2 = 0.7275$ | | |
| Standard deviation $= \sqrt{0.7275} = 0.853$ | A1 | AWRT 0.853 |
## Question 3(c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\text{Var}(9A-2) = 9^2\,\text{Var}(A)$ | M1 | Uses correct formula for $\text{Var}(9A-2)$ with their variance; condone substitution of their standard deviation for $\text{Var}(A)$ provided formula stated |
| $\text{Var}(9A-2) = 9^2 \times 0.7275 = 58.9275$ | A1F | AWRT 58.9 oe; FT their variance or their standard deviation squared multiplied by 81 from 3(b) to at least 3 s.f. |
---3 The discrete random variable $A$ has the following probability distribution function
$$\mathrm { P } ( A = a ) = \begin{cases} 0.45 & a = 0 \\ 0.25 & a = 1 \\ 0.3 & a = 2 \\ 0 & \text { otherwise } \end{cases}$$
3
\begin{enumerate}[label=(\alph*)]
\item Find the median of $A$\\
3
\item Find the standard deviation of $A$, giving your answer to three significant figures.\\
3
\item $\quad$ Find $\operatorname { Var } ( 9 A - 2 )$
\end{enumerate}
\hfill \mbox{\textit{AQA Further AS Paper 2 Statistics 2022 Q3 [7]}}