| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/3 (Pre-U Mathematics Paper 3) |
| Year | 2016 |
| Session | June |
| Marks | 11 |
| Topic | Measures of Location and Spread |
| Type | Standard combined mean and SD |
| Difficulty | Moderate -0.3 This is a straightforward application of formulas for combining datasets and linear transformations of random variables. Part (i) requires calculating pooled mean and standard deviation using standard formulas (requiring knowledge that variance combines additively). Part (ii) applies the linear transformation rules (mean scales and shifts, SD only scales). While it involves multiple steps and careful arithmetic, it's entirely procedural with no problem-solving insight required—slightly easier than a typical A-level statistics question. |
| Spec | 2.02g Calculate mean and standard deviation5.02c Linear coding: effects on mean and variance |
| Answer | Marks | Guidance |
|---|---|---|
| \(\sum x = 20 \times 0.7 + 4 = 18\) | M1, A1 | Find \(\sum x\) for \(n = 20\). c.a.o. |
| \(\sum x^2 = 20(0.9^2 + 0.7^2) + 1 + 4 + 1 = 32\) | M1, A1 | Find \(\sum x^2\) for \(n = 20\). c.a.o. Allow M1A0 for either 25.19 or 31.19 seen, i.e. for use of unbiased sd provided used consistently later. |
| \(\bar{x} = \frac{18}{25} = 0.72\) kg | B1 | c.a.o. |
| \(s = \sqrt{\frac{32}{25} - 0.72^2}\) | M1 | Use of correct formula for standard deviation; may be implied. |
| \(\therefore s = \sqrt{0.7616} = 0.872(69\ldots) = 0.873\) kg | A1 | c.a.o. Allow M1A0 for unbiased sd (0.8715...) provided used consistently earlier. |
| Answer | Marks | Guidance |
|---|---|---|
| \(y = 65 + 25x\), \(\bar{y} = 65 + 25 \times 0.72 = 83\) | M1, A1 | \(y\) = total payment, \(x\) = goals scored. ft c's mean in (i). |
| \(s_y = 25 \times 0.873 = 21.825\) | M1, A1 | ft c's sd in (i). Accept 21.80 or 21.82 – 21.83, i.e. to nearest penny or better. |
### (i)
With no working shown allow only correct answers.
$\sum x = 20 \times 0.7 + 4 = 18$ | M1, A1 | Find $\sum x$ for $n = 20$. c.a.o.
$\sum x^2 = 20(0.9^2 + 0.7^2) + 1 + 4 + 1 = 32$ | M1, A1 | Find $\sum x^2$ for $n = 20$. c.a.o. Allow M1A0 for either 25.19 or 31.19 seen, i.e. for use of unbiased sd provided used consistently later.
$\bar{x} = \frac{18}{25} = 0.72$ kg | B1 | c.a.o.
$s = \sqrt{\frac{32}{25} - 0.72^2}$ | M1 | Use of correct formula for standard deviation; may be implied.
$\therefore s = \sqrt{0.7616} = 0.872(69\ldots) = 0.873$ kg | A1 | c.a.o. Allow M1A0 for unbiased sd (0.8715...) provided used consistently earlier. | [7]
### (ii)
$y = 65 + 25x$, $\bar{y} = 65 + 25 \times 0.72 = 83$ | M1, A1 | $y$ = total payment, $x$ = goals scored. ft c's mean in (i).
$s_y = 25 \times 0.873 = 21.825$ | M1, A1 | ft c's sd in (i). Accept 21.80 or 21.82 – 21.83, i.e. to nearest penny or better. | [4]
---Chris plays for his local hockey club. In his first 20 games for the club, the mean number of goals per game he has scored is $0.7$, with a standard deviation of $0.9$.
In the next 5 games he scores $0, 1, 0, 2, 1$ goals.
\begin{enumerate}[label=(\roman*)]
\item Find the mean and standard deviation for the number of goals per game Chris has scored in all 25 games. [7]
\item A sponsor pays Chris £65 each time he plays for the club and a further £25 for each goal he scores. Find the mean and standard deviation of the amount per game he earns from the sponsor for all 25 games. [4]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/3 2016 Q3 [11]}}