| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2004 |
| Session | November |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Bivariate data |
| Type | Calculate summary statistics (Sxx, Syy, Sxy) |
| Difficulty | Easy -1.2 This is a routine S1 question testing standard formulas for summary statistics (Sxx, Syy, Sxy), correlation coefficient, and basic normal distribution. All parts require direct application of memorized formulas with straightforward arithmetic—no problem-solving insight or novel reasoning required. Easier than average A-level. |
| Spec | 2.02f Measures of average and spread2.02g Calculate mean and standard deviation2.04e Normal distribution: as model N(mu, sigma^2)5.08a Pearson correlation: calculate pmcc5.08b Linear coding: effect on pmcc5.08c Pearson: measure of straight-line fit |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(S_{xx} = 10164 - \frac{272^2}{8} = 916\) | M1A1 | Any one method, cao |
| \(S_{yy} = 13464 - \frac{320^2}{8} = 664\) | A1 | cao |
| \(S_{xy} = 11222 - \frac{272 \times 320}{8} = 342\) | A1 | cao (Or 114.5, 83 & 42.75) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(r = \frac{342}{\sqrt{916 \times 664}} = 0.43852...\) | M1A1\(\int\)A1 | formula all correct \((\sqrt{608224})\); 0.439 |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| Slight/weak evidence | B1 | |
| Students perform similarly in pressups and situps; context for +ve | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(\bar{x} = \frac{272}{8} = 34\) | M1A1 | |
| \(s = \sqrt{\frac{10164}{8} - 34^2} = \sqrt{114.5} = 10.700...\) | M1A1 | method includes \(\sqrt{\phantom{x}}\); awrt 10.7 |
| OR divisor \((n-1)\) awrt 11.4 |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(a = 1.96 \times 10.700... = 20.9729...\) (or 22.4 divisor \((n-1)\)) | B1, M1A1 | 1.96 seen; \(1.96 \times s\); 21.0 or 22.4 |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| Pressups discrete, Normal continuous | B1 | |
| Not a very good assumption | B1 dep |
## Question 6:
### Part (a):
| Working | Mark | Guidance |
|---------|------|----------|
| $S_{xx} = 10164 - \frac{272^2}{8} = 916$ | M1A1 | Any one method, cao |
| $S_{yy} = 13464 - \frac{320^2}{8} = 664$ | A1 | cao |
| $S_{xy} = 11222 - \frac{272 \times 320}{8} = 342$ | A1 | cao (Or 114.5, 83 & 42.75) |
### Part (b):
| Working | Mark | Guidance |
|---------|------|----------|
| $r = \frac{342}{\sqrt{916 \times 664}} = 0.43852...$ | M1A1$\int$A1 | formula all correct $(\sqrt{608224})$; 0.439 |
### Part (c):
| Working | Mark | Guidance |
|---------|------|----------|
| Slight/weak evidence | B1 | |
| Students perform similarly in pressups and situps; context for +ve | B1 | |
### Part (d):
| Working | Mark | Guidance |
|---------|------|----------|
| $\bar{x} = \frac{272}{8} = 34$ | M1A1 | |
| $s = \sqrt{\frac{10164}{8} - 34^2} = \sqrt{114.5} = 10.700...$ | M1A1 | method includes $\sqrt{\phantom{x}}$; awrt 10.7 |
| OR divisor $(n-1)$ awrt 11.4 | | |
### Part (e):
| Working | Mark | Guidance |
|---------|------|----------|
| $a = 1.96 \times 10.700... = 20.9729...$ (or 22.4 divisor $(n-1)$) | B1, M1A1 | 1.96 seen; $1.96 \times s$; 21.0 or 22.4 |
### Part (f):
| Working | Mark | Guidance |
|---------|------|----------|
| Pressups discrete, Normal continuous | B1 | |
| Not a very good assumption | B1 dep | |
---6. Students in Mr Brawn's exercise class have to do press-ups and sit-ups. The number of press-ups $x$ and the number of sit-ups $y$ done by a random sample of 8 students are summarised below.
$$\begin{array} { l l }
\Sigma x = 272 , & \Sigma x ^ { 2 } = 10164 , \quad \Sigma x y = 11222 , \\
\Sigma y = 320 , & \Sigma y ^ { 2 } = 13464 .
\end{array}$$
\begin{enumerate}[label=(\alph*)]
\item Evaluate $S _ { x x } , S _ { y y }$ and $S _ { x y }$.
\item Calculate, to 3 decimal places, the product moment correlation coefficient between $x$ and $y$.
\item Give an interpretation of your coefficient.
\item Calculate the mean and the standard deviation of the number of press-ups done by these students.
Mr Brawn assumes that the number of press-ups that can be done by any student can be modelled by a normal distribution with mean $\mu$ and standard deviation $\sigma$. Assuming that $\mu$ and $\sigma$ take the same values as those calculated in part (d),
\item find the value of $a$ such that $\mathrm { P } ( \mu - a < X < \mu + a ) = 0.95$.
\item Comment on Mr Brawn's assumption of normality.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 2004 Q6 [18]}}