| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2010 |
| Session | January |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Bivariate data |
| Type | Calculate summary statistics (Sxx, Syy, Sxy) |
| Difficulty | Moderate -0.8 This is a standard S1 bivariate data question requiring routine application of formulas for Stt, Spp, Stp, correlation coefficient, and regression line. All summations are provided, making it purely computational with no problem-solving or interpretation challenges beyond basic recall of standard procedures. |
| Spec | 2.02c Scatter diagrams and regression lines5.08a Pearson correlation: calculate pmcc5.09a Dependent/independent variables5.09b Least squares regression: concepts |
| Patient | A | B | C | D | E | F | G |
| \(t\) | 42 | 74 | 48 | 35 | 56 | 26 | 60 |
| \(p\) | 98 | 130 | 120 | 88 | 182 | 80 | 135 |
| Answer | Marks |
|---|---|
| Marks: M1 A1 A1 A1 | (4) |
| Answer | Marks |
|---|---|
| Marks: M1 A1ft A1 | (3) |
| Answer | Marks |
|---|---|
| Marks: B1 | (1) |
| Answer | Marks |
|---|---|
| Marks: B2 B1ft B1 B1ft B1 | (3) for (d); (2+2) for (f) |
| Answer | Marks |
|---|---|
| Marks: M1 A1 M1 M1 A1 | (4) |
| Answer | Marks |
|---|---|
| Marks: M1 A1 | (2) |
## Part (a)
**Answer:** $S_{pp} = 106397 - \frac{833^2}{7} = 7270$
$S_{tp} = 42948 - \frac{341 \times 833}{7} = 2369$, $S_{tt} = 18181 - \frac{341^2}{7} = 1569.42857...$ or $\frac{10986}{7}$
**Marks:** M1 A1 A1 A1 | (4)
**Guidance:**
- M1 for at least one correct expression
- 1st A1 for $S_{pp} = 7270$
- 2nd A1 for $S_{tp} = 2369$ or 2370
- 3rd A1 for $S_{tt} = $ awrt 1570
## Part (b)
**Answer:** $r = \frac{2369}{\sqrt{7270 \times 1569.42857...}} = 0.7013375$ awrt (0.701)
**Marks:** M1 A1ft A1 | (3)
**Guidance:**
- M1 for attempt at correct formula and at least one correct value (or correct ft) M0 for $\frac{42948}{\sqrt{106397 \times 18181}}$
- A1ft All values correct or correct ft. Allow for an answer of 0.7 or 0.70
- Answer only: awrt 0.701 is 3/3, answer of 0.7 or 0.70 is 2/3
## Part (c)
**Answer:** (Pmcc shows positive correlation.) Older patients have higher blood pressure
**Marks:** B1 | (1)
**Guidance:**
- B1 for comment in context that interprets the fact that correlation is positive, as in scheme. Must mention age and blood pressure in words, not just "r" and "p"
## Part (d) + (f)
**Answer:** (d) Points plotted correctly on graph: -1 each error or omission (within one square of correct position). * see diagram below for correct points
(f) Line drawn with correct intercept, and gradient
**Marks:** B2 B1ft B1 B1ft B1 | (3) for (d); (2+2) for (f)
**Guidance:**
- Record 1 point incorrect as B1B0 on open. [NB overlay for (60, 135) is slightly wrong]
## Part (e)
**Answer:** $b = \frac{2369}{1569.42857...} = 1.509466...$
$a = \frac{833}{7} - b \times \frac{341}{7} = 45.467413...$
$p = 45.5 + 1.5l r$
**Marks:** M1 A1 M1 M1 A1 | (4)
**Guidance:**
- 1st M1 for use of the correct formula for $b$, fit their values from (a)
- 1st A1 allow 1.5 or better
- 2nd M1 for use of $\bar{y} - b\bar{x}$ with their values
- 2nd A1 for full equation with $a = $ awrt 45.5 and $b = $ awrt 1.51. Must be $p$ in terms of $t$, not $x$ and $y$
## Part (g)
**Answer:** $t = 40$, $p = 105.84...$ from equation or graph.
**Marks:** M1 A1 | (2)
**Guidance:**
- M1 for clear use of their equation with $t = 40$ or correct value from their graph
- A1 for awrt 106. Correct answer only (2/2) otherwise look for evidence on graph to award M1
---The blood pressures, $p$ mmHg, and the ages, $t$ years, of 7 hospital patients are shown in the table below.
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
Patient & A & B & C & D & E & F & G \\
\hline
$t$ & 42 & 74 & 48 & 35 & 56 & 26 & 60 \\
\hline
$p$ & 98 & 130 & 120 & 88 & 182 & 80 & 135 \\
\hline
\end{tabular}
[$\sum t = 341$, $\sum p = 833$, $\sum t^2 = 18181$, $\sum p^2 = 106397$, $\sum tp = 42948$]
\begin{enumerate}[label=(\alph*)]
\item Find $S_{tt}$, $S_{pp}$ and $S_t$ for these data. [4]
\item Calculate the product moment correlation coefficient for these data. [3]
\item Interpret the correlation coefficient. [1]
\item On the graph paper on page 17, draw the scatter diagram of blood pressure against age for these 7 patients. [2]
\item Find the equation of the regression line of $p$ on $t$. [4]
\item Plot your regression line on your scatter diagram. [2]
\item Use your regression line to estimate the blood pressure of a 40 year old patient. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 2010 Q6 [18]}}