| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2005 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Bivariate data |
| Type | Hypothesis test for correlation |
| Difficulty | Standard +0.3 This is a standard S3 correlation question following a textbook template: draw scatter diagram, calculate correlation coefficient using given summations, and perform a hypothesis test. All steps are routine applications of formulae with no problem-solving or interpretation challenges beyond basic commentary. |
| Spec | 2.02c Scatter diagrams and regression lines5.08a Pearson correlation: calculate pmcc5.08d Hypothesis test: Pearson correlation |
| \(s\) | 1.2 | 1.9 | 3.2 | 3.9 | 2.5 | 4.5 | 5.7 | 4.0 | 1.1 | 5.9 |
| \(d\) | 3.8 | 7.0 | 11.0 | 12.0 | 9.0 | 12.0 | 13.5 | 12.2 | 2.0 | 13.9 |
| Answer | Marks | Guidance |
|---|---|---|
| Content | Marks | Guidance |
| (a) Scatter diagram with scales and labels | B1, B2 | Points (8,9 point) |
| (b) Linear association between \(x\) and \(d\) | B1 (i) | |
| (c) \(S_{xx} = 141.51 - \frac{35.3^2}{10} = 26.589\); \(S_{dx} = 152.44\); \(S_{dd} = 59.524\) | B1; B1; B1 (3) | |
| (d) \(T = \frac{59.524}{\sqrt{152.44 \times 26.569}}\) \(= 0.93494...\) AWET \(0.939\) | M1, A1 (2) | |
| (e) \(H_0: \rho = 0\); \(H_1: \rho > 0\) | B1, B1 | |
| B1 (3) | ||
| Critical Value at \(H_0 = 0.7155\) Reject \(H_0\); Level of serum \(\lambda\) disease on positively correlated | B1 (3) | |
| (f) Linear correlation significant but scatter diagram looks non-linear. | B1 (i) |
| Content | Marks | Guidance |
|---------|-------|----------|
| **(a) Scatter diagram with scales and labels** | B1, B2 | Points (8,9 point) |
| **(b) Linear association between $x$ and $d$** | B1 (i) | |
| **(c)** $S_{xx} = 141.51 - \frac{35.3^2}{10} = 26.589$; $S_{dx} = 152.44$; $S_{dd} = 59.524$ | B1; B1; B1 (3) | |
| **(d)** $T = \frac{59.524}{\sqrt{152.44 \times 26.569}}$ $= 0.93494...$ **AWET** $0.939$ | M1, A1 (2) | |
| **(e)** $H_0: \rho = 0$; $H_1: \rho > 0$ | B1, B1 | |
| | B1 (3) | |
| **Critical Value at $H_0 = 0.7155$** Reject $H_0$; Level of serum $\lambda$ disease on positively correlated | B1 (3) | |
| **(f)** Linear correlation significant but scatter diagram looks non-linear. | B1 (i) | |
---Over a period of time, researchers took 10 blood samples from one patient with a blood disease. For each sample, they measured the levels of serum magnesium, $s$ mg/dl, in the blood and the corresponding level of the disease protein, $d$ mg/dl. The results are shown in the table.
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
$s$ & 1.2 & 1.9 & 3.2 & 3.9 & 2.5 & 4.5 & 5.7 & 4.0 & 1.1 & 5.9 \\
\hline
$d$ & 3.8 & 7.0 & 11.0 & 12.0 & 9.0 & 12.0 & 13.5 & 12.2 & 2.0 & 13.9 \\
\hline
\end{tabular}
\end{center}
[Use $\sum s^2 = 141.51$, $\sum d^2 = 1081.74$ and $\sum sd = 386.32$]
\begin{enumerate}[label=(\alph*)]
\item Draw a scatter diagram to represent these data. [3]
\item State what is measured by the product moment correlation coefficient. [1]
\item Calculate $S_{ss}$, $S_{dd}$ and $S_{sd}$. [3]
\item Calculate the value of the product moment correlation coefficient $r$ between $s$ and $d$. [2]
\item Stating your hypotheses clearly, test, at the 1\% significance level, whether or not the correlation coefficient is greater than zero. [3]
\item With reference to your scatter diagram, comment on your result in part (e). [1]
\end{enumerate}
(Total 13 marks)
\hfill \mbox{\textit{Edexcel S3 2005 Q4 [13]}}