| Exam Board | AQA |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2006 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of Pearson’s product-moment correlation coefficient |
| Type | One-tailed test for positive correlation |
| Difficulty | Standard +0.3 This is a straightforward application of standard S3 procedures: calculate PMCC using the formula (or calculator), then perform a one-tailed hypothesis test by comparing to critical values from tables. While it requires careful arithmetic and knowledge of the test procedure, it involves no conceptual challenges or novel problem-solving—just methodical execution of a textbook technique, making it slightly easier than average. |
| Spec | 5.08a Pearson correlation: calculate pmcc5.08d Hypothesis test: Pearson correlation |
| Patient | \(\mathbf { 1 }\) | \(\mathbf { 2 }\) | \(\mathbf { 3 }\) | \(\mathbf { 4 }\) | \(\mathbf { 5 }\) | \(\mathbf { 6 }\) | \(\mathbf { 7 }\) | \(\mathbf { 8 }\) | \(\mathbf { 9 }\) | \(\mathbf { 1 0 }\) |
| \(\boldsymbol { x }\) | 83 | 86 | 88 | 92 | 94 | 98 | 101 | 111 | 115 | 121 |
| \(\boldsymbol { y }\) | 157 | 172 | 161 | 154 | 171 | 169 | 179 | 180 | 192 | 182 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(r = 0.819\) to \(0.82\) | B3 | AWFW |
| or \(r = 0.81\) to \(0.83\) | (B2) | AWFW |
| or \(r = 0.8\) to \(0.85\) | (B1) | AWFW |
| Attempt at \(\Sigma x,\ \Sigma x^2,\ \Sigma y,\ \Sigma y^2,\ \Sigma xy\) or attempt at \(S_{xx},\ S_{yy},\ S_{xy}\) | (M1) | Values: 989, 99321, 1717, 296101, 170956 or 1508.9, 1292.1, 1144.7 |
| Attempt at correct formula for \(r\) | (m1) | |
| \(r = 0.819\) to \(0.82\) | (A1) | AWFW |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(H_0: \rho = 0\), \(H_1: \rho > 0\) | B1 | Both |
| CV \(r = 0.7155\) at \(\alpha = 0.01\) (1%), \(n = 10\) | B1 | AWFW 0.715 to 0.716 |
| Calculated \(r >\) Tabulated \(r\) | M1 | Comparison |
| Evidence (at 1% level) of a positive correlation between heart rate and systolic blood pressure | A1\(\checkmark\) | \(\checkmark\) on \(r\) and CV |
# Question 2:
## Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $r = 0.819$ to $0.82$ | B3 | AWFW |
| or $r = 0.81$ to $0.83$ | (B2) | AWFW |
| or $r = 0.8$ to $0.85$ | (B1) | AWFW |
| Attempt at $\Sigma x,\ \Sigma x^2,\ \Sigma y,\ \Sigma y^2,\ \Sigma xy$ or attempt at $S_{xx},\ S_{yy},\ S_{xy}$ | (M1) | Values: 989, 99321, 1717, 296101, 170956 or 1508.9, 1292.1, 1144.7 |
| Attempt at correct formula for $r$ | (m1) | |
| $r = 0.819$ to $0.82$ | (A1) | AWFW |
## Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $H_0: \rho = 0$, $H_1: \rho > 0$ | B1 | Both |
| CV $r = 0.7155$ at $\alpha = 0.01$ (1%), $n = 10$ | B1 | AWFW 0.715 to 0.716 |
| Calculated $r >$ Tabulated $r$ | M1 | Comparison |
| Evidence (at 1% level) of a positive correlation between heart rate and systolic blood pressure | A1$\checkmark$ | $\checkmark$ on $r$ and CV |
---2 The table below shows the heart rates, $x$ beats per minute, and the systolic blood pressures, $y$ milligrams of mercury, of a random sample of 10 patients undergoing kidney dialysis.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | c | c | c | }
\hline
Patient & $\mathbf { 1 }$ & $\mathbf { 2 }$ & $\mathbf { 3 }$ & $\mathbf { 4 }$ & $\mathbf { 5 }$ & $\mathbf { 6 }$ & $\mathbf { 7 }$ & $\mathbf { 8 }$ & $\mathbf { 9 }$ & $\mathbf { 1 0 }$ \\
\hline
$\boldsymbol { x }$ & 83 & 86 & 88 & 92 & 94 & 98 & 101 & 111 & 115 & 121 \\
\hline
$\boldsymbol { y }$ & 157 & 172 & 161 & 154 & 171 & 169 & 179 & 180 & 192 & 182 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Calculate the value of the product moment correlation coefficient for these data.
\item Assuming that these data come from a bivariate normal distribution, investigate, at the $1 \%$ level of significance, the claim that, for patients undergoing kidney dialysis, there is a positive correlation between heart rate and systolic blood pressure.
\end{enumerate}
\hfill \mbox{\textit{AQA S3 2006 Q2 [7]}}