| Exam Board | OCR MEI |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2009 |
| Session | January |
| Marks | 20 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of Spearman’s rank correlation coefficien |
| Type | Hypothesis test for association |
| Difficulty | Moderate -0.3 This is a straightforward application of standard S2 procedures: ranking data and calculating Spearman's coefficient using the formula, performing a routine hypothesis test with critical values, and computing a regression line from given summaries. All steps are mechanical with no conceptual challenges or novel problem-solving required, making it slightly easier than average for A-level statistics. |
| Spec | 5.08e Spearman rank correlation5.08f Hypothesis test: Spearman rank5.09a Dependent/independent variables5.09b Least squares regression: concepts5.09c Calculate regression line |
| \(x\) | 18 | 43 | 52 | 94 | 98 | 206 | 784 | 1530 |
| \(y\) | 1.15 | 0.97 | 1.26 | 1.35 | 1.28 | 1.42 | 1.32 | 1.64 |
| \(t\) | 20 | 30 | 40 | 50 | 60 | 70 |
| \(w\) | 2.49 | 2.41 | 2.38 | 2.14 | 1.97 | 2.03 |
| Answer | Marks | Guidance |
|---|---|---|
| (A) For \(t = 80\), predicted speed \(= -0.011 \times 80 + 2.73 = 1.85\) | M1, A1 FT provided \(b < 0\) | — |
| (B) The relationship relates to adults, but a ten year old will not be fully grown so may walk more slowly. | E1 extrapolation o.e., E1 sensible contextual comment | — |
| TOTAL | 20 |
## Part (iv)
(A) For $t = 80$, predicted speed $= -0.011 \times 80 + 2.73 = 1.85$ | M1, A1 FT provided $b < 0$ | —
(B) The relationship relates to adults, but a ten year old will not be fully grown so may walk more slowly. | E1 extrapolation o.e., E1 sensible contextual comment | —
| | | **TOTAL** | **20** |
---1 A researcher is investigating whether there is a relationship between the population size of cities and the average walking speed of pedestrians in the city centres. Data for the population size, $x$ thousands, and the average walking speed of pedestrians, $y \mathrm {~m} \mathrm {~s} ^ { - 1 }$, of eight randomly selected cities are given in the table below.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | c | }
\hline
$x$ & 18 & 43 & 52 & 94 & 98 & 206 & 784 & 1530 \\
\hline
$y$ & 1.15 & 0.97 & 1.26 & 1.35 & 1.28 & 1.42 & 1.32 & 1.64 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\roman*)]
\item Calculate the value of Spearman's rank correlation coefficient.
\item Carry out a hypothesis test at the $5 \%$ significance level to determine whether there is any association between population size and average walking speed.
In another investigation, the researcher selects a random sample of six adult males of particular ages and measures their maximum walking speeds. The data are shown in the table below, where $t$ years is the age of the adult and $w \mathrm {~m} \mathrm {~s} ^ { - 1 }$ is the maximum walking speed. Also shown are summary statistics and a scatter diagram on which the regression line of $w$ on $t$ is drawn.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | }
\hline
$t$ & 20 & 30 & 40 & 50 & 60 & 70 \\
\hline
$w$ & 2.49 & 2.41 & 2.38 & 2.14 & 1.97 & 2.03 \\
\hline
\end{tabular}
\end{center}
$$n = 6 \quad \Sigma t = 270 \quad \Sigma w = 13.42 \quad \Sigma t ^ { 2 } = 13900 \quad \Sigma w ^ { 2 } = 30.254 \quad \Sigma t w = 584.6$$
\includegraphics[max width=\textwidth, alt={}, center]{77b97142-afb6-41d6-8fec-e982b7a7501b-2_728_1091_1379_529}
\item Calculate the equation of the regression line of $w$ on $t$.
\item (A) Use this equation to calculate an estimate of maximum walking speed of an 80 -year-old male.\\
(B) Explain why it might not be appropriate to use the equation to calculate an estimate of maximum walking speed of a 10 -year-old male.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI S2 2009 Q1 [20]}}