| Exam Board | OCR |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2016 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of Spearman’s rank correlation coefficien |
| Type | Sketch scatter diagram scenarios |
| Difficulty | Moderate -0.3 This question tests understanding of Spearman's rank correlation through visual interpretation and basic calculation. Part (i) requires recognizing perfect rank correlation from a diagram (straightforward), part (ii) tests conceptual understanding of when r_s ≠ r (requires insight but is a standard S1 concept), and part (iii) involves a routine calculation of r_s from given data. While it requires understanding the distinction between correlation coefficients, the calculations are mechanical and the concepts are core S1 material, making it slightly easier than average. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities5.08e Spearman rank correlation5.08f Hypothesis test: Spearman rank5.08g Compare: Pearson vs Spearman |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(-1\) | B1 [1] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| 5 pts (or line or curve or zigzag) such that: grad always +ve (not vertical); not in st line | B1, B1 dep [2] | Allow \(\geq 4\) pts; dep 1st B1; Must be clearly intended not to be st line; eg st line, +ve grad, B1B0; If crosses and curve or line, mark crosses; SC Some segments vertical (not all) B0B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(x\): 1 2 3 4 5 or 1 2 3 4 5; \(y\): 1 2 3 5 4 \(\quad\) 2 1 3 4 5; Allow both sets reversed | M1, A1 | Attempt ranks; Correct ranks; \(\Sigma x=\Sigma y=15\); \(\Sigma x^2=\Sigma y^2=55\); \(\Sigma xy=54\); \(S_{xx}=S_{yy}=55-(15^2\div5)=10\); \(S_{xy}=54-(15^2\div5)=9\); correct method for one \(S\) M1; \(r_s=\frac{9}{\sqrt{10\times10}}\) fully correct method M1 |
| \(\Sigma d^2 = 2\) | M1 | dep 1st M1; Correct method for \(\Sigma d^2\); ft their ranks; NB \(\Sigma d^2 = 2^2 = 4\): M0 |
| \(r_s = 1 - \frac{6\times\text{"2"}}{5(5^2-1)}\) | M1 | Sub their \(\Sigma d^2\) into correct formula, dep M1M1, eg not using \(\Sigma d^2=2^2=4\) |
| \(= 0.9\) | A1 [5] | ans 0.9, no wking: full marks; ans \(-0.9\) with wking may get M1M1M1; ans \(-0.9\), no wking: no marks |
# Question 4:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $-1$ | B1 **[1]** | |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| 5 pts (or line or curve or zigzag) such that: grad always +ve (not vertical); not in st line | B1, B1 dep **[2]** | Allow $\geq 4$ pts; dep 1st B1; Must be clearly intended not to be st line; eg st line, +ve grad, B1B0; If crosses **and** curve or line, mark crosses; SC Some segments vertical (not all) B0B1 |
## Part (iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x$: 1 2 3 4 5 or 1 2 3 4 5; $y$: 1 2 3 5 4 $\quad$ 2 1 3 4 5; Allow both sets reversed | M1, A1 | Attempt ranks; Correct ranks; $\Sigma x=\Sigma y=15$; $\Sigma x^2=\Sigma y^2=55$; $\Sigma xy=54$; $S_{xx}=S_{yy}=55-(15^2\div5)=10$; $S_{xy}=54-(15^2\div5)=9$; correct method for one $S$ M1; $r_s=\frac{9}{\sqrt{10\times10}}$ fully correct method M1 |
| $\Sigma d^2 = 2$ | M1 | dep 1st M1; Correct method for $\Sigma d^2$; ft their ranks; NB $\Sigma d^2 = 2^2 = 4$: M0 |
| $r_s = 1 - \frac{6\times\text{"2"}}{5(5^2-1)}$ | M1 | Sub their $\Sigma d^2$ into correct formula, dep M1M1, eg not using $\Sigma d^2=2^2=4$ |
| $= 0.9$ | A1 **[5]** | ans 0.9, no wking: full marks; ans $-0.9$ with wking may get M1M1M1; ans $-0.9$, no wking: no marks |
---4 In this question the product moment correlation coefficient is denoted by $r$ and Spearman's rank correlation coefficient is denoted by $r _ { s }$.\\
(i) The scatter diagram in Fig. 1 shows the results of an experiment involving some bivariate data.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{b5ce3230-7528-439c-9e85-ef159a49cba3-4_597_595_434_733}
\captionsetup{labelformat=empty}
\caption{Fig. 1}
\end{center}
\end{figure}
Write down the value of $r _ { s }$ for these data.\\
(ii) On the diagram in the Answer Booklet, draw five points such that $r _ { s } = 1$ and $r \neq 1$.\\
(iii) The scatter diagram in Fig. 2 shows the results of another experiment involving 5 items of bivariate data.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{b5ce3230-7528-439c-9e85-ef159a49cba3-4_604_608_1484_731}
\captionsetup{labelformat=empty}
\caption{Fig. 2}
\end{center}
\end{figure}
Calculate the value of $r _ { s }$.\\
(i) A random variable $X$ has the distribution $\mathrm { B } ( 25,0.6 )$. Find
\begin{enumerate}[label=(\alph*)]
\item $\mathrm { P } ( X \leqslant 14 )$,
\item $\mathrm { P } ( X = 14 )$,
\item $\quad \operatorname { Var } ( X )$.\\
(ii) A random variable $Y$ has the distribution $\mathrm { B } ( 24,0.3 )$. Write down an expression for $\mathrm { P } ( Y = y )$ and evaluate this probability in the case where $y = 8$.\\
(iii) A random variable $Z$ has the distribution $\mathrm { B } ( 2,0.2 )$. Find the probability that two randomly chosen values of $Z$ are equal.\\
(a) Find the number of ways in which 12 people can be divided into three groups containing 5 people, 4 people and 3 people, without regard to order.\\
(b) The diagram shows 7 cards, each with a letter on it.
$$\mathrm { A } \mathrm {~A} \mathrm {~A} \mathrm {~B} \text { } \mathrm { B } \text { } \mathrm { R } \text { } \mathrm { R }$$
The 7 cards are arranged in a random order in a straight line.
\begin{enumerate}[label=(\roman*)]
\item Find the number of possible arrangements of the 7 letters.
\item Find the probability that the 7 letters form the name BARBARA.
The 7 cards are shuffled. Now 4 of the 7 cards are chosen at random and arranged in a random order in a straight line.
\item Find the probability that the letters form the word ABBA .
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR S1 2016 Q4 [8]}}