OCR S1 2007 June — Question 2 5 marks

Exam BoardOCR
ModuleS1 (Statistics 1)
Year2007
SessionJune
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicHypothesis test of Spearman’s rank correlation coefficien
TypeCalculate Spearman's coefficient only
DifficultyEasy -1.2 This is a straightforward application of Spearman's rank correlation formula with n=5 and no tied ranks. Students simply need to calculate differences in ranks, square them, and substitute into the formula rs = 1 - (6Σd²)/(n(n²-1)). It requires only routine calculation with no problem-solving or interpretation, making it easier than average.
Spec5.08e Spearman rank correlation
2 Two judges each placed skaters from five countries in rank order.
Position1st2nd3rd4th5th
Judge 1UKFranceRussiaPolandCanada
Judge 2RussiaCanadaFranceUKPoland
Calculate Spearman's rank correlation coefficient, \(\mathrm { r } _ { \mathbf { s } ^ { \prime } }\) for the two judges' rankings.
Question 2:
AnswerMarks Guidance
Answer/WorkingMark Guidance
UK Fr Ru Po Ca; \(1\ 2\ 3\ 4\ 5\) or \(5\ 4\ 3\ 2\ 1\); \(4\ 3\ 1\ 5\ 2\ \ \ 2\ 3\ 5\ 1\ 4\)M1 Consistent attempt rank
\(\Sigma d^2\)A1
\((= 24)\)M1
\(r_s = 1 - \dfrac{6\times\text{"24"}}{5\times(5^2-1)}\)M1 All 5 \(d^2\) attempted & added. Dep ranks att'd
\(= -\frac{1}{5}\) or \(-0.2\)A1 Dep 2nd M1: \(\dfrac{43-15^2/5}{\sqrt{((55-15^2/5)(55-15^2/5))}}\)
Total5
Question 3i:
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(^{15}C_7\) or \(^{15!}/_{7!8!}\)M1
\(6435\)A1
2
Question 3ii:
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(^6C_3 \times\ ^9C_4\) or \(^{6!}/_{3!3!} \times\ ^{9!}/_{4!5!}\)M1 Alone except allow \(\div\ ^{15}C_7\); Or \(^6P_3\times\ ^9P_4\) or \(^{6!}/_{3!}\times\ ^{9!}/_{5!}\). Allow \(\div\ ^{15}P_7\); NB not \(^{6!}/_{3!}\times^{9!}/_{4!}\)
\(2520\)A1 \(362880\)
2
Question 4ia:
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(\frac{1}{3}\) oeB1 B\(\leftrightarrow\)W MR: max (a)B0(b)M1M1(c)B1M1
1
Question 4b:
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(P(BB) + P(WB)\) attemptedM1 Or \(\frac{4}{10}\times\frac{3}{9}\) OR \(\frac{6}{10}\times\frac{4}{9}\) correct
\(= \frac{4}{10}\times\frac{3}{9} + \frac{6}{10}\times\frac{4}{9}\) or \(\frac{2}{15}+\frac{4}{15}\)M1 NB \(\frac{4}{10}\times\frac{4}{10}+\frac{6}{10}\times\frac{4}{10} = \frac{2}{5}\): M1M0A0
\(= \frac{2}{5}\) oeA1
3
Question 4c:
AnswerMarks Guidance
Answer/WorkingMark Guidance
Denoms 9 & 8 seen or impliedB1 Or \(\frac{2}{15}\) as numerator
\(\frac{3}{9}\times\frac{2}{8} + \frac{6}{9}\times\frac{3}{8}\)M1 Or \(\frac{2}{15}\)
\(= \frac{1}{3}\) oeA1 May not see working
3
Question 4ii:
AnswerMarks Guidance
Answer/WorkingMark Guidance
P(Blue) not constant or discs not indep, so noB1 Prob changes as discs removed; Limit to no. of discs; Fixed no. of discs; Discs will run out; Context essential: "disc" or "blue"; NOT fixed no. of trials; NOT because without replacement. Ignore extra
1
Question 5i:
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(1991\)B1 ind Or fewer in 2001
\(100\,000\) to \(110\,000\)B1 ind Allow digits 100 to 110
2
Question 5iia:
AnswerMarks Guidance
Answer/WorkingMark Guidance
Median \(= 29\) to \(29.9\)B1
Quartiles 33 to 34, 24.5 to 26M1 Or one correct quartile and subtr; NOT from incorrect working
\(= 7.5\) to \(9.5\)A1 \(\times1000\), but allow without
\(140\) to \(155\)M1
\(23\) to \(26.3\%\)A1 Rounded to 1 dp or integer 73.7 to 77%: SC1
5
Question 5b:
AnswerMarks Guidance
Answer/WorkingMark Guidance
OlderB1 Or 1991 younger
Median (or ave) greaterB1 Any two
% older mothers greater oeB1 Or 1991 steeper so more younger: B2; NOT mean greater
% younger mothers less oe Ignore extra
3
Question 6ia:
AnswerMarks Guidance
Answer/WorkingMark Guidance
Correct subst in \(\geq\) two \(S\) formulaeM1 Any version
\(\dfrac{767 - \dfrac{60\times72}{8}}{\sqrt{(1148-\dfrac{60^2}{8})(810-\dfrac{72^2}{8})}}\) or \(\dfrac{227}{\sqrt{698}\sqrt{162}}\)M1 All correct. Or \(\dfrac{767-8\times7.5\times9}{\sqrt{(1148-8\times7.5^2)(810-8\times9^2)}}\); or correct subst in any correct formula for \(r\)
\(= 0.675\) (3 sfs)A1
3
Question 6b:
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(1\)B1
\(y\) always increases with \(x\) or ranks same oeB1 +ve grad throughout. Increase in steps. Same order. Both ascending order; Perfect RANK corr'n; Ignore extra; NOT increasing proportionately
2
Question 6iia:
AnswerMarks Guidance
Answer/WorkingMark Guidance
Closer to 1, or increasesB1
because nearer to straight lineB1 Corr'n stronger. Fewer outliers. "They" are outliers; Ignore extra
2
Question 6b:
AnswerMarks Guidance
Answer/WorkingMark Guidance
None, or remains at 1B1
Because \(y\) still increasing with \(x\) oeB1 \(\Sigma d^2\) still 0. Still same order. Ignore extra; NOT differences still the same; NOT ft (i)(b)
2
Question 6iii:
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(13.8\) to \(14.0\)B1
1
Question 6iv:
AnswerMarks Guidance
Answer/WorkingMark Guidance
(iii) or graph or diag or my estB1 Must be clear which est. Can be implied. "This est" probably \(\Rightarrow\) using eqn of line
Takes account of curveB1 Straight line is not good fit. Not linear. Corr'n not strong.
2
Question 7i:
AnswerMarks Guidance
Answer/WorkingMark Guidance
P(contains voucher) constant oeB1 Context essential
Packets indep oeB1 NOT vouchers indep
2
Question 7ii:
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(0.9857\) or \(0.986\) (3 sfs)B2 B1 for \(0.9456\) or \(0.946\) or \(0.997(2)\); or for 7 terms correct, allow one omit or extra; NOT \(1-0.9857=0.0143\) (see (iii))
2
Question 7iii:
AnswerMarks Guidance
Answer/WorkingMark Guidance
\((1-0.9857) = 0.014(3)\) (2 sfs)B1ft Allow 1 - their (ii) correctly calc'd
1
Question 7iv:
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(B(11, 0.25)\) or 6 in 11 weeks stated or impliedB1 or \(0.75^a\times0.25^b\ (a+b=11)\) or \(^{11}C_6\)
\(^{11}C_6\times0.75^5\times0.25^6\ (=0.0267663)\)M1 dep B1
\(P(6\text{ from }11)\times0.25\)M1
\(= 0.00669\) or \(6.69\times10^{-3}\) (3 sfs)A1
4
Question 8i:
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(\sqrt{0.04}\ (=0.2)\)M1
\((1-\text{their }\sqrt{0.04})^2\)M1
\(= 0.64\)A1
3
Question 8ii:
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(1-p\) seenB1
\(2p(1-p)=0.42\) or \(p(1-p)=0.21\) oeM1 \(2pq=0.42\) or \(pq=0.21\). Allow \(pq=0.42\)
\(2p^2-2p+0.42(=0)\) or \(p^2-p+0.21(=0)\)M1 or opp signs, correct terms any order \((=0)\)
\(\dfrac{2\pm\sqrt{((-2)^2-4\times0.42)}}{2\times2}\) or \(\dfrac{1\pm\sqrt{((-1)^2-4\times0.21)}}{2\times1}\)M1 oe Correct; Dep B1M1M1 Any corr subst'n or fact'n
\((p-0.7)(p-0.3)=0\) or \((10p-7)(10p-3)=0\)M1
\(p=0.7\) or \(0.3\)A1
5
Question 9ia:
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(1/\frac{1}{5}\)M1
\(= 5\)A1
2
Question 9b:
AnswerMarks Guidance
Answer/WorkingMark Guidance
\((\frac{4}{5})^3\times\frac{1}{5}\)M1
\(= \frac{64}{625}\) or \(0.102\) (3 sfs)A1
2
Question 9c:
AnswerMarks Guidance
Answer/WorkingMark Guidance
\((\frac{4}{5})^3\)M1 or \(1-(\frac{1}{5}+\frac{4}{5}\times\frac{1}{5}+(\frac{4}{5})^2\times\frac{1}{5}+(\frac{4}{5})^3\times\frac{1}{5})\); NOT \(1-(\frac{4}{5})^4\)
\(= \frac{256}{625}\) or a.r.t \(0.410\) (3 sfs) or \(0.41\)A1
2
Question 9iia:
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(P(Y=1)=p,\ P(Y=3)=q^2p,\ P(Y=5)=q^4p\)B1 \(P(Y=1)+P(Y=3)+P(Y=5)=p+q^2p+q^4p\); \(p,\ p(1-p)^2,\ p(1-p)^4\); \(q^{1-1},\ q^{3-1},\ q^{5-1}\); or any of these with \(1-p\) instead of \(q\); "Always \(q\) to even power \(\times p\)"; Either associate each term with relevant prob; Or give indication of how terms derived \(>\) two terms
1
Question 9b:
AnswerMarks Guidance
Answer/WorkingMark Guidance
Recog that c.r. \(= q^2\) or \((1-p)^2\)M1 or e.g. \(r=q^2p/p\)
\(S_\infty = \dfrac{p}{1-q^2}\) or \(\dfrac{p}{1-(1-p)^2}\)M1
\(P(\text{odd}) = \dfrac{1-q}{1-q^2}\)M1 \(\left(=\dfrac{p}{2p-p^2}\right) = \dfrac{p}{p(2-p)}\)
\(= \dfrac{1-q}{(1-q)(1+q)}\) Must see this step for A1A1 \(\left(=\dfrac{1}{2-p}\right) = \dfrac{1}{2-(1-q)}\)
\(\left(= \dfrac{1}{1+q}\right)\) AG
4 
## Question 2:

| Answer/Working | Mark | Guidance |
|---|---|---|
| UK Fr Ru Po Ca; $1\ 2\ 3\ 4\ 5$ or $5\ 4\ 3\ 2\ 1$; $4\ 3\ 1\ 5\ 2\ \ \ 2\ 3\ 5\ 1\ 4$ | M1 | Consistent attempt rank |
| $\Sigma d^2$ | A1 | |
| $(= 24)$ | M1 | |
| $r_s = 1 - \dfrac{6\times\text{"24"}}{5\times(5^2-1)}$ | M1 | All 5 $d^2$ attempted & added. Dep ranks att'd |
| $= -\frac{1}{5}$ or $-0.2$ | A1 | Dep 2nd M1: $\dfrac{43-15^2/5}{\sqrt{((55-15^2/5)(55-15^2/5))}}$ |
| **Total** | **5** | |

---

## Question 3i:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $^{15}C_7$ or $^{15!}/_{7!8!}$ | M1 | |
| $6435$ | A1 | |
| | **2** | |

## Question 3ii:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $^6C_3 \times\ ^9C_4$ or $^{6!}/_{3!3!} \times\ ^{9!}/_{4!5!}$ | M1 | Alone except allow $\div\ ^{15}C_7$; Or $^6P_3\times\ ^9P_4$ or $^{6!}/_{3!}\times\ ^{9!}/_{5!}$. Allow $\div\ ^{15}P_7$; NB not $^{6!}/_{3!}\times^{9!}/_{4!}$ |
| $2520$ | A1 | $362880$ |
| | **2** | |

---

## Question 4ia:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{1}{3}$ oe | B1 | B$\leftrightarrow$W MR: max (a)B0(b)M1M1(c)B1M1 |
| | **1** | |

## Question 4b:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $P(BB) + P(WB)$ attempted | M1 | Or $\frac{4}{10}\times\frac{3}{9}$ OR $\frac{6}{10}\times\frac{4}{9}$ correct |
| $= \frac{4}{10}\times\frac{3}{9} + \frac{6}{10}\times\frac{4}{9}$ or $\frac{2}{15}+\frac{4}{15}$ | M1 | NB $\frac{4}{10}\times\frac{4}{10}+\frac{6}{10}\times\frac{4}{10} = \frac{2}{5}$: M1M0A0 |
| $= \frac{2}{5}$ oe | A1 | |
| | **3** | |

## Question 4c:

| Answer/Working | Mark | Guidance |
|---|---|---|
| Denoms 9 & 8 seen or implied | B1 | Or $\frac{2}{15}$ as numerator |
| $\frac{3}{9}\times\frac{2}{8} + \frac{6}{9}\times\frac{3}{8}$ | M1 | Or $\frac{2}{15}$ |
| $= \frac{1}{3}$ oe | A1 | May not see working |
| | **3** | |

## Question 4ii:

| Answer/Working | Mark | Guidance |
|---|---|---|
| P(Blue) not constant or discs not indep, so no | B1 | Prob changes as discs removed; Limit to no. of discs; Fixed no. of discs; Discs will run out; Context essential: "disc" or "blue"; NOT fixed no. of trials; NOT because without replacement. Ignore extra |
| | **1** | |

---

## Question 5i:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $1991$ | B1 ind | Or fewer in 2001 |
| $100\,000$ to $110\,000$ | B1 ind | Allow digits 100 to 110 |
| | **2** | |

## Question 5iia:

| Answer/Working | Mark | Guidance |
|---|---|---|
| Median $= 29$ to $29.9$ | B1 | |
| Quartiles 33 to 34, 24.5 to 26 | M1 | Or one correct quartile and subtr; NOT from incorrect working |
| $= 7.5$ to $9.5$ | A1 | $\times1000$, but allow without |
| $140$ to $155$ | M1 | |
| $23$ to $26.3\%$ | A1 | Rounded to 1 dp or integer 73.7 to 77%: SC1 |
| | **5** | |

## Question 5b:

| Answer/Working | Mark | Guidance |
|---|---|---|
| Older | B1 | Or 1991 younger |
| Median (or ave) greater | B1 | Any two |
| % older mothers greater oe | B1 | Or 1991 steeper so more younger: B2; NOT mean greater |
| % younger mothers less oe | | Ignore extra |
| | **3** | |

---

## Question 6ia:

| Answer/Working | Mark | Guidance |
|---|---|---|
| Correct subst in $\geq$ two $S$ formulae | M1 | Any version |
| $\dfrac{767 - \dfrac{60\times72}{8}}{\sqrt{(1148-\dfrac{60^2}{8})(810-\dfrac{72^2}{8})}}$ or $\dfrac{227}{\sqrt{698}\sqrt{162}}$ | M1 | All correct. Or $\dfrac{767-8\times7.5\times9}{\sqrt{(1148-8\times7.5^2)(810-8\times9^2)}}$; or correct subst in any correct formula for $r$ |
| $= 0.675$ (3 sfs) | A1 | |
| | **3** | |

## Question 6b:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $1$ | B1 | |
| $y$ always increases with $x$ or ranks same oe | B1 | +ve grad throughout. Increase in steps. Same order. Both ascending order; Perfect RANK corr'n; Ignore extra; NOT increasing proportionately |
| | **2** | |

## Question 6iia:

| Answer/Working | Mark | Guidance |
|---|---|---|
| Closer to 1, or increases | B1 | |
| because nearer to straight line | B1 | Corr'n stronger. Fewer outliers. "They" are outliers; Ignore extra |
| | **2** | |

## Question 6b:

| Answer/Working | Mark | Guidance |
|---|---|---|
| None, or remains at 1 | B1 | |
| Because $y$ still increasing with $x$ oe | B1 | $\Sigma d^2$ still 0. Still same order. Ignore extra; NOT differences still the same; NOT ft (i)(b) |
| | **2** | |

## Question 6iii:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $13.8$ to $14.0$ | B1 | |
| | **1** | |

## Question 6iv:

| Answer/Working | Mark | Guidance |
|---|---|---|
| (iii) or graph or diag or my est | B1 | Must be clear which est. Can be implied. "This est" probably $\Rightarrow$ using eqn of line |
| Takes account of curve | B1 | Straight line is not good fit. Not linear. Corr'n not strong. |
| | **2** | |

---

## Question 7i:

| Answer/Working | Mark | Guidance |
|---|---|---|
| P(contains voucher) constant oe | B1 | Context essential |
| Packets indep oe | B1 | NOT vouchers indep |
| | **2** | |

## Question 7ii:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $0.9857$ or $0.986$ (3 sfs) | B2 | B1 for $0.9456$ or $0.946$ or $0.997(2)$; or for 7 terms correct, allow one omit or extra; NOT $1-0.9857=0.0143$ (see (iii)) |
| | **2** | |

## Question 7iii:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $(1-0.9857) = 0.014(3)$ (2 sfs) | B1ft | Allow 1 - their (ii) correctly calc'd |
| | **1** | |

## Question 7iv:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $B(11, 0.25)$ or 6 in 11 weeks stated or implied | B1 | or $0.75^a\times0.25^b\ (a+b=11)$ or $^{11}C_6$ |
| $^{11}C_6\times0.75^5\times0.25^6\ (=0.0267663)$ | M1 | dep B1 |
| $P(6\text{ from }11)\times0.25$ | M1 | |
| $= 0.00669$ or $6.69\times10^{-3}$ (3 sfs) | A1 | |
| | **4** | |

---

## Question 8i:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $\sqrt{0.04}\ (=0.2)$ | M1 | |
| $(1-\text{their }\sqrt{0.04})^2$ | M1 | |
| $= 0.64$ | A1 | |
| | **3** | |

## Question 8ii:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $1-p$ seen | B1 | |
| $2p(1-p)=0.42$ or $p(1-p)=0.21$ oe | M1 | $2pq=0.42$ or $pq=0.21$. Allow $pq=0.42$ |
| $2p^2-2p+0.42(=0)$ or $p^2-p+0.21(=0)$ | M1 | or opp signs, correct terms any order $(=0)$ |
| $\dfrac{2\pm\sqrt{((-2)^2-4\times0.42)}}{2\times2}$ or $\dfrac{1\pm\sqrt{((-1)^2-4\times0.21)}}{2\times1}$ | M1 | oe Correct; Dep B1M1M1 Any corr subst'n or fact'n |
| $(p-0.7)(p-0.3)=0$ or $(10p-7)(10p-3)=0$ | M1 | |
| $p=0.7$ or $0.3$ | A1 | |
| | **5** | |

---

## Question 9ia:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $1/\frac{1}{5}$ | M1 | |
| $= 5$ | A1 | |
| | **2** | |

## Question 9b:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $(\frac{4}{5})^3\times\frac{1}{5}$ | M1 | |
| $= \frac{64}{625}$ or $0.102$ (3 sfs) | A1 | |
| | **2** | |

## Question 9c:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $(\frac{4}{5})^3$ | M1 | or $1-(\frac{1}{5}+\frac{4}{5}\times\frac{1}{5}+(\frac{4}{5})^2\times\frac{1}{5}+(\frac{4}{5})^3\times\frac{1}{5})$; NOT $1-(\frac{4}{5})^4$ |
| $= \frac{256}{625}$ or a.r.t $0.410$ (3 sfs) or $0.41$ | A1 | |
| | **2** | |

## Question 9iia:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $P(Y=1)=p,\ P(Y=3)=q^2p,\ P(Y=5)=q^4p$ | B1 | $P(Y=1)+P(Y=3)+P(Y=5)=p+q^2p+q^4p$; $p,\ p(1-p)^2,\ p(1-p)^4$; $q^{1-1},\ q^{3-1},\ q^{5-1}$; or any of these with $1-p$ instead of $q$; "Always $q$ to even power $\times p$"; Either associate each term with relevant prob; Or give indication of how terms derived $>$ two terms |
| | **1** | |

## Question 9b:

| Answer/Working | Mark | Guidance |
|---|---|---|
| Recog that c.r. $= q^2$ or $(1-p)^2$ | M1 | or e.g. $r=q^2p/p$ |
| $S_\infty = \dfrac{p}{1-q^2}$ or $\dfrac{p}{1-(1-p)^2}$ | M1 | |
| $P(\text{odd}) = \dfrac{1-q}{1-q^2}$ | M1 | $\left(=\dfrac{p}{2p-p^2}\right) = \dfrac{p}{p(2-p)}$ |
| $= \dfrac{1-q}{(1-q)(1+q)}$ Must see this step for A1 | A1 | $\left(=\dfrac{1}{2-p}\right) = \dfrac{1}{2-(1-q)}$ |
| $\left(= \dfrac{1}{1+q}\right)$ **AG** | | |
| | **4** | |
2 Two judges each placed skaters from five countries in rank order.

\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
Position & 1st & 2nd & 3rd & 4th & 5th \\
\hline
Judge 1 & UK & France & Russia & Poland & Canada \\
\hline
Judge 2 & Russia & Canada & France & UK & Poland \\
\hline
\end{tabular}
\end{center}

Calculate Spearman's rank correlation coefficient, $\mathrm { r } _ { \mathbf { s } ^ { \prime } }$ for the two judges' rankings.

\hfill \mbox{\textit{OCR S1 2007 Q2 [5]}}
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