Questions - OCR S1 - A-Level Maths

8876	1	66677889
76553321	2	1334578899
98443	3	23347
521	4	018
90	5	0

\end{table} Key: 1 | 4 | 0 represents a male aged 41 and a female aged 40.

Find the median and interquartile range for the males.
The median and interquartile range for the females are 27 and 15 respectively. Make two comparisons between the ages of the males and the ages of the females.
The mean age of the males is 30.7 and the mean age of the females is 27.5 , each correct to 1 decimal place. Give one advantage of using the median rather than the mean to compare the ages of the males with the ages of the females. A record was kept of the number of hours, $X$, spent by each member at the club in a year. The results were summarised by $$n = 49 , \quad \Sigma ( x - 200 ) = 245 , \quad \Sigma ( x - 200 ) ^ { 2 } = 9849 .$$
Calculate the mean and standard deviation of $X$.

OCR S1 2008 January Q9

9 It is thought that the pH value of sand (a measure of the sand's acidity) may affect the extent to which a particular species of plant will grow in that sand. A botanist wished to determine whether there was any correlation between the pH value of the sand on certain sand dunes, and the amount of each of two plant species growing there. She chose random sections of equal area on each of eight sand dunes and measured the pH values. She then measured the area within each section that was covered by each of the two species. The results were as follows.

\cline { 2 - 10 } \multicolumn{1}{c|}{}

\cline { 2 - 10 } \multicolumn{1}{c|}{}

Area, $y \mathrm {~cm} ^ { 2 }$

The results for species $P$ can be summarised by $$n = 8 , \quad \Sigma x = 66.5 , \quad \Sigma x ^ { 2 } = 558.75 , \quad \Sigma y = 1935 , \quad \Sigma y ^ { 2 } = 711275 , \quad \Sigma x y = 17082.5 .$$

Give a reason why it might be appropriate to calculate the equation of the regression line of $y$ on $x$ rather than $x$ on $y$ in this situation.
Calculate the equation of the regression line of $y$ on $x$ for species $P$, in the form $y = a + b x$, giving the values of $a$ and $b$ correct to 3 significant figures.
Estimate the value of $y$ for species $P$ on sand where the pH value is 7.0 . The values of the product moment correlation coefficient between $x$ and $y$ for species $P$ and $Q$ are $r _ { P } = 0.828$ and $r _ { Q } = 0.0302$.
Describe the relationship between the area covered by species $Q$ and the pH value.
State, with a reason, whether the regression line of $y$ on $x$ for species $P$ will provide a reliable estimate of the value of $y$ when the pH value is
(a) 8,
(b) 4 .
Assume that the equation of the regression line of $y$ on $x$ for species $Q$ is also known. State, with a reason, whether this line will provide a reliable estimate of the value of $y$ when the pH value is 8 .

OCR S1 2005 June Q1

Calculate the value of Spearman's rank correlation coefficient between the two sets of rankings, $A$ and $B$, shown in Table 1. \begin{table}[h]
$A$ 1 2 3 4 5
$B$ 4 1 3 2 5
\captionsetup{labelformat=empty} \caption{Table 1}
\end{table}
The value of Spearman's rank correlation coefficient between the set of rankings $B$ and a third set of rankings, $C$, is known to be - 1 . Copy and complete Table 2 showing the set of rankings $C$. \begin{table}[h]
$B$ 4 1 3 2 5
$C$
\captionsetup{labelformat=empty} \caption{Table 2}
\end{table}

OCR S1 2005 June Q2

2 The probability that a certain sample of radioactive material emits an alpha-particle in one unit of time is 0.14 . In one unit of time no more than one alpha-particle can be emitted. The number of units of time up to and including the first in which an alpha-particle is emitted is denoted by $T$.

Find the value of
(a) $\mathrm { P } ( T = 5 )$,
(b) $\mathrm { P } ( T < 8 )$.
State the value of $\mathrm { E } ( T )$.

OCR S1 2005 June Q3

3 In a supermarket the proportion of shoppers who buy washing powder is denoted by $p .16$ shoppers are selected at random.

Given that $p = 0.35$, use tables to find the probability that the number of shoppers who buy washing powder is
(a) at least 8,
(b) between 4 and 9 inclusive.
Given instead that $p = 0.38$, find the probability that the number of shoppers who buy washing powder is exactly 6 .

OCR S1 2005 June Q4

4 The table shows the latitude, $x$ (in degrees correct to 3 significant figures), and the average rainfall $y$ (in cm correct to 3 significant figures) of five European cities.

City	$x$	$y$
Berlin	52.5	58.2
Bucharest	44.4	58.7
Moscow	55.8	53.3
St Petersburg	60.0	47.8
Warsaw	52.3	56.6

$$\left[ n = 5 , \Sigma x = 265.0 , \Sigma y = 274.6 , \Sigma x ^ { 2 } = 14176.54 , \Sigma y ^ { 2 } = 15162.22 , \Sigma x y = 14464.10 . \right]$$

Calculate the product moment correlation coefficient.
The values of $y$ in the table were in fact obtained from measurements in inches and converted into centimetres by multiplying by 2.54 . State what effect it would have had on the value of the product moment correlation coefficient if it had been calculated using inches instead of centimetres.
It is required to estimate the annual rainfall at Bergen, where $x = 60.4$. Calculate the equation of an appropriate line of regression, giving your answer in simplified form, and use it to find the required estimate.

OCR S1 2005 June Q5

5 The examination marks obtained by 1200 candidates are illustrated on the cumulative frequency graph, where the data points are joined by a smooth curve.
\includegraphics[max width=\textwidth, alt={}, center]{5faf0d93-4037-4958-8665-1008477a79de-4_1344_1335_386_425} Use the curve to estimate

the interquartile range of the marks,
$x$, if $40 \%$ of the candidates scored more than $x$ marks,
the number of candidates who scored more than 68 marks. Five of the candidates are selected at random, with replacement.
Estimate the probability that all five scored more than 68 marks. It is subsequently discovered that the candidates' marks in the range 35 to 55 were evenly distributed - that is, roughly equal numbers of candidates scored $35,36,37 , \ldots , 55$.
What does this information suggest about the estimate of the interquartile range found in part (i)?

OCR S1 2005 June Q6

6 Two bags contain coloured discs. At first, bag $P$ contains 2 red discs and 2 green discs, and bag $Q$ contains 3 red discs and 1 green disc. A disc is chosen at random from bag $P$, its colour is noted and it is placed in bag $Q$. A disc is then chosen at random from bag $Q$, its colour is noted and it is placed in bag $P$. A disc is then chosen at random from bag $P$. The tree diagram shows the different combinations of three coloured discs chosen.
\includegraphics[max width=\textwidth, alt={}, center]{5faf0d93-4037-4958-8665-1008477a79de-5_863_986_559_612}

Write down the values of $a , b , c , d , e$ and $f$. The total number of red discs chosen, out of 3, is denoted by $R$. The table shows the probability distribution of $R$.
$r$ 0 1 2 3
$\mathrm { P } ( R = r )$ $\frac { 1 } { 10 }$ $k$ $\frac { 9 } { 20 }$ $\frac { 1 } { 5 }$
Show how to obtain the value $\mathrm { P } ( R = 2 ) = \frac { 9 } { 20 }$.
Find the value of $k$.
Calculate the mean and variance of $R$.

OCR S1 2006 June Q1

1 Some observations of bivariate data were made and the equations of the two regression lines were found to be as follows. $$\begin{array} { c c } y \text { on } x : & y = - 0.6 x + 13.0
x \text { on } y : & x = - 1.6 y + 21.0 \end{array}$$

State, with a reason, whether the correlation between $x$ and $y$ is negative or positive.
Neither variable is controlled. Calculate an estimate of the value of $x$ when $y = 7.0$.
Find the values of $\bar { x }$ and $\bar { y }$.

OCR S1 2006 June Q2

2 A bag contains 5 black discs and 3 red discs. A disc is selected at random from the bag. If it is red it is replaced in the bag. If it is black, it is not replaced. A second disc is now selected at random from the bag. Find the probability that

the second disc is black, given that the first disc was black,
the second disc is black,
the two discs are of different colours.

OCR S1 2006 June Q3

3 Each of the 7 letters in the word DIVIDED is printed on a separate card. The cards are arranged in a row.

How many different arrangements of the letters are possible?
In how many of these arrangements are all three Ds together? The 7 cards are now shuffled and 2 cards are selected at random, without replacement.
Find the probability that at least one of these 2 cards has D printed on it.

OCR S1 2006 June Q4

The random variable $X$ has the distribution $\mathrm { B } ( 25,0.2 )$. Using the tables of cumulative binomial probabilities, or otherwise, find $\mathrm { P } ( X \geqslant 5 )$.
The random variable $Y$ has the distribution $\mathrm { B } ( 10,0.27 )$. Find $\mathrm { P } ( Y = 3 )$.
The random variable $Z$ has the distribution $\mathrm { B } ( n , 0.27 )$. Find the smallest value of $n$ such that $\mathrm { P } ( Z \geqslant 1 ) > 0.95$.

OCR S1 2006 June Q5

5 The probability distribution of a discrete random variable, $X$, is given in the table.

$x$	0	1	2	3
$\mathrm { P } ( X = x )$	$\frac { 1 } { 3 }$	$\frac { 1 } { 4 }$	$p$	$q$

It is given that the expectation, $\mathrm { E } ( X )$, is $1 \frac { 1 } { 4 }$.

Calculate the values of $p$ and $q$.
Calculate the standard deviation of $X$.

OCR S1 2006 June Q6

6 The table shows the total distance travelled, in thousands of miles, and the amount of commission earned, in thousands of pounds, by each of seven sales agents in 2005.

Agent	$A$	$B$	$C$	$D$	$E$	$F$	$G$
Distance travelled	18	15	12	14	16	24	13
Commission earned	18	45	19	24	27	22	23

(a) Calculate Spearman's rank correlation coefficient, $r _ { s }$, for these data.
(b) Comment briefly on your value of $r _ { s }$ with reference to this context.
(c) After these data were collected, agent $A$ found that he had made a mistake. He had actually travelled 19000 miles in 2005. State, with a reason, but without further calculation, whether the value of Spearman's rank correlation coefficient will increase, decrease or stay the same. The agents were asked to indicate their level of job satisfaction during 2005. A score of 0 represented no job satisfaction, and a score of 10 represented high job satisfaction. Their scores, $y$, together with the data for distance travelled, $x$, are illustrated in the scatter diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{b37239e4-5d63-47e4-844a-f01b79f8dd67-3_691_981_1231_587}
For this scatter diagram, what can you say about the value of
(a) Spearman's rank correlation coefficient,
(b) the product moment correlation coefficient?

OCR S1 2006 June Q7

7 In a UK government survey in 2000, smokers were asked to estimate the time between their waking and their having the first cigarette of the day. For heavy smokers, the results were as follows.

Percentage of smokers

Times are given correct to the nearest minute.

Assuming that 'At least 60 minutes' means 'At least 60 minutes but less than 240 minutes', calculate estimates for the mean and standard deviation of the time between waking and first cigarette for these smokers.
Find an estimate for the interquartile range of the time between waking and first cigarette for these smokers. Give your answer correct to the nearest minute.
The meaning of 'At least 60 minutes' is now changed to 'At least 60 minutes but less than 480 minutes'. Without further calculation, state whether this would cause an increase, a decrease or no change in the estimated value of
(a) the mean,
(b) the standard deviation,
(c) the interquartile range.

OCR S1 2006 June Q8

8 Henry makes repeated attempts to light his gas fire. He makes the modelling assumption that the probability that the fire will light on any attempt is $\frac { 1 } { 3 }$. Let $X$ be the number of attempts at lighting the fire, up to and including the successful attempt.

Name the distribution of $X$, stating a further modelling assumption needed. In the rest of this question, you should use the distribution named in part (i).
Calculate
(a) $\mathrm { P } ( X = 4 )$,
(b) $\mathrm { P } ( X < 4 )$.
State the value of $\mathrm { E } ( X )$.
Henry has to light the fire once a day, starting on March 1st. Calculate the probability that the first day on which fewer than 4 attempts are needed to light the fire is March 3rd.

OCR S1 2007 June Q1

1 The table shows the probability distribution for a random variable X.

x	0	1	2	3
$\mathrm { P } ( \mathrm { X } = \mathrm { x } )$	0.1	0.2	0.3	0.4

Calculate $\mathrm { E } ( \mathrm { X } )$ and $\operatorname { Var } ( \mathrm { X } )$.

OCR S1 2007 June Q2

2 Two judges each placed skaters from five countries in rank order.

Position	1st	2nd	3rd	4th	5th
Judge 1	UK	France	Russia	Poland	Canada
Judge 2	Russia	Canada	France	UK	Poland

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

OCR S1 2007 June Q3

How many different teams of 7 people can be chosen, without regard to order, from a squad of 15 ?
The squad consists of 6 forwards and 9 defenders. How many different teams containing 3 forwards and 4 defenders can be chosen?

OCR S1 2007 June Q4

4 A bag contains 6 white discs and 4 blue discs. Discs are removed at random, one at a time, without replacement.

Find the probability that
(a) the second disc is blue, given that the first disc was blue,
(b) the second disc is blue,
(c) the third disc is blue, given that the first disc was blue.
The random variable $X$ is the number of discs which are removed up to and including the first blue disc. State whether the variable X has a geometric distribution. Explain your answer briefly.

OCR S1 2007 June Q5

5 The numbers of births, in thousands, to mothers of different ages in England and Wales, in 1991 and 2001 are illustrated by the cumulative frequency curves. Cumulative frequency (000's)
\includegraphics[max width=\textwidth, alt={}, center]{dfad6626-75ca-4dbd-9c45-42f809c163f3-3_949_1338_461_479}

In which of these two years were there more births? How many more births were there in this year?
The following quantities were estimated from the diagram.
Year
M edian age
(years)
Interquartile
range (years)
Proportion of mothers
giving birth aged below 25
Proportion of mothers
giving birth aged 35 or above
1991 27.5 7.3 $33 \%$ $9 \%$
2001 $18 \%$
(a) Find the values missing from the table.
(b) Did the women who gave birth in 2001 tend to be younger or older or about the same age as the women who gave birth in 1991? Using the table and your values from part (a), give two reasons for your answer.

OCR S1 2007 June Q6

6 A machine with artificial intelligence is designed to improve its efficiency rating with practice. The table shows the values of the efficiency rating, y , after the machine has carried out its task various numbers of times, $x$

x	0	1	2	3	4	7	13	30
y	0	4	8	10	11	12	13	14

$$\left[ n = 8 , \Sigma x = 60 , \Sigma y = 72 , \Sigma x ^ { 2 } = 1148 , \Sigma y ^ { 2 } = 810 , \Sigma x y = 767 . \right]$$ These data are illustrated in the scatter diagram.
\includegraphics[max width=\textwidth, alt={}, center]{dfad6626-75ca-4dbd-9c45-42f809c163f3-4_769_1328_760_411}

(a) Calculate the value of r , the product moment correlation coefficient.
(b) Without calculation, state with a reason the value of $\mathrm { r } _ { \mathrm { s } ^ { \prime } }$ Spearman's rank correlation coefficient.
A researcher suggests that the data for $\mathrm { x } = 0$ and $\mathrm { x } = 1$ should be ignored. Without cal culation, state with a reason what effect this would have on the value of
(a) $r$,
(b) $r _ { s }$.
Use the diagram to estimate the value of y when $\mathrm { x } = 29$.
Jack finds the equation of the regression line of y on xf for all the data, and uses it to estimate the value of $y$ when $x = 29$. Without calculation, state with a reason whether this estimate or the one found in part (iii) will be the more reliable.

OCR S1 2007 June Q7

7 On average, $25 \%$ of the packets of a certain kind of soup contain a voucher. Kim buys one packet of soup each week for 12 weeks. The number of vouchers she obtains is denoted by X .

State two conditions needed for X to be modelled by the distribution $\mathrm { B } ( 12,0.25 )$. In the rest of this question you should assume that these conditions are satisfied.
Find $\mathrm { P } ( \mathrm { X } \leqslant 6 )$. In order to claim a free gift, 7 vouchers are needed.
Find the probability that Kim will be able to claim a free gift at some time during the 12 weeks.
Find the probability that Kim will be able to claim a free gift in the 12th week but not before.

OCR S1 2007 June Q8

A biased coin is thrown twice. The probability that it shows heads both times is 0.04 . Find the probability that it shows tails both times.
A nother coin is biased so that the probability that it shows heads on any throw is p . The probability that the coin shows heads exactly once in two throws is 0.42 . Find the two possible values of p.

OCR S1 2007 June Q9

A random variable $X$ has the distribution $\operatorname { Geo } \left( \frac { 1 } { 5 } \right)$. Find
(a) $\mathrm { E } ( \mathrm { X } )$,
(b) $\mathrm { P } ( \mathrm { X } = 4 )$,
(c) $P ( X > 4 )$.
A random variable $Y$ has the distribution $\operatorname { Geo } ( p )$, and $q = 1 - p$.
(a) Show that $P ( Y$ is odd $) = p + q ^ { 2 } p + q ^ { 4 } p + \ldots$.
(b) Use the formula for the sum to infinity of a geometric progression to show that $$P ( Y \text { is odd } ) = \frac { 1 } { 1 + q }$$ \href{http://physicsandmathstutor.com}{physicsandmathstutor.com}
7

\(r\)	0	1	2	3
\(\mathrm { P } ( R = r )\)	\(\frac { 1 } { 10 }\)	\(k\)	\(\frac { 9 } { 20 }\)	\(\frac { 1 } { 5 }\)

\(x\)	0	1	2	3
\(\mathrm { P } ( X = x )\)	\(\frac { 1 } { 3 }\)	\(\frac { 1 } { 4 }\)	\(p\)	\(q\)

City	\(x\)	\(y\)
Berlin	52.5	58.2
Bucharest	44.4	58.7
Moscow	55.8	53.3
St Petersburg	60.0	47.8
Warsaw	52.3	56.6

Agent	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)
Distance travelled	18	15	12	14	16	24	13
Commission earned	18	45	19	24	27	22	23

x	0	1	2	3
\(\mathrm { P } ( \mathrm { X } = \mathrm { x } )\)	0.1	0.2	0.3	0.4

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