Questions - OCR S4 - A-Level Maths

Find $\mathrm { P } ( A \cap B )$ in terms of $c$.
Find $\mathrm { P } ( B \mid A )$ and deduce that $0.1 \leqslant c \leqslant 0.4$.

OCR S4 2007 June Q2

2 Of 9 randomly chosen students attending a lecture, 4 were found to be smokers and 5 were nonsmokers. During the lecture their pulse-rates were measured, with the following results in beats per minute.

Smokers	77	85	90	98
Non-smokers	59	64	68	80	88

It may be assumed that these two groups of students were random samples from the student populations of smokers and non-smokers. Using a suitable Wilcoxon test at the $10 \%$ significance level, test whether there is a difference in the median pulse-rates of the two populations.

OCR S4 2007 June Q3

3 The discrete random variables $X$ and $Y$ have the joint probability distribution given in the following table.

	$X$
\cline { 2 - 5 } \multicolumn{1}{l}{}	- 1	0	1
1	0.24	0.22	0.04
2	0.26	0.18	0.06

Show that $\operatorname { Cov } ( X , Y ) = 0$.
Find the conditional distribution of $X$ given that $Y = 2$.

OCR S4 2007 June Q4

4 The levels of impurity in a particular alloy were measured using a random sample of 20 specimens. The results, in suitable units, were as follows.

3.00	2.05	3.15	2.65	3.50	3.25	2.85	3.35	2.65	2.75
2.90	2.20	2.95	3.05	3.65	3.45	2.55	2.15	2.80	2.60

Use the sign test, at the $5 \%$ significance level, to decide if there is evidence that the population median level of impurity is greater than 2.70 .
State what other test might have been used, and give one advantage and one disadvantage this other test has over the sign test.

OCR S4 2007 June Q5

5 The continuous random variable $X$ has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} \frac { 1 } { ( \alpha - 1 ) ! } x ^ { \alpha - 1 } \mathrm { e } ^ { - x } & x \geqslant 0
0 & x < 0 \end{cases}$$ where $\alpha$ is a positive integer.

Explain how you can deduce that $\int _ { 0 } ^ { \infty } x ^ { \alpha - 1 } \mathrm { e } ^ { - x } \mathrm {~d} x = ( \alpha - 1 )$ !.
Write down an integral for the moment generating function $\mathrm { M } _ { X } ( t )$ of $X$ and show, by using the substitution $x = \frac { u } { 1 - t }$, that $\mathrm { M } _ { X } ( t ) = ( 1 - t ) ^ { - \alpha }$.
Use the moment generating function to find, in terms of $\alpha$,
(a) $\mathrm { E } ( X )$,
(b) $\operatorname { Var } ( X )$.

OCR S4 2007 June Q6

6 The discrete random variable $X$ takes the values 0 and 1 with $\mathrm { P } ( X = 0 ) = q$ and $\mathrm { P } ( X = 1 ) = p$, where $p + q = 1$.

Write down the probability generating function of $X$. The sum of $n$ independent observations of $X$ is denoted by $S$.
Write down the probability generating function of $S$, and name the distribution of $S$.
Use the probability generating function of $S$ to find $\mathrm { E } ( S )$ and $\operatorname { Var } ( S )$.
The independent random variables $Y$ and $Z$ are such that $Y$ has the distribution $\mathrm { B } \left( 10 , \frac { 1 } { 2 } \right)$, and $Z$ has probability generating function $\mathrm { e } ^ { - ( 1 - t ) }$. Find the probability that the sum of one random observation of $Y$ and one random observation of $Z$ is equal to 2 .

OCR S4 2007 June Q7

7 The continuous random variable $X$ has a uniform distribution over the interval $[ 0 , \theta ]$ so that the probability density function is given by $$f ( x ) = \begin{cases} \frac { 1 } { \theta } & 0 \leqslant x \leqslant \theta
0 & \text { otherwise } \end{cases}$$ where $\theta$ is a positive constant. A sample of $n$ independent observations of $X$ is taken and the sample mean is denoted by $\bar { X }$.

The estimator $T _ { 1 }$ is defined by $T _ { 1 } = 2 \bar { X }$. Show that $T _ { 1 }$ is an unbiased estimator of $\theta$. It is given that the probability density function of the largest value, $U$, in the sample is $$g ( u ) = \begin{cases} \frac { n u ^ { n - 1 } } { \theta ^ { n } } & 0 \leqslant u \leqslant \theta
0 & \text { otherwise } \end{cases}$$
Find $\mathrm { E } ( U )$ and show that $\operatorname { Var } ( U ) = \frac { n \theta ^ { 2 } } { ( n + 1 ) ^ { 2 } ( n + 2 ) }$.
The estimator $T _ { 2 }$ is defined by $T _ { 2 } = \frac { n + 1 } { n } U$. Given that $T _ { 2 }$ is also an unbiased estimator of $\theta$, show that $T _ { 2 }$ is a more efficient estimator than $T _ { 1 }$ for $n > 1$.

OCR S4 2008 June Q1

1 For the mutually exclusive events $A$ and $B , \mathrm { P } ( A ) = \mathrm { P } ( B ) = x$, where $x \neq 0$.

Show that $x \leqslant \frac { 1 } { 2 }$.
Show that $A$ and $B$ are not independent. The event $C$ is independent of $A$ and also independent of $B$, and $\mathrm { P } ( C ) = 2 x$.
Show that $\mathrm { P } ( A \cup B \cup C ) = 4 x ( 1 - x )$.

OCR S4 2008 June Q2

2 Part of Helen’s psychology dissertation involved the reaction times to a certain stimulus. She measured the reaction times of 30 randomly selected students, in seconds correct to 2 decimal places. The results are shown in the following stem-and-leaf diagram.

14	1	2
15	2	4
16	0	3	6
17	1	5	7
18	3	4	5	7	9
19	2	4	6	7	8	9
20	0	1	3	4	5	7	8	9
21	7

Key: 18 | 3 means 1.83 seconds Helen wishes to test whether the population median time exceeds 1.80 seconds.

Give a reason why the Wilcoxon signed-rank test should not be used.
Carry out a suitable non-parametric test at the $5 \%$ significance level.

OCR S4 2008 June Q3

3 From the records of Mulcaster United Football Club the following distribution was suggested as a probability model for future matches. $X$ and $Y$ denoted the numbers of goals scored by the home team and the away team respectively.

	$X$
\cline { 2 - 5 } \multicolumn{1}{c}{}	0	1	2	3
0	0.11	0.04	0.06	0.08
1	0.08	0.05	0.12	0.05
2	0.05	0.08	0.07	0.03
3	0.03	0.06	0.07	0.02

Use the model to find

$\mathrm { E } ( X )$,
the probability that the away team wins a randomly chosen match,
the probability that the away team wins a randomly chosen match, given that the home team scores. One of the directors, an amateur statistician, finds that $\operatorname { Cov } ( X , Y ) = 0.007$. He states that, as this value is very close to zero, $X$ and $Y$ may be considered to be independent.
Comment on the director's statement.

OCR S4 2008 June Q4

4 William takes a bus regularly on the same journey, sometimes in the morning and sometimes in the afternoon. He wishes to compare morning and afternoon journey times. He records the journey times on 7 randomly chosen mornings and 8 randomly chosen afternoons. The results, each correct to the nearest minute, are as follows, where M denotes a morning time and A denotes an afternoon time.

M	A	A	M	M	M	M	M	M	A	A	A	A	A	A
19	20	22	24	25	26	28	30	31	33	35	37	38	39	42

William wishes to test for a difference between the average times of morning and afternoon journeys.

Given that $s _ { M } ^ { 2 } = 16.5$ and $s _ { A } ^ { 2 } = 64.5$, with the usual notation, explain why a $t$-test is not appropriate in this case.
William chooses a non-parametric test at the $5 \%$ significance level. Carry out the test, stating the rejection region.

OCR S4 2008 June Q5

5 The discrete random variable $X$ has moment generating function $\frac { 1 } { 4 } \mathrm { e } ^ { 2 t } + a \mathrm { e } ^ { 3 t } + b \mathrm { e } ^ { 4 t }$, where $a$ and $b$ are constants. It is given that $\mathrm { E } ( X ) = 3 \frac { 3 } { 8 }$.

Show that $a = \frac { 1 } { 8 }$, and find the value of $b$.
Find $\operatorname { Var } ( X )$.
State the possible values of $X$.

OCR S4 2008 June Q6

6 The continuous random variable $Y$ has cumulative distribution function given by $$\mathrm { F } ( y ) = \begin{cases} 0 & y < a ,
1 - \frac { a ^ { 3 } } { y ^ { 3 } } & y \geqslant a , \end{cases}$$ where $a$ is a positive constant. A random sample of 3 observations, $Y _ { 1 } , Y _ { 2 } , Y _ { 3 }$, is taken, and the smallest is denoted by $S$.

Show that $\mathrm { P } ( S > s ) = \left( \frac { a } { s } \right) ^ { 9 }$ and hence obtain the probability density function of $S$.
Show that $S$ is not an unbiased estimator of $a$, and construct an unbiased estimator, $T _ { 1 }$, based on $S$. It is given that $T _ { 2 }$, where $T _ { 2 } = \frac { 2 } { 9 } \left( Y _ { 1 } + Y _ { 2 } + Y _ { 3 } \right)$, is another unbiased estimator of $a$.
Given that $\operatorname { Var } ( Y ) = \frac { 3 } { 4 } a ^ { 2 }$ and $\operatorname { Var } ( S ) = \frac { 9 } { 448 } a ^ { 2 }$, determine which of $T _ { 1 }$ and $T _ { 2 }$ is the more efficient estimator.
The values of $Y$ for a particular sample are 12.8, 4.5 and 7.0. Find the values of $T _ { 1 }$ and $T _ { 2 }$ for this sample, and give a reason, unrelated to efficiency, why $T _ { 1 }$ gives a better estimate of $a$ than $T _ { 2 }$ in this case.

OCR S4 2008 June Q7

7 The probability generating function of the random variable $X$ is given by $$\mathrm { G } ( t ) = \frac { 1 + a t } { 4 - t }$$ where $a$ is a constant.

Find the value of $a$.
Find $\mathrm { P } ( X = 3 )$. The sum of 3 independent observations of $X$ is denoted by $Y$. The probability generating function of $Y$ is denoted by $\mathrm { H } ( t )$.
Use $\mathrm { H } ( t )$ to find $\mathrm { E } ( Y )$.
By considering $\mathrm { H } ( - 1 ) + \mathrm { H } ( 1 )$, show that $\mathrm { P } ( Y$ is an even number $) = \frac { 62 } { 125 }$. \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }

OCR S4 2011 June Q1

1 The random variable $X$ has the distribution $\mathrm { B } ( n , p )$.

Show, from the definition, that the probability generating function of $X$ is $( q + p t ) ^ { n }$, where $q = 1 - p$.
The independent random variable $Y$ has the distribution $\mathrm { B } ( 2 n , p )$ and $T = X + Y$. Use probability generating functions to determine the distribution of $T$, giving its parameters.

OCR S4 2011 June Q2

2 A botanist believes that some species of plants produce more flowers at high altitudes than at low altitudes. In order to investigate this belief the botanist randomly samples 11 species of plants each of which occurs at both altitudes. The numbers of flowers on the plants are shown in the table.

Species	1	2	3	4	5	6	7	8	9	10	11
Number of flowers at low altitude	5	3	4	7	2	9	6	5	4	11	2
Number of flowers at high altitude	1	6	10	8	14	16	20	21	15	2	12

Use the Wilcoxon signed rank test at the 5\% significance level to test the botanist's belief.
Explain why the Wilcoxon rank sum test should not be used for this test.

OCR S4 2011 June Q3

3 For the events $A$ and $B , \mathrm { P } ( A ) = \mathrm { P } ( B ) = \frac { 3 } { 4 }$ and $\mathrm { P } \left( A \mid B ^ { \prime } \right) = \frac { 1 } { 2 }$.

Find $\mathrm { P } ( A \cap B )$. For a third event $C , \mathrm { P } ( C ) = \frac { 1 } { 4 }$ and $C$ is independent of the event $A \cap B$.
Find $\mathrm { P } ( A \cap B \cap C )$.
Given that $\mathrm { P } ( C \mid A ) = \lambda$ and $\mathrm { P } ( B \mid C ) = 3 \lambda$, and that no event occurs outside $A \cup B \cup C$, find the value of $\lambda$.

OCR S4 2011 June Q4

4 The discrete random variable $X$ has moment generating function $\left( \frac { 1 } { 4 } + \frac { 3 } { 4 } \mathrm { e } ^ { t } \right) ^ { 3 }$.

Find $\mathrm { E } ( X )$.
Find $\mathrm { P } ( X = 2 )$.
Show that $X$ can be expressed as a sum of 3 independent observations of a random variable $Y$. Obtain the probability distribution of $Y$, and the variance of $Y$.

OCR S4 2011 June Q5

5 A test was carried out to compare the breaking strengths of two brands of elastic band, $A$ and $B$, of the same size. Random samples of 6 were selected from each brand and the breaking strengths were measured. The results, in suitable units and arranged in ascending order for each brand, are as follows.

Brand $A :$	5.6	8.7	9.2	10.7	11.2	12.6
Brand $B :$	10.1	11.6	12.0	12.2	12.9	13.5

Give one advantage that a non-parametric test might have over a parametric test in this context.
Carry out a suitable Wilcoxon test at the $5 \%$ significance level of whether there is a difference between the average breaking strengths of the two brands.
An extra elastic band of brand $B$ was tested and found to have a breaking strength exceeding all of the other 12 bands. Determine whether this information alters the conclusion of your test.

OCR S4 2011 June Q6

6 A City Council comprises 16 Labour members, 14 Conservative members and 6 members of Other parties. A sample of two members was chosen at random to represent the Council at an event. The number of Labour members and the number of Conservative members in this sample are denoted by $L$ and $C$ respectively. The joint probability distribution of $L$ and $C$ is given in the following table. $C$

$L$
	0	1	2
0	$\frac { 1 } { 42 }$	$\frac { 16 } { 105 }$	$\frac { 4 } { 21 }$
1	$\frac { 2 } { 15 }$	$\frac { 16 } { 45 }$	0
2	$\frac { 13 } { 90 }$	0	0

Verify the two non-zero probabilities in the table for which $C = 1$.
Find the expected number of Conservatives in the sample.
Find the expected number of Other members in the sample.
Explain why $L$ and $C$ are not independent, and state what can be deduced about $\operatorname { Cov } ( L , C )$.

OCR S4 2011 June Q7

7 The continuous random variable $U$ has unknown mean $\mu$ and known variance $\sigma ^ { 2 }$. In order to estimate $\mu$, two random samples, one of 4 observations of $U$ and the other of 6 observations of $U$, are taken. The sample means are denoted by $\bar { U } _ { 4 }$ and $\bar { U } _ { 6 }$ respectively. One estimator $S$, given by $S = \frac { 1 } { 2 } \left( \bar { U } _ { 4 } + \bar { U } _ { 6 } \right)$, is proposed.

Show that $S$ is unbiased and find $\operatorname { Var } ( S )$ in terms of $\sigma ^ { 2 }$. A second estimator $T$ of the form $a \bar { U } _ { 4 } + b \bar { U } _ { 6 }$ is proposed, where $a$ and $b$ are chosen such that $T$ is an unbiased estimator for $\mu$ with the smallest possible variance.
Find the values of $a$ and $b$ and the corresponding variance of $T$.
State, giving a reason, which of $S$ and $T$ is the better estimator.
Compare the efficiencies of this preferred estimator and the mean of all 10 observations.

OCR S4 2012 June Q1

1 Independent random variables $X$ and $Y$ have distributions $\mathrm { B } ( 7 , p )$ and $\mathrm { B } ( 8 , p )$ respectively.

Explain why $X + Y \sim \mathrm {~B} ( 15 , p )$.
Find $\mathrm { P } ( X = 2 \mid X + Y = 5 )$.

OCR S4 2012 June Q2

2 The continuous random variable $X$ has probability density function given by $$f ( x ) = \begin{cases} 4 x e ^ { - 2 x } & x \geqslant 0
0 & \text { otherwise } \end{cases}$$

Show that the moment generating function ( mgf ) of $X$ is $$\frac { 4 } { ( 2 - t ) ^ { 2 } } , \text { where } | t | < 2$$
Explain why the $\operatorname { mgf }$ of $- X$ is $\frac { 4 } { ( 2 + t ) ^ { 2 } }$.
Two random observations of $X$ are denoted by $X _ { 1 }$ and $X _ { 2 }$. What is the $\operatorname { mgf }$ of $X _ { 1 } - X _ { 2 }$ ?

OCR S4 2012 June Q3

3 Because of the large number of students enrolled for a university geography course and the limited accommodation in the lecture theatre, the department provides a filmed lecture. Students are randomly assigned to two groups, one to attend the lecture theatre and the other the film. At the end of term the two groups are given the same examination. The geography professor wishes to test whether there is a difference in the performance of the two groups and selects the marks of two random samples of students, 6 from the group attending the lecture theatre and 7 from the group attending the films. The marks for the two samples, ordered for convenience, are shown below.

Lecture theatre:	30	36	48	51	59	62
Filmed lecture:	40	49	52	56	63	64	68

Stating a necessary assumption, carry out a suitable non-parametric test, at the $10 \%$ significance level, for a difference between the median marks of the two groups.
State conditions under which a two-sample $t$-test could have been used.
Assuming that the tests in parts (i) and (ii) are both valid, state with a reason which test would be preferable.

OCR S4 2012 June Q4

4 The random variable $U$ has the distribution $\operatorname { Geo } ( p )$.

Show, from the definition, that the probability generating function ( pgf ) of $U$ is given by $$G _ { U } ( t ) = \frac { p t } { 1 - q t } , \text { for } | t | < \frac { 1 } { q } ,$$ where $q = 1 - p$.
Explain why the condition $| t | < \frac { 1 } { q }$ is necessary.
Use the pgf to obtain $\mathrm { E } ( U )$. Each packet of Corn Crisp cereal contains a voucher and $20 \%$ of the vouchers have a gold star. When 4 gold stars have been collected a gift can be claimed. Let $X$ denote the number of packets bought by a family up to and including the one from which the $4 ^ { \text {th } }$ gold star is obtained.
Obtain the pgf of $X$.
Find $\mathrm { P } ( X = 6 )$.

\(L\)
	0	1	2
0	\(\frac { 1 } { 42 }\)	\(\frac { 16 } { 105 }\)	\(\frac { 4 } { 21 }\)
1	\(\frac { 2 } { 15 }\)	\(\frac { 16 } { 45 }\)	0
2	\(\frac { 13 } { 90 }\)	0	0

	\(X\)
\cline { 2 - 5 } \multicolumn{1}{l}{}	- 1	0	1
1	0.24	0.22	0.04
2	0.26	0.18	0.06

Brand \(A :\)	5.6	8.7	9.2	10.7	11.2	12.6
Brand \(B :\)	10.1	11.6	12.0	12.2	12.9	13.5

Questions — OCR S4 (86 questions)

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