Questions - OCR S4 - A-Level Maths

Show by calculation that the test can never result in rejection of the null hypothesis when $n = 4$. The coach of a college swimming team expects Elena, the best 50 m freestyle swimmer, to have a median time less than 30 seconds. Elena found from records of her previous 72 swims that 44 were less than 30 seconds and 28 were greater than 30 seconds.
Stating a necessary assumption, test at the $5 \%$ significance level whether Elena's median time for the 50 m freestyle is less than 30 seconds.

OCR S4 2012 June Q6

6 The random variables $S$ and $T$ are independent and have joint probability distribution given in the table.

	$S$
\cline { 2 - 5 }		0	1	2
\cline { 2 - 5 }	1	$a$	0.18	$b$
2	0.08	0.12	0.20
\cline { 2 - 5 }
\cline { 2 - 5 }

Show that $a = 0.12$ and find the value of $b$.
Find $\mathrm { P } ( T - S = 1 )$.
Find $\operatorname { Var } ( T - S )$.

OCR S4 2012 June Q7

7 The continuous random variable $X$ has probability density function given by $$f ( x ) = \begin{cases} \frac { 1 } { 4 } ( 1 + a x ) & - 2 \leqslant x \leqslant 2
0 & \text { otherwise } \end{cases}$$ where $a$ is a constant.

Show that $| a | \leqslant \frac { 1 } { 2 }$.
Find $\mathrm { E } ( X )$ in terms of $a$.
Construct an unbiased estimator $T _ { 1 }$ of $a$ based on one observation $X _ { 1 }$ of $X$.
A second observation $X _ { 2 }$ is taken. Show that $T _ { 2 }$, where $T _ { 2 } = \frac { 3 } { 8 } \left( X _ { 1 } + X _ { 2 } \right)$, is also an unbiased estimator of a.
Given that $\operatorname { Var } ( X ) = \sigma ^ { 2 }$, determine which of $T _ { 1 }$ and $T _ { 2 }$ is the better estimator.

OCR S4 2012 June Q8

8 Events $A$ and $B$ are such that $\mathrm { P } ( A ) = 0.3$ and $\mathrm { P } ( A \mid B ) = 0.6$.

Show that $\mathrm { P } ( B ) \leqslant 0.5$.
Given also that $\mathrm { P } ( A \cup B ) = x$, find $\mathrm { P } ( B )$ in terms of $x$.

OCR S4 2013 June Q1

	$S$
	0	1	2
0	$\frac { 1 } { 8 }$	$\frac { 1 } { 8 }$	0
1	$\frac { 1 } { 8 }$	$\frac { 1 } { 4 }$	$\frac { 1 } { 8 }$
2	0	$\frac { 1 } { 8 }$	$\frac { 1 } { 8 }$

An unbiased coin is tossed three times. The random variables $F$ and $S$ denote the total number of heads that occur in the first two tosses and the total number of heads that occur in the last two tosses respectively. The table above shows the joint probability distribution of $F$ and $S$.

Show how the entry $\frac { 1 } { 4 }$ in the table is obtained.
Find $\operatorname { Cov } ( F , S )$.

OCR S4 2013 June Q2

2 Two drugs, I and II, for alleviating hay fever are trialled in a hospital on each of 12 volunteer patients. Each received drug I on one day and drug II on a different day. After receiving a drug, the number of times each patient sneezed over a period of one hour was noted. The results are given in the table.

Patient	1	2	3	4	5	6	7	8	9	10	11	12
Drug I	11	34	19	16	10	29	6	17	20	13	4	25
Drug II	12	20	10	18	3	21	9	13	10	19	9	12

The patients may be considered to be a random sample of all hay fever sufferers.
A researcher believes that patients taking drug II sneeze less than patients taking drug I.
Test this belief using the Wilcoxon signed rank test at the $5 \%$ significance level.

OCR S4 2013 June Q3

3 The continuous random variable $X$ has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} \frac { 1 } { 4 } x \mathrm { e } ^ { - \frac { 1 } { 2 } x } & x \geqslant 0
0 & \text { otherwise } . \end{cases}$$

Show that the moment generating function of $X$ is $( 1 - 2 t ) ^ { - 2 }$ for $t < \frac { 1 } { 2 }$, and state why the condition $t < \frac { 1 } { 2 }$ is necessary.
Use the moment generating function to find $\operatorname { Var } ( X )$.

OCR S4 2013 June Q4

4 The effect of water salinity on the growth of a type of grass was studied by a biologist. A random sample of 22 seedlings was divided into two groups $A$ and $B$, each of size 11 .
Group $A$ was treated with water of $0 \%$ salinity and group $B$ was treated with water of $0.5 \%$ salinity. After three weeks the height (in cm) of each seedling was measured with the following results, which are ordered for convenience.

Group $A$	8.6	9.4	9.7	9.8	10.1	10.5	11.0	11.2	11.8	12.7
Group $B$	7.4	8.4	8.5	8.8	9.2	9.3	9.5	9.9	10.0	11.1

Jeffery was asked to test whether the two treatments resulted, on average, in a difference in growth. He chose the Wilcoxon rank sum test.

Justify Jeffery's choice of test.
Carry out the test at the $5 \%$ significance level.

OCR S4 2013 June Q5

5 The discrete random variable $U$ has probability distribution given by $$\mathrm { P } ( U = r ) = \begin{cases} \frac { 1 } { 16 } \binom { 4 } { r } & r = 0,1,2,3,4
0 & \text { otherwise } \end{cases}$$

Find and simplify the probability generating function (pgf) of $U$.
Use the pgf to find $\mathrm { E } ( U )$ and $\operatorname { Var } ( U )$.
Identify the distribution of $U$, giving the values of any parameters.
Obtain the pgf of $Y$, where $Y = U ^ { 2 }$.
State, giving a reason, whether you can obtain the pgf of $U + Y$ by multiplying the pgf of $U$ by the pgf of $Y$.

OCR S4 2013 June Q6

6 The continuous random variable $X$ has mean $\mu$ and variance $\sigma ^ { 2 }$, and the independent continuous random variable $Y$ has mean $2 \mu$ and variance $3 \sigma ^ { 2 }$. Two observations of $X$ and three observations of $Y$ are taken and are denoted by $X _ { 1 } , X _ { 2 } , Y _ { 1 } , Y _ { 2 }$ and $Y _ { 3 }$ respectively.

Find the expectation of the sum of these 5 observations and hence construct an unbiased estimator, $T _ { 1 }$, of $\mu$.
The estimator $T _ { 2 }$, where $T _ { 2 } = X _ { 1 } + X _ { 2 } + c \left( Y _ { 1 } + Y _ { 2 } + Y _ { 3 } \right)$, is an unbiased estimator of $\mu$. Find the value of the constant $c$.
Determine which of $T _ { 1 }$ and $T _ { 2 }$ is more efficient.
Find the values of the constants $a$ and $b$ for which $$a \left( X _ { 1 } ^ { 2 } + X _ { 2 } ^ { 2 } \right) + b \left( Y _ { 1 } ^ { 2 } + Y _ { 2 } ^ { 2 } + Y _ { 3 } ^ { 2 } \right)$$ is an unbiased estimator of $\sigma ^ { 2 }$.

OCR S4 2013 June Q7

7 Each question on a multiple-choice examination paper has $n$ possible responses, only one of which is correct. Joni takes the paper and has probability $p$, where $0 < p < 1$, of knowing the correct response to any question, independently of any other. If she knows the correct response she will choose it, otherwise she will choose randomly from the $n$ possibilities. The events $K$ and $A$ are 'Joni knows the correct response' and 'Joni answers correctly' respectively.

Show that $\mathrm { P } ( A ) = \frac { q + n p } { n }$, where $q = 1 - p$.
Find $P ( K \mid A )$. A paper with 100 questions has $n = 4$ and $p = 0.5$. Each correct response scores 1 and each incorrect response scores - 1 .
(a) Joni answers all the questions on the paper and scores 40 . How many questions did she answer correctly?
(b) By finding the distribution of the number of correct answers, or otherwise, find the probability that Joni scores at least 40 on the paper using her strategy.

OCR S4 2014 June Q1

1 A teacher believes that the calculator paper in a GCSE Mathematics examination was easier than the non-calculator paper. The marks of a random sample of ten students are shown in the table.

Student	A	B	C	D	E	F	G	H	I	J
Mark on paper 1 (non-calculator)	66	79	58	87	67	55	75	62	50	84
Mark on paper 2 (calculator)	57	84	70	90	75	42	82	72	65	82

Use a Wilcoxon signed-rank test, at the $5 \%$ significance level, to test the teacher's belief.
State the assumption necessary for this test to be applied.

OCR S4 2014 June Q2

2 During an outbreak of a disease, it is known that $68 \%$ of people do not have the disease. Of people with the disease, $96 \%$ react positively to a test for diagnosing it, as do $m \%$ of people who do not have the disease.

In the case $m = 8$, find the probability that a randomly chosen person has the disease, given that the person reacts positively to the test.
What value of $m$ would be required for the answer to part (i) to be 0.95 ?

OCR S4 2014 June Q3

3 The discrete random variable $X$ has probability generating function $\frac { t } { a - b t }$, where $a$ and $b$ are constants.

Find a relationship between $a$ and $b$.
Use the probability generating function to find $\mathrm { E } ( X )$ in terms of $a$, giving your answer as simply as possible.
Expand the probability generating function as a power series, as far as the term in $t ^ { 3 }$, giving the coefficients in terms of $a$ and $b$.
Name the distribution for which $\frac { t } { a - b t }$ is the probability generating function, and state its parameter(s) in terms of $a$.

OCR S4 2014 June Q4

4 The continuous random variable $X$ has probability density function $$f ( x ) = \left\{ \begin{array} { c c } x & 0 \leqslant x \leqslant 1
2 - x & 1 \leqslant x \leqslant 2
0 & \text { otherwise } \end{array} \right.$$

Show that the moment generating function of $X$ is $\frac { \left( \mathrm { e } ^ { t } - 1 \right) ^ { 2 } } { t ^ { 2 } }$.
$Y _ { 1 }$ and $Y _ { 2 }$ are independent observations of a random variable $Y$. The moment generating function of $Y _ { 1 } + Y _ { 2 }$ is $\frac { \left( \mathrm { e } ^ { t } - 1 \right) ^ { 2 } } { t ^ { 2 } }$.
Write down the moment generating function of $Y$.
Use the expansion of $\mathrm { e } ^ { t }$ to find $\operatorname { Var } ( Y )$.
Deduce the value of $\operatorname { Var } ( X )$.

OCR S4 2014 June Q5

5 Two discrete random variables $X$ and $Y$ have a joint probability distribution defined by $$\mathrm { P } ( X = x , Y = y ) = a ( x + y + 1 ) \quad \text { for } x = 0,1,2 \text { and } y = 0,1,2 ,$$ where $a$ is a constant.

Show that $a = \frac { 1 } { 27 }$.
Find $\mathrm { E } ( X )$.
Find $\operatorname { Cov } ( X , Y )$.
Are $X$ and $Y$ independent? Give a reason for your answer.
Find $\mathrm { P } ( X = 1 \mid Y = 2 )$.

OCR S4 2014 June Q6

6 A Wilcoxon rank-sum test with samples of sizes 11 and 12 is carried out.

What is the least possible value of the test statistic $W$ ?
The null hypothesis is that the two samples came from identical populations. Given that the null hypothesis was rejected at the $1 \%$ level using a 2 -tail test, find the set of possible values of $W$.

OCR S4 2014 June Q7

7 The continuous random variable $X$ has probability density function $$f ( x ) = \left\{ \begin{array} { c l } \frac { k } { ( x + \theta ) ^ { 5 } } & \text { for } x \geqslant 0
0 & \text { otherwise } \end{array} \right.$$ where $k$ is a positive constant and $\theta$ is a parameter taking positive values.

Find an expression for $k$ in terms of $\theta$.
Show that $\mathrm { E } ( X ) = \frac { 1 } { 3 } \theta$. You are given that $\operatorname { Var } ( X ) = \frac { 2 } { 9 } \theta ^ { 2 }$. A random sample $X _ { 1 } , X _ { 2 } , \ldots , X _ { n }$ of $n$ observations of $X$ is obtained. The estimator $T _ { 1 }$ is defined as $T _ { 1 } = \frac { 3 } { n } \sum _ { i = 1 } ^ { n } X _ { i }$.
Show that $T _ { 1 }$ is an unbiased estimator of $\theta$, and find the variance of $T _ { 1 }$.
A second unbiased estimator $T _ { 2 }$ is defined by $T _ { 2 } = \frac { 1 } { 3 } \left( X _ { 1 } + 3 X _ { 2 } + 5 X _ { 3 } \right)$. For the case $n = 3$, which of $T _ { 1 }$ and $T _ { 2 }$ is more efficient? \section*{OCR}

OCR S4 2016 June Q1

1 Ten archers shot at targets with two types of bow. Their scores out of 100 are shown in the table.

Archer	$A$	$B$	$C$	$D$	$E$	$F$	$G$	$H$	$I$	$J$
Bow type $P$	95	97	92	85	87	92	90	89	98	77
Bow type $Q$	91	91	88	90	80	88	93	85	94	84

Use the sign test, at the $5 \%$ level of significance, to test the hypothesis that bow type $P$ is better than bow type $Q$.
Why would a Wilcoxon signed rank test, if valid, be a better test than the sign test?

OCR S4 2016 June Q2

2 Low density lipoprotein (LDL) cholesterol is known as 'bad' cholesterol.
15 randomly chosen patients, each with an LDL level of 190 mg per decilitre of blood, were given one of two treatments, chosen at random. After twelve weeks their LDL levels, in mg per decilitre, were as follows.

Treatment $A$	189	168	176	186	183	187	188
Treatment $B$	177	179	173	180	178	170	175	174

Use a Wilcoxon rank sum test, at the $5 \%$ level of significance, to test whether the LDL levels of patients given treatment $B$ are lower than the LDL levels of patients given treatment $A$.

OCR S4 2016 June Q3

3 The table shows the joint probability distribution of two random variables $X$ and $Y$.

\cline { 2 - 5 } \multicolumn{2}{c\|}{}	$Y$
\cline { 2 - 5 } \multicolumn{2}{c\|}{}	0	1	2
\multirow{3}{*}{$X$}	0	0.07	0.07	0.16
\cline { 2 - 5 }	1	0.06	0.09	0.15
\cline { 2 - 5 }	2	0.07	0.14	0.19

Find $\operatorname { Cov } ( X , Y )$.
Are $X$ and $Y$ independent? Give a reason for your answer.
Find $\mathrm { P } ( X = 1 \mid X Y = 2 )$.

OCR S4 2016 June Q4

4 The continuous random variable $Y$ has a uniform (rectangular) distribution on $[ a , b ]$, where $a$ and $b$ are constants.

Show that the moment generating function $\mathrm { M } _ { Y } ( \mathrm { t } )$ of $Y$ is $\frac { \left( \mathrm { e } ^ { b t } - \mathrm { e } ^ { a t } \right) } { t ( b - a ) }$.
Use the series expansion of $\mathrm { e } ^ { x }$ to show that the mean and variance of $Y$ are $\frac { 1 } { 2 } ( a + b )$ and $\frac { 1 } { 12 } ( b - a ) ^ { 2 }$, respectively.

OCR S4 2016 June Q5

5 Events $A$ and $B$ are such that $\mathrm { P } ( A ) = 0.5 , \mathrm { P } ( B ) = 0.6$ and $\mathrm { P } \left( A \mid B ^ { \prime } \right) = 0.75$.

Find $\mathrm { P } ( A \cap B )$ and $\mathrm { P } ( A \cup B )$.
Determine, giving a reason in each case,
(a) whether $A$ and $B$ are mutually exclusive,
(b) whether $A$ and $B$ are independent.
A further event $C$ is such that $\mathrm { P } ( A \cup B \cup C ) = 1$ and $\mathrm { P } ( A \cap B \cap C ) = 0.05$. It is also given that $\mathrm { P } \left( A \cap B ^ { \prime } \cap C \right) = \mathrm { P } \left( A ^ { \prime } \cap B \cap C \right) = x$ and $\mathrm { P } \left( A \cap B ^ { \prime } \cap C ^ { \prime } \right) = 2 x$.
Find $\mathrm { P } ( C )$.

OCR S4 2016 June Q6

6 Andrew has five coins. Three of them are unbiased. The other two are biased such that the probability of obtaining a head when one of them is tossed is $\frac { 3 } { 5 }$. Andrew tosses all five coins. It is given that the probability generating function of $X$, the number of heads obtained on the unbiased coins, is $\mathrm { G } _ { X } ( t )$, where $$\mathrm { G } _ { X } ( t ) = \frac { 1 } { 8 } + \frac { 3 } { 8 } t + \frac { 3 } { 8 } t ^ { 2 } + \frac { 1 } { 8 } t ^ { 3 }$$

Find $G _ { Y } ( \mathrm { t } )$, the probability generating function of $Y$, the number of heads on the biased coins.
The random variable $Z$ is the total number of heads obtained when Andrew tosses all five coins. Find the probability generating function of $Z$, giving your answer as a polynomial.
Find $\mathrm { E } ( Z )$ and $\operatorname { Var } ( Z )$.
Write down the value of $\mathrm { P } ( Z = 3 )$.

OCR S4 2016 June Q7

7 A continuous random variable $Y$ has cumulative distribution function $$\mathrm { F } ( y ) = \left\{ \begin{array} { c c } 0 & y < a
1 - \frac { a ^ { 5 } } { y ^ { 5 } } & y \geqslant a \end{array} \right.$$ where $a$ is a parameter.
Two independent observations of $Y$ are denoted by $Y _ { 1 }$ and $Y _ { 2 }$. The smaller of them is denoted by S .

Show that $P ( S > \mathrm { s } ) = \frac { a ^ { 10 } } { s ^ { 10 } }$ and hence find the probability density function of $S$.
Show that $S$ is not an unbiased estimator of $a$, and construct an unbiased estimator of $a , T _ { 1 }$ based on $S$.
Construct another unbiased estimator of $a , T _ { 2 }$, of the form $k \left( Y _ { 1 } + Y _ { 2 } \right)$, where $k$ is a constant to be found.
Without further calculation, explain how you would decide which of $T _ { 1 }$ and $T _ { 2 }$ is the more efficient estimator.

	\(S\)
	0	1	2
0	\(\frac { 1 } { 8 }\)	\(\frac { 1 } { 8 }\)	0
1	\(\frac { 1 } { 8 }\)	\(\frac { 1 } { 4 }\)	\(\frac { 1 } { 8 }\)
2	0	\(\frac { 1 } { 8 }\)	\(\frac { 1 } { 8 }\)

Group \(A\)	8.6	9.4	9.7	9.8	10.1	10.5	11.0	11.2	11.8	12.7
Group \(B\)	7.4	8.4	8.5	8.8	9.2	9.3	9.5	9.9	10.0	11.1

Archer	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)	\(I\)	\(J\)
Bow type \(P\)	95	97	92	85	87	92	90	89	98	77
Bow type \(Q\)	91	91	88	90	80	88	93	85	94	84

Treatment \(A\)	189	168	176	186	183	187	188
Treatment \(B\)	177	179	173	180	178	170	175	174

\cline { 2 - 5 } \multicolumn{2}{c\|}{}	\(Y\)
\cline { 2 - 5 } \multicolumn{2}{c\|}{}	0	1	2
\multirow{3}{*}{\(X\)}	0	0.07	0.07	0.16
\cline { 2 - 5 }	1	0.06	0.09	0.15
\cline { 2 - 5 }	2	0.07	0.14	0.19

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