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OCR S3 2014 June Q1

5 marks Standard +0.3

1 The independent random variables $X$ and $Y$ have Poisson distributions with parameters 16 and 2 respectively, and $Z = \frac { 1 } { 2 } X - Y$.

Find $\mathrm { E } ( Z )$ and $\operatorname { Var } ( Z )$.
State whether $Z$ has a Poisson distribution, giving a reason for your answer.

Calculate an approximate $99.9 \%$ confidence interval for the proportion of voters intending to vote Conservative. The actual proportion voting Conservative was above the upper limit of the confidence interval.
Give two possible reasons for this occurrence.
What sample size would be required to produce a $99.9 \%$ confidence interval of width 0.05 ?

OCR S3 2014 June Q6

8 marks Standard +0.3

6 The continuous random variable $X$ has probability density function given by $$\mathrm { f } ( x ) = \left\{ \begin{array} { c l } k \sin x & 0 \leqslant x \leqslant \frac { 1 } { 2 } \pi , \\ k \left( 2 - \frac { 2 x } { \pi } \right) & \frac { 1 } { 2 } \pi \leqslant x \leqslant \pi , \\ 0 & \text { otherwise, } \end{array} \right.$$ where $k$ is a constant.

Show that $k = \frac { 4 } { 4 + \pi }$.
Find $\mathrm { E } ( X )$, correct to 3 significant figures, showing all necessary working.

OCR S3 2014 June Q7

9 marks Standard +0.3

7 A random sample of 100 adults with a chronic disease was chosen. Each adult was randomly assigned to one of three different treatments. After six months of treatment, each adult was then assessed and classified as 'much improved', 'improved', 'slightly improved' or 'no change'. The results are summarised in Table 1. \begin{table}[h]

	Treatment $A$	Treatment $B$	Treatment $C$
Much improved	12	16	4
Improved	13	12	6
Slightly improved	7	6	7
No change	5	3	9

\captionsetup{labelformat=empty} \caption{Table 1}

\end{table} A $\chi ^ { 2 }$ test, at the $5 \%$ significance level, is to be carried out.

State suitable hypotheses. Combining the last two rows of Table 1 gives Table 2. \begin{table}[h]
Treatment $A$ Treatment $B$ Treatment $C$
Much improved 12 16 4
Improved 13 12 6
Slightly improved/ No change 12 9 16
\captionsetup{labelformat=empty} \caption{Table 2}
\end{table}
By considering the expected frequencies for Treatment $C$ in Table 1, explain why it was necessary to combine rows.
Show that the contribution to the $\chi ^ { 2 }$ value for the cell 'slightly improved/no change, Treatment $C$ ' is 4.231 , correct to 3 decimal places. You are given that the $\chi ^ { 2 }$ test statistic is 10.51 , correct to 2 decimal places.
Carry out the test.

OCR S3 2014 June Q8

10 marks Standard +0.3

8 A random sample of 20 plots of land, each of equal area, was used to test whether the addition of phosphorus would increase the yield of corn. 10 plots were treated with phosphorus and 10 plots were untreated. The yields of corn, in litres, on a treated plot and on an untreated plot are denoted by $X$ and $Y$ respectively. You are given that $$\sum x = 2112 , \quad \sum y = 2008$$ You are also given that an unbiased estimate for the variance of treated plots is 87.96 and an unbiased estimate for the variance of untreated plots is 31.96 , both correct to 4 significant figures.

You may assume that the population variance estimates are sufficiently similar for the assumption of common variance to be made. What other assumption needs to be made for a $t$-test to be valid?
Carry out a suitable $t$-test at the $1 \%$ significance level, to test whether the use of phosphorus increases the yield of corn.

OCR S3 2014 June Q9

11 marks Challenging +1.2

9 A rectangle of area $A \mathrm {~m} ^ { 2 }$ has a perimeter of 20 m and each of the two shorter sides are of length $X \mathrm {~m}$, where $X$ is uniformly distributed between 0 and 2 .

Write down an expression for $A$ in terms of $X$, and hence show that $A = 25 - ( X - 5 ) ^ { 2 }$.
Write down the probability density function of $X$.
Show that the cumulative distribution function of $A$ is $$\mathrm { F } ( a ) = \left\{ \begin{array} { l r } 0 & a < 0 , \\ \frac { 1 } { 2 } ( 5 - \sqrt { 25 - a } ) & 0 \leqslant a \leqslant 16 , \\ 1 & a > 16 . \end{array} \right.$$
Find the probability density function of $A$. \section*{END OF QUESTION PAPER} \section*{$\mathrm { OCR } ^ { \text {勾 } }$}

OCR S3 2015 June Q1

6 marks Moderate -0.8

1 A laminate consists of 4 layers of material $C$ and 3 layers of material $D$. The thickness of a layer of material $C$ has a normal distribution with mean 1 mm and standard deviation 0.1 mm , and the thickness of a layer of material $D$ has a normal distribution with mean 8 mm and standard deviation 0.2 mm . The layers are independent of one another.

Find the mean and variance of the total thickness of the laminate.
What total thickness is exceeded by $1 \%$ of the laminates?

OCR S3 2015 June Q2

7 marks Standard +0.3

2 In a poll of people aged 18-21, 46 out of 200 randomly chosen university students agreed with a proposition. 51 out of 300 randomly chosen others who were not university students agreed with it. Test, at the $5 \%$ significance level, whether the proportion of university students who agree with the proposition differs from the proportion of those who are not university students.

OCR S3 2015 June Q3

12 marks Standard +0.3

3 A tutor gave an assessment to 6 randomly chosen new eleven-year-old students. After each student had received 30 hours of tuition, they were given a second assessment. The scores are shown in the table.

Student	$A$	$B$	$C$	$D$	$E$	$F$
1st assessment	124	121	111	113	118	119
2nd assessment	127	119	114	110	120	122

Show that, at the $5 \%$ significance level, there is insufficient evidence that students' scores are higher, on average, after tuition than before tuition. State a necessary assumption.
Disappointed by this result, the tutor looked again at the first assessment. She discovered that the first assessment was too easy, in fact being a test for ten-year-olds, not eleven-year-olds. She decided to reduce each score for the first assessment by a constant integer $k$. Find the least value of $k$ for which there is evidence at the $5 \%$ significance level that the students' scores have, on average, improved.

OCR S3 2015 June Q4

9 marks Standard +0.3

4 A set of bathroom scales is known to operate with an error which is normally distributed. One morning a man weighs himself 4 times. The 4 values for his mass, in kg , which can be considered to be a random sample are as follows. $$\begin{array} { l l l l } 62.6 & 62.8 & 62.1 & 62.5 \end{array}$$

Find a $95 \%$ confidence interval for his mass. Give the end-points of the interval correct to 3 decimal places.
Based on these results, a $y \%$ confidence interval has width 0.482 . Find $y$.

OCR S3 2015 June Q5

11 marks Challenging +1.2

5 Two guesthouses, the Albion and the Blighty, have 8 and 6 rooms respectively. The demand for rooms at the Albion has a Poisson distribution with mean 6.5 and the demand for rooms at the Blighty has an independent Poisson distribution with mean 5.5. The owners have agreed that if their guesthouse is full, they will re-direct guests to the other.

Find the probability that, on any particular night, the two guesthouses together do not have enough rooms to meet demand.
The Albion charges $\pounds 60$ per room per night, and the Blighty $\pounds 80$. Find the probability, that on a particular night, the total income of the two guesthouses is exactly $\pounds 400$.
If $A$ is the number of rooms demanded at the Albion each night, and $B$ the number of rooms demanded at the Blighty each night, find the mean and variance of the variable $C = 60 A + 80 B$. State whether $C$ has a Poisson distribution, giving a reason for your answer.

OCR S3 2015 June Q6

13 marks Standard +0.3

6 In each of 38 randomly selected weeks of the English Premier Football League there were 10 matches. Table 1 summarises the number of home wins in 10 matches, $X$, and the corresponding number of weeks. \begin{table}[h]

Number of home wins	0	1	2	3	4	5	6	7	8	9	10
Number of weeks	0	1	2	8	8	9	7	1	2	0	0

\captionsetup{labelformat=empty} \caption{Table 1}

\end{table} A researcher investigates whether $X$ can be modelled by the distribution $\mathrm { B } ( 10 , p )$. He calculates the expected frequencies using a value of $p$ obtained from the sample mean.

Show that $p = 0.45$. Table 2 shows the observed and expected number of weeks. \begin{table}[h]
Number of home wins 0 1 2 3 4 5 6 7 8 9 10 Totals
Observed number of weeks 0 1 2 8 8 9 7 1 2 0 0 38
Expected number of weeks 0.096 0.788 2.899 6.326 9.058 8.893 6.064 2.835 0.870 0.158 0.013 38
\captionsetup{labelformat=empty} \caption{Table 2
Show how the value of 2.835 for 7 home wins is obtained.}

\end{table} The researcher carries out a test, at the $5 \%$ significance level, of whether the distribution $\mathrm { B } ( 10 , p )$ fits the data.

Explain why it is necessary to combine classes.

Carry out the test.

OCR S3 2015 June Q7

14 marks Standard +0.3

7 A continuous random variable $X$ has probability density function $$f ( x ) = \left\{ \begin{array} { c c } k x & 0 \leqslant x < 2 \\ \frac { k ( 4 - x ) ^ { 2 } } { 2 } & 2 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{array} \right.$$ where $k$ is a constant.

Show that $k = \frac { 3 } { 10 }$.
Find $\mathrm { E } ( X )$.
Find the cumulative distribution function of $X$.
Find the upper quartile of $X$, correct to 3 significant figures. \section*{END OF QUESTION PAPER}

OCR S4 2016 June Q1

8 marks Moderate -0.8

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

8 marks Standard +0.3

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

9 marks Standard +0.3

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

9 marks Standard +0.8

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

11 marks Standard +0.8

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

13 marks

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

14 marks Challenging +1.8

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.

OCR S4 2017 June Q1

4 marks Standard +0.3

1 A meteorologist claims that the median daily rainfall in London is 2.2 mm . A single sample sign test is to be used to test the claim, using the following hypotheses:
$\mathrm { H } _ { 0 }$ : a sample comes from a population with median 2.2,
$\mathrm { H } _ { 1 }$ : the sample does not come from a population with median 2.2.
30 randomly selected observations of daily rainfall in London are compared with 2.2, and given a '+' sign if greater than 2.2 and a '-' sign if less than 2.2. (You may assume that no data values are exactly equal to 2.2.) The test is to be carried out at the $5 \%$ level of significance. Let the number of ' + ' signs be $k$. Find, in terms of $k$, the critical region for the test showing the values of any relevant probabilities.

OCR S4 2017 June Q2

11 marks Challenging +1.2

2 The independent discrete random variables $X$ and $Y$ can take the values 0,1 and 2 with probabilities as given in the tables.

$x$	0	1	2
$\mathrm { P } ( X = x )$	0.5	0.3	0.2

$\quad$

$y$	0	1	2
$\mathrm { P } ( Y = y )$	0.5	0.3	0.2

The random variables $U$ and $V$ are defined as follows: $$U = X Y , V = | X - Y | .$$

In the Printed Answer Book complete the table giving the joint distribution of $U$ and $V$.
Find $\operatorname { Cov } ( U , V )$.
Find $\mathrm { P } ( U V = 0 \mid V = 2 )$.

	Treatment \(A\)	Treatment \(B\)	Treatment \(C\)
Much improved	12	16	4
Improved	13	12	6
Slightly improved	7	6	7
No change	5	3	9

Student	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)
1st assessment	124	121	111	113	118	119
2nd assessment	127	119	114	110	120	122

Number of home wins	0	1	2	3	4	5	6	7	8	9	10	Totals
Observed number of weeks	0	1	2	8	8	9	7	1	2	0	0	38
Expected number of weeks	0.096	0.788	2.899	6.326	9.058	8.893	6.064	2.835	0.870	0.158	0.013	38

\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

\(x\)	0	1	2
\(\mathrm { P } ( X = x )\)	0.5	0.3	0.2

\(y\)	0	1	2
\(\mathrm { P } ( Y = y )\)	0.5	0.3	0.2

Questions — OCR (4619 questions)

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OCR S3 2014 June Q1

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