Questions Further Statistics

Find the probability distribution of the amount of money won by the contestant.
The contestant pays $\pounds 5$ for every time the dice is thrown. Find the standard deviation of the loss made by the contestant in 120 throws of the dice.

OCR Further Statistics Specimen Q4

4 A psychologist investigated the scores of pairs of twins on an aptitude test. Seven pairs of twins were chosen randomly, and the scores are given in the following table.

Elder twin	65	37	60	79	39	40	88
Younger twin	58	39	61	62	50	26	84

Carry out an appropriate Wilcoxon test at the $10 \%$ significance level to investigate whether there is evidence of a difference in test scores between the elder and the younger of a pair of twins.
Explain the advantage in this case of a Wilcoxon test over a sign test.

OCR Further Statistics Specimen Q5

5 The number of goals scored by the home team in a randomly chosen hockey match is denoted by $X$.

In order for $X$ to be modelled by a Poisson distribution it is assumed that goals scored are random events. State two other conditions needed for $X$ to be modelled by a Poisson distribution in this context. Assume now that $X$ can be modelled by the distribution $\operatorname { Po } ( 1.9 )$.
(a) Write down an expression for $\mathrm { P } ( X = r )$.
(b) Hence find $\mathrm { P } ( X = 3 )$.
Assume also that the number of goals scored by the away team in a randomly chosen hockey match has an independent Poisson distribution with mean $\lambda$ between 1.31 and 1.32. Find an estimate for the probability that more than 3 goals are scored altogether in a randomly chosen match.

OCR Further Statistics Specimen Q6

6 A bag contains 3 green counters, 3 blue counters and $w$ white counters. Counters are selected at random, one at a time, with replacement, until a white counter is drawn.
The total number of counters selected, including the white counter, is denoted by $X$.

In the case when $w = 2$,
(a) write down the distribution of $X$,
(b) find $P ( 3 < X \leq 7 )$.
In the case when $\mathrm { E } ( X ) = 2$, determine the value of $w$.
In the case when $w = 2$ and $X = 6$, find the probability that the first five counters drawn alternate in colour.

OCR Further Statistics Specimen Q7

7 Sweet pea plants grown using a standard plant food have a mean height of 1.6 m . A new plant food is used for a random sample of 49 randomly chosen plants and the heights, $x$ metres, of this sample can be summarised by the following. $$\begin{aligned} n & = 49
\Sigma x & = 74.48
\Sigma x ^ { 2 } & = 120.8896 \end{aligned}$$ Test, at the $5 \%$ significance level, whether, when the new plant food is used, the mean height of sweet pea plants is less than 1.6 m .

OCR Further Statistics Specimen Q8

8 A continuous random variable $X$ has probability density function given by $$\mathrm { f } ( x ) = \left\{ \begin{array} { c c } 0.8 \mathrm { e } ^ { - 0.8 x } & x \geq 0
0 & x < 0 \end{array} \right.$$

Find the mean and variance of $X$. The lifetime of a certain organism is thought to have the same distribution as $X$. The lifetimes in days of a random sample of 60 specimens of the organism were found. The observed frequencies, together with the expected frequencies correct to 3 decimal places, are given in the table.
Range $0 \leq x < 1$ $1 \leq x < 2$ $2 \leq x < 3$ $3 \leq x < 4$ $x \geq 4$
Observed 24 22 10 3 1
Expected 33.040 14.846 6.671 2.997 2.446
Show how the expected frequency for $1 \leq x < 2$ is obtained.
Carry out a goodness of fit test at the $5 \%$ significance level.

OCR Further Statistics Specimen Q9

9 The continuous random variable $X$ has cumulative distribution function given by $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 0
\frac { 1 } { 16 } x ^ { 2 } & 0 \leq x \leq 4
1 & x > 4 \end{array} \right.$$

The random variable $Y$ is defined by $Y = \frac { 1 } { X ^ { 2 } }$. Find the cumulative distribution function of $Y$.
Show that $\mathrm { E } ( Y )$ is not defined. \section*{END OF QUESTION PAPER}

OCR Further Statistics 2022 June Q3

3 In this question you must show detailed reasoning. A discrete random variable $V$ has the following probability distribution, where $p$ and $q$ are constants.

$v$	0	1	2	3
$\mathrm { P } ( \mathrm { V } = \mathrm { v } )$	$p$	$q$	0.12	0.2

It is given that $\mathrm { E } ( V ) = \operatorname { Var } ( V )$. Determine the value of $p$ and the value of $q$.

OCR Further Statistics 2018 March Q1

1 The numbers of customers arriving at a ticket desk between 8 a.m. and 9 a.m. on a Monday morning and on a Tuesday morning are denoted by $X$ and $Y$ respectively. It is given that $X \sim \operatorname { Po } ( 17 )$ and $Y \sim \operatorname { Po } ( 14 )$.

Find
(a) $\mathrm { P } ( X + Y ) > 40$,
(b) $\operatorname { Var } ( 2 X - Y )$.
State a necessary assumption for your calculations in part (i) to be valid.

OCR Further Statistics 2018 March Q2

2 A continuous random variable $X$ has cumulative distribution function given by $$\mathrm { F } ( x ) = \begin{cases} 1 - \frac { 1 } { x ^ { 3 } } & x \geqslant 1
0 & \text { otherwise } \end{cases}$$ Find $E ( \sqrt { } X )$.

OCR Further Statistics 2018 March Q3

3 Adila has a pack of 50 cards.

Each of the 50 cards is numbered with a different integer from 1 to 50 . Adila selects 5 cards at random without replacement.
(a) Find the probability that exactly 3 of the 5 cards have numbers which are 10 or less.
(b) Adila arranges the 5 cards in a line in a random order. Find the probability that the 5 cards are arranged in numerically increasing order. 10 of the 50 cards are blue and the rest are green.
Adila randomly selects three sets of 10 cards each, without replacement. The sets are labelled $A , B$ and $C$. Given that $A$ contains 3 blue cards and 7 green cards, find the probability that $B$ contains exactly 2 blue cards and $C$ contains exactly 3 blue cards.

OCR Further Statistics 2018 March Q4

4 Sheena travels to school by bus. She records the number of minutes, $T$, that her bus is late on each of 32 days. She believes that on average $T$ is greater than 5, and she carries out a significance test at the $5 \%$ level.

State a condition needed for a Wilcoxon test to be valid in this case. Assume now that this condition is satisfied.
State an advantage of using a Wilcoxon test rather than a sign test.
Calculate the critical region for the test, in terms of a variable which should be defined.

OCR Further Statistics 2018 March Q5

5 A spinner has 5 edges. Each edge is numbered with a different integer from 1 to 5 . When the spinner is spun, it is equally likely to come to rest on any one of the edges. The spinner is spun 100 times. The number of times on which the spinner comes to rest on the edge numbered 5 is denoted by $X$.

$\mathrm { E } ( X )$,
$\operatorname { Var } ( X )$.
1. Write down
2. Use a normal distribution with the same mean and variance as in your answers to part (i) to estimate the smallest value of $n$ such that $\mathrm { P } ( X \geqslant n ) < 0.02$.
3. Use the binomial distribution to find exactly the smallest value of $n$ such that $\mathrm { P } ( X \geqslant n ) < 0.02$. Show the values of all relevant calculations.

OCR Further Statistics 2018 March Q6

6 The captain of a sports team analyses the team's results according to the weather conditions, classified as "sunny" and "not sunny". The frequencies are shown in the following table.

\cline { 3 - 5 } \multicolumn{2}{c\|}{}	Results
\cline { 3 - 5 } \multicolumn{2}{c\|}{}	Win	Draw	Lose
\multirow{2}{*}{Weather}	Sunny	12	3	5
\cline { 2 - 5 }	Not sunny	8	12	10

Test at the $5 \%$ significance level whether the team's performances are associated with weather conditions.
(a) Identify the cell that gives the largest contribution to the test statistic.
(b) Interpret your answer to part (ii)(a).

OCR Further Statistics 2018 March Q7

7 The function $\mathrm { f } ( x )$ is defined by $$f ( x ) = \begin{cases} \frac { 1 } { 4 } x \left( 4 - x ^ { 2 } \right) & 0 \leqslant x \leqslant 2
0 & \text { otherwise } \end{cases}$$

Show that $\mathrm { f } ( x )$ satisfies the conditions for a probability density function.
Find the value of $a$ such that $\mathrm { P } ( X < a ) = \frac { 15 } { 16 }$.

OCR Further Statistics 2018 March Q8

8 At a wine-tasting competition, two judges give marks out of 100 to 7 wines as follows.

Wine	$A$	$B$	$C$	$D$	$E$	$F$	$G$
Judge I	86.3	87.5	87.6	88.8	89.4	89.9	90.5
Judge II	85.3	88.1	82.7	87.7	89.0	89.4	91.5

A spectator claims that there is a high level of agreement between the rank orders of the marks given by the two judges. Test the spectator's claim at the $1 \%$ significance level.
A competitor ranks the wines in a random order. The value of Spearman's rank correlation coefficient between the competitor and Judge I is $r _ { s }$.
(a) Find the probability that $r _ { s } = 1$.
(b) Show that $r _ { s }$ cannot take the value $\frac { 55 } { 56 }$.

OCR Further Statistics 2018 March Q9

9 The values of a set of bivariate data $\left( x _ { i } , y _ { i } \right)$ can be summarised by $$n = 50 , \sum x = 1270 , \sum y = 5173 , \sum x ^ { 2 } = 42767 , \sum y ^ { 2 } = 701301 , \sum x y = 173161 .$$ Ten independent observations of $Y$ are obtained, all corresponding to $x = 20$. It may be assumed that the variance of $Y$ is 1.9 , independently of the value of $x$. Find a $95 \%$ confidence interval for the mean $\bar { Y }$ of the 10 observations of $Y$. \section*{END OF QUESTION PAPER}

OCR Further Statistics 2018 September Q1

1 An experiment involves releasing a coin on a sloping plane so that it slides down the slope and then slides along a horizontal plane at the bottom of the slope before coming to rest. The angle $\theta ^ { \circ }$ of the sloping plane is varied, and for each value of $\theta$, the distance $d \mathrm {~cm}$ the coin slides on the horizontal plane is recorded. A scatter diagram to illustrate the results of the experiment is shown below, together with the least squares regression line of $d$ on $\theta$.
\includegraphics[max width=\textwidth, alt={}, center]{28c6a0d9-09a6-4743-af0e-fe2e43e256c9-2_639_972_561_548}

State which two of the following correctly describe the variable $\theta$.
Controlled variable Correlation coefficient
Dependent variable Independent variable
Response variable Regression coefficient
The least squares regression line of $d$ on $\theta$ has equation $d = 1.96 + 0.11 \theta$.
Use the diagram in the Printed Answer Booklet to explain the term "least squares".
State what difference, if any, it would make to the equation of the regression line if $d$ were measured in inches rather than centimetres. ( 1 inch $\approx 2.54 \mathrm {~cm}$ ).

OCR Further Statistics 2018 September Q2

2 Shooting stars occur randomly, independently of one another and at a constant average rate of 12.0 per hour. On each of a series of randomly chosen clear nights I look for shooting stars for 20 minutes at a time. A successful night is a night on which I see at least 8 shooting stars in a 20 -minute period.
From tomorrow, I will count the number, $X$, of nights on which I look for shooting stars, up to and including the first successful night. Find $\mathrm { E } ( X )$.

OCR Further Statistics 2018 September Q3

3 A discrete random variable $X$ has the distribution $\mathrm { U } ( 11 )$.
The mean of 50 observations of $X$ is denoted by $\bar { X }$.
Use an approximate method, which should be justified, to find $\mathrm { P } ( \bar { X } \leqslant 6.10 )$.

OCR Further Statistics 2018 September Q4

4 A survey is carried out into the length of time for which customers wait for a response on a telephone helpline. A statistician who is analysing the results of the survey starts by modelling the waiting time, $x$ minutes, by an exponential distribution with probability density function (PDF) $$\mathrm { f } ( x ) = \begin{cases} \lambda \mathrm { e } ^ { - \lambda x } & x \geqslant 0
0 & x < 0 \end{cases}$$ \section*{(i) In this question you must show detailed reasoning.} The mean waiting time is found to be 5.0 minutes. Show that $\lambda = 0.2$.
(ii) Use the model to calculate the probability that a customer has to wait longer than 20 minutes for a response. In practice it is found that no customer waits for more than 15 minutes for a response. The statistician constructs an improved model to incorporate this fact.
(iii) On the diagram in the Printed Answer Booklet, sketch the following, labelling the curves clearly:

the PDF of the model using the exponential distribution,
a possible PDF for the improved model.

OCR Further Statistics 2018 September Q5

5 Hal designs a 4-edged spinner with edges labelled 1, 2, 3 and 4. He intends that the probability that the spinner will land on any edge should be proportional to the number on that edge. He spins the spinner 20 times and on each spin he records the number of the edge on which it lands. The results are shown in the table.

Edge number	1	2	3	4
Frequency	3	7	4	6

Test at the $10 \%$ significance level whether the results are consistent with the intended probabilities.

OCR Further Statistics 2018 September Q6

6 A bag contains 7 red counters and 5 blue counters.

Fred chooses 4 counters at random, without replacement. Show that the probability that Fred chooses exactly 2 red counters is $\frac { 14 } { 33 }$.
Lina chooses 4 counters at random from the bag, records whether or not exactly 2 red counters are chosen, and returns the counters to the bag. She carries out this experiment 99 times.
(a) Find the mean of the number of experiments that result in choosing exactly 2 red counters.
(b) Find the variance of the number of experiments that result in choosing exactly 2 red counters.
Alex arranges all 12 counters in a random order in a straight line. A is the event: no two blue counters are next to one another. B is the event: all the blue counters are next to one another. Find $\mathrm { P } ( A \cup B )$.

OCR Further Statistics 2018 September Q7

7 The table shows the values of 5 observations of bivariate data $( x , y )$.

$x$	4.6	5.9	6.5	7.8	8.3
$y$	15.6	10.8	10.4	10.1	9.7

$$n = 5 , \Sigma x = 33.1 , \Sigma y = 56.6 , \Sigma x ^ { 2 } = 227.95 , \Sigma y ^ { 2 } = 664.26 , \Sigma x y = 362.37$$

Calculate Pearson's product-moment correlation coefficient $r$ for the data.
State what this value of $r$ tells you about a scatter diagram illustrating the data.
Test at the $5 \%$ significance level whether there is association between $x$ and $y$.
State the value of Spearman's rank correlation coefficient $r _ { s }$ for the data.
State whether $r , r _ { s }$, or both or neither is changed when the values of $x$ are replaced by
(a) $3 x - 2$,
(b) $\sqrt { x }$.

OCR Further Statistics 2018 September Q8

8 In an experiment to investigate the effect of background music in carrying out work, ten students were each given a task. Five of the students did the task in silence and the other five did the task with background music. The scores on the tasks were as follows.

Silence	43	46	55	58	61
Background music	19	31	38	52	70

Use a Wilcoxon rank-sum test to test at the 10\% level whether the presence of background music affects scores.
A statistician suggests that the experiment is redesigned so that each student takes one task in silence and another task with background music. The differences in the test scores would then be analysed using a paired-sample method. State an advantage in redesigning the experiment in this way.

\(v\)	0	1	2	3
\(\mathrm { P } ( \mathrm { V } = \mathrm { v } )\)	\(p\)	\(q\)	0.12	0.2

Range	\(0 \leq x < 1\)	\(1 \leq x < 2\)	\(2 \leq x < 3\)	\(3 \leq x < 4\)	\(x \geq 4\)
Observed	24	22	10	3	1
Expected	33.040	14.846	6.671	2.997	2.446

Wine	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)
Judge I	86.3	87.5	87.6	88.8	89.4	89.9	90.5
Judge II	85.3	88.1	82.7	87.7	89.0	89.4	91.5

Controlled variable	Correlation coefficient
Dependent variable	Independent variable
Response variable	Regression coefficient

Questions Further Statistics (100 questions)

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