Questions Further Statistics A AS

State a distribution which can be used to model the number of particles emitted by the source in a 10-second period.
State the variance of this distribution.
Find the probability that at least 6 particles are detected in a period of 10 seconds.
Find the probability that at least 36 particles are detected in a period of 60 seconds.
Another radioactive source emits particles randomly and independently at a constant average rate of 1.7 particles per 5 seconds. Find the probability that at least 10 but no more than 15 particles are detected altogether from the two sources in a period of 10 seconds.

OCR MEI Further Statistics A AS 2018 June Q2

2 In a quiz, competitors have to match 5 landmarks to the 5 British counties which the landmarks are in. The random variable $X$ represents the number of correct matches that a competitor gets, assuming that the competitor guesses randomly. The probability distribution of $X$ is given in the following table.

$r$	0	1	2	3	4	5
$\mathrm { P } ( X = r )$	$\frac { 11 } { 30 }$	$\frac { 3 } { 8 }$	$\frac { 1 } { 6 }$	$\frac { 1 } { 12 }$	0	$\frac { 1 } { 120 }$

Explain why $\mathrm { P } ( X = 4 )$ must be 0 .
Explain how the value $\frac { 1 } { 120 }$ for $\mathrm { P } ( X = 5 )$ is calculated.
Draw a graph to illustrate the distribution.
Find each of the following.
- $\mathrm { E } ( X )$
- $\operatorname { Var } ( X )$
- Find $\mathrm { P } ( X > \mathrm { E } ( X ) )$.
- There are 12 competitors in the quiz. Assuming that they all guess randomly, find the probability that at least one of them gets all five matches correct.

OCR MEI Further Statistics A AS 2018 June Q3

3 Samples of water are taken from 10 randomly chosen wells in an area of a country. A researcher is investigating whether there is any relationship between the levels of dissolved oxygen, $x$, and the amounts of radium, $y$, in the water from the wells. Both quantities are measured in suitable units. The table and the scatter diagram in Fig. 3 show the values of $x$ and $y$ for the ten wells.

$x$	45.9	48.3	52.2	64.6	66.6	67.6	69.3	75.0	77.4	82.8
$y$	25.4	23.9	26.6	18.8	18.9	19.0	16.8	16.3	17.8	17.2

\begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e3ac0ba0-9692-4018-894e-2b04b07eaf32-3_865_786_657_635} \captionsetup{labelformat=empty} \caption{Fig. 3}

\end{figure}

Explain why it may not be appropriate to carry out a hypothesis test based on the product moment correlation coefficient.
Calculate Spearman's rank correlation coefficient for these data.
Using this value of Spearman's rank correlation coefficient, carry out a hypothesis test at the 1\% significance level to investigate whether there is any association between $x$ and $y$.
Explain the meaning of the term 'significance level' in the context of the test carried out in part (iii).

OCR MEI Further Statistics A AS 2018 June Q4

4 The probability that an expert darts player hits the bullseye on any throw is 0.12 , independently of any other throw. The player throws darts at the bullseye until she hits it.

Find the probability that the player has to throw exactly six darts.
Find the probability that the player has to throw more than six darts.
(A) Find the mean number of darts that the player has to throw.
(B) Find the variance of the number of darts that the player has to throw. The player continues to throw more darts at the bullseye after she has hit it for the first time.
Find the probability that the player hits the bullseye at least twice in the first ten throws.
Find the probability that the player hits the bullseye for the second time on the tenth throw.

OCR MEI Further Statistics A AS 2018 June Q5

5 A random sample of workers for a large company were asked whether they are smokers, ex-smokers or have never smoked. The responses were classified by the type of worker: Managerial, Production line or Administrative. Fig. 5 is a screenshot showing part of the spreadsheet used to analyse the data. Some values in the spreadsheet have been deliberately omitted. \begin{table}[h]

	A	B	C	D	E	F
1	Observed frequencies
2		Smoker	Ex-smoker	Never smoked	Totals
3	Managerial	2	10	5	17
4	Production line	18	15	21	54
5	Administrative	13	6	14	33
6	Totals	33	31	40	104
7
8		Expected frequencies
9		5.3942	5.0673	6.5385
10		17.1346		20.7692
11		10.4712	9.8365	12.6923
12
13		Contributions to the test statistic
14		2.1358	4.8017	0.3620
15		0.0437		0.0026
16			1.4964	0.1347
17				Test statistic	9.66
18

\captionsetup{labelformat=empty} \caption{Fig. 5}

\end{table}

(A) State the sample size.
(B) State the null and alternative hypotheses for a test to investigate whether there is any association between type of worker and smoking status.
Showing your calculations, find the missing values in each of the following cells.
- C 10
- C 15
- B 16
- Complete the hypothesis test at the $10 \%$ level of significance.
- Discuss briefly what the data suggest about smoking status for different types of workers. You should make a comment for each type of worker.

OCR MEI Further Statistics A AS 2018 June Q6

6 A researcher is investigating various bodily characteristics of frogs of various species. She collects data on length, $x \mathrm {~mm}$, and head width, $y \mathrm {~mm}$, of a random sample of 14 frogs of a particular species. A scatter diagram of the data is shown in Fig. 6, together with the equation of the regression line of $y$ on $x$ and also the value of $r ^ { 2 }$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e3ac0ba0-9692-4018-894e-2b04b07eaf32-6_949_1616_450_228} \captionsetup{labelformat=empty} \caption{Fig. 6}

\end{figure}

(A) Use the equation of the regression line to estimate the mean head width for frogs of each of the following lengths.
- 45 mm
- 60 mm
  (B) Comment briefly on each of the estimates in part (i)(A).
- Explain how the mean length of frogs with head width 16 mm should be estimated.
- Calculate the value of the product moment correlation coefficient.
- In the light of the information in the scatter diagram, comment on the goodness of fit of the regression line.

OCR MEI Further Statistics A AS 2019 June Q1

1 The discrete random variable $X$ has probability distribution defined by $$\mathrm { P } ( X = r ) = k \left( r ^ { 2 } + 3 r \right) \text { for } r = 1,2,3,4,5 \text {, where } k \text { is a constant. }$$

Complete the table below, using the copy in the Printed Answer Booklet giving the probabilities in terms of $k$.
$r$ 1 2 3 4 5
$\mathrm { P } ( X = r )$ $4 k$ $10 k$
Show that the value of $k$ is 0.01 .
Draw a graph to illustrate the distribution.
Describe the shape of the distribution.
Find each of the following.
- $\mathrm { E } ( X )$
- $\operatorname { Var } ( X )$

OCR MEI Further Statistics A AS 2019 June Q2

2 Almost all plants of a particular species have red flowers. However on average 1 in every 1500 plants of this species have white flowers. A random sample of 2000 plants of this species is selected. The random variable $X$ represents the number of plants in the sample that have white flowers.

Name two distributions which could be used to model the distribution of $X$, stating the parameters of each of these distributions. You may use either of the distributions you have named in the rest of this question.
Calculate each of the following.
- $\mathrm { P } ( X = 2 )$
- $\mathrm { P } ( X > 2 )$
- A random sample of 20000 plants of this species is selected.
Calculate the probability that there are at least 10 plants in the sample that have white flowers.

OCR MEI Further Statistics A AS 2019 June Q3

3 A fair 8 -sided dice has faces labelled 10, 20, 30, ..., 80 .

State the distribution of the score when the dice is rolled once.
Write down the probability that, when the dice is rolled once, the score is at least 40 .
The dice is rolled three times.
1. Find the variance of the total score obtained.
2. Find the probability that on one of the rolls the score is less than 30 , on another it is between 30 and 50 inclusive and on the other it is greater than 50 .

OCR MEI Further Statistics A AS 2019 June Q4

4 A student is investigating correlations between various personality traits, two of which are conscientiousness and openness to new experiences.
She selects a random sample of 10 students at her university and uses standard tests to measure their conscientiousness and their openness. The product moment correlation coefficient between these two variables for the 10 students is 0.476 .

Assuming that the underlying population has a bivariate Normal distribution, carry out a hypothesis test at the $10 \%$ significance level to investigate whether there is any correlation between openness and conscientiousness in students. Table 4.1 below shows the values of the product moment correlation coefficients between 5 different personality traits for a much larger sample of students. Those correlations that are significant at the $5 \%$ level are denoted by a * after the value of the correlation. \begin{table}[h]
Neuroticism Extroversion Openness Agreeableness Conscientiousness
Neuroticism 1
Extroversion -0.296* 1
Openness -0.044 0.405* 1
Agreeableness -0.190* 0.061 0.042 1
Conscientiousness -0.485* 0.145 0.235* 0.112 1
\captionsetup{labelformat=empty} \caption{Table 4.1}
\end{table} The student analyses these factors for effect size.
Guidelines often used when considering effect size are given in Table 4.2 below. \begin{table}[h]
Product moment
correlation coefficient
Effect size
0.1 Small
0.3 Medium
0.5 Large
\captionsetup{labelformat=empty} \caption{Table 4.2}
\end{table}
The student notes that, despite the result of the test in part (a), the correlation between openness and conscientiousness is significant at the $5 \%$ level with this second sample. Comment briefly on why this may be the case.
The student intends to summarise her findings about relationships between these factors, including effect sizes, in a report.
Use the information in Tables 4.1 and 4.2 to identify two summary points the student could make.

OCR MEI Further Statistics A AS 2019 June Q5

5 A researcher is investigating births of females and males in a particular species of animal which very often produces litters of 7 offspring.
The table shows some data about the number of females per litter in 200 litters of 7 offspring. The researcher thinks that a binomial distribution $\mathrm { B } ( 7 , p )$ may be an appropriate model for these data. (c) Complete the test at the $5 \%$ significance level. Fig. 5 shows the probability distribution $\mathrm { B } ( 7,0.35 )$ together with the relative frequencies of the observed data (the numbers of litters each divided by 200). \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{fd496303-10f1-450e-bbeb-421ab6f4de21-5_659_1285_342_319} \captionsetup{labelformat=empty} \caption{Fig. 5}

\end{figure} (d) Comment on the result of the test completed in part (c) by considering Fig. 5.

OCR MEI Further Statistics A AS 2019 June Q6

6 A meteorologist is investigating the relationship between altitude $x$ metres and mean annual temperature $y ^ { \circ } \mathrm { C }$ in an American state.
She selects 12 locations at various altitudes and then stations a remote monitoring device at each of them to measure the temperature over the course of a year. Fig. 6 illustrates the data which she obtains. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{fd496303-10f1-450e-bbeb-421ab6f4de21-6_686_1477_486_292} \captionsetup{labelformat=empty} \caption{Fig. 6}

\end{figure}

Explain why it would not be appropriate to carry out a hypothesis test for correlation based on the product moment correlation coefficient.
Explain why altitude has been plotted on the horizontal axis in Fig. 6. Summary statistics for $x$ and $y$ are as follows. $$\sum x = 21200 \quad \sum y = 105.4 \quad \sum x ^ { 2 } = 39100000 \quad \sum y ^ { 2 } = 1004 \quad \sum x y = 176090$$
Calculate the equation of the regression line of $y$ on $x$.
Use the equation of the regression line to predict the values of the mean annual temperature at each of the following altitudes.
- 2000 metres
- 3000 metres
- Comment on the reliability of your predictions in part (d).
- Calculate the value of the residual for the data point ( $1600,8.1$ ).

OCR MEI Further Statistics A AS 2022 June Q1

1 A fair five-sided spinner has sectors labelled 1, 2, 3, 4, 5. In a game at a stall at a charity event, the spinner is spun twice. The random variable $X$ represents the lower of the two scores. The probability distribution of $X$ is given by the formula
$\mathrm { P } ( \mathrm { X } = \mathrm { r } ) = \mathrm { k } ( 11 - 2 \mathrm { r } )$ for $r = 1,2,3,4,5$,
where $k$ is a constant.

Complete the copy of this table in the Printed Answer Booklet.
$r$ 1 2 3 4 5
$\mathrm { P } ( X = r )$ $7 k$ $3 k$
Determine the value of $k$.
Find each of the following.
- $\mathrm { E } ( X )$
- $\operatorname { Var } ( X )$
- The stall-holder charges a player $C$ pence to play the game, and then pays the player $50 X$ pence, where $X$ is the player's score.
Given that the average profit that the stall-holder makes on one game is 25 pence, find the value of $C$.

OCR MEI Further Statistics A AS 2022 June Q2

2 On a car assembly line, a robot is used for a particular task.

State the conditions under which a Poisson distribution is an appropriate model for the number of breakdowns of the robot in a week. It is given that the average number of breakdowns of the robot in a week is 1.7 . For the remainder of this question, you should assume that a Poisson distribution is an appropriate model for the number of breakdowns of the robot in a week.
1. Find the probability that the number of breakdowns of the robot in a week is exactly 4.
2. Determine the probability that the number of breakdowns of the robot in a week is at least 2 .
Determine the probability that the number of breakdowns of the robot in 52 weeks is less than 100.

OCR MEI Further Statistics A AS 2022 June Q3

3 A biology student is doing an experiment in which plants are inoculated with a particular microorganism in an attempt to help them grow. She is investigating whether there is any association between the percentage of roots which have been colonised by the microorganism and the dry weight of the plant shoots. After the plants have grown for a few weeks, the student takes a random sample of 10 plants and measures the percentage of roots which have been colonised by the microorganism and the dry weight of the plant shoots. The spreadsheet output shows the data, together with a scatter diagram to illustrate the data.
\includegraphics[max width=\textwidth, alt={}, center]{8f1e0c68-a334-4657-823e-386ab0994c02-3_722_1648_635_244}

The student decides that a test based on Pearson’s product moment correlation coefficient may not be valid. Explain why she comes to this conclusion.
Calculate the value of Spearman’s rank correlation coefficient.
Carry out a test based on this coefficient, at the $5 \%$ significance level, to investigate whether there is any association between percentage colonisation and shoot dry weight.

OCR MEI Further Statistics A AS 2022 June Q4

4 A random number generator generates integers between 1 and 50 inclusive, with each number having an equal probability of being generated.

State the probability distribution of the numbers generated.
Determine the probability that a number generated is within one standard deviation of the mean.

OCR MEI Further Statistics A AS 2022 June Q5

5 A researcher is investigating whether there is any relationship between the overall performance of a student at GCSE and their grade in A Level Mathematics. Their A Level Mathematics grade is classified as A* or A, B, C or lower, and their overall performance at GCSE is classified as Low, Middle, High. Data are collected for a sample of 80 students in a particular area. The researcher carries out a chi-squared test. The screenshot below shows part of a spreadsheet used to analyse the data. Some values in the spreadsheet have been deliberately omitted.

\multirow[b]{2}{*}{Contribution to the test statistic}

State what needs to be known about the sample for the test to be valid. For the remainder of this question, you should assume that the test is valid.
Determine the missing values in each of the following cells.
- C11
- C18
- In this question you must show detailed reasoning.
Carry out a hypothesis test at the $10 \%$ significance level to investigate whether there is any association between level of performance at GCSE and A Level Mathematics grade.
Discuss briefly what the data suggest about A Level Mathematics grade for different levels of performance at GCSE.
State one disadvantage of using a 10\% significance level rather than a 5\% significance level in a hypothesis test.

OCR MEI Further Statistics A AS 2022 June Q6

6 Tom has read in a newspaper that you can tell the air temperature by counting how often a cricket chirps in a period of 20 seconds. (A cricket is a type of insect.) He wants to know exactly how the temperature can be predicted. On 8 randomly selected days, when Tom can hear crickets chirping, he records the number of chirps, $x$, made by a cricket in a 20-second interval, and also the temperature, $y ^ { \circ } \mathrm { C }$, at that time. The data are summarised as follows.
$n = 8 \quad \sum x = 268 \quad \sum y = 141.9 \quad \sum x ^ { 2 } = 9618 \quad \sum y ^ { 2 } = 2630.55 \quad \sum \mathrm { xy } = 5009.1$
These data are illustrated below.
\includegraphics[max width=\textwidth, alt={}, center]{8f1e0c68-a334-4657-823e-386ab0994c02-5_661_1035_699_242}

Determine the equation of the regression line of $y$ on $x$. Give your answer in the form $\mathrm { y } = \mathrm { ax } + \mathrm { b }$, giving the values of $a$ and $b$ correct to $\mathbf { 3 }$ significant figures.
Use the equation of the regression line to predict the temperature for the following values of $x$.
- 35
- 10
- Comment on the reliability of your predictions in part (b).
- State the coordinates of the point of intersection of the line whose equation you have calculated with the regression line of $x$ on $y$.

OCR MEI Further Statistics A AS 2022 June Q7

7 On average one in five packets of a breakfast cereal contains a voucher for a discount on the next packet bought. Whether or not a packet contains a voucher is independent of other packets, and can only be determined by opening the packet.

State the distribution of the number of packets that need to be opened in order to find one which contains a voucher.
Determine the probability that exactly 4 packets have to be opened in order to find one which contains a voucher.
Determine the probability that exactly 10 packets have to be opened in order to find two which contain a voucher.
I have $n$ packets, and I open them one by one until I find a voucher or until all the packets are open. Given that the probability that I find a voucher is greater than 0.99 , determine the least possible value of $n$.

OCR MEI Further Statistics A AS 2023 June Q1

1 Ryan has 6 one-pound coins and 4 two-pound coins. Ryan decides to select 3 of these coins at random to donate to a charity. The total value, in pounds, of these 3 coins is denoted by the random variable $X$.

Show that $\mathrm { P } ( X = 3 ) = \frac { 1 } { 6 }$. The table below shows the probability distribution of $X$.
$r$ 3 4 5 6
$\mathrm { P } ( \mathrm { X } = \mathrm { r } )$ $\frac { 1 } { 6 }$ $\frac { 1 } { 2 }$ $\frac { 3 } { 10 }$ $\frac { 1 } { 30 }$
Draw a graph to illustrate the distribution.
In this question you must show detailed reasoning. Find each of the following.
- $\mathrm { E } ( X )$
- $\operatorname { Var } ( X )$
Ryan's friend Sasha decides to give the same amount as Ryan does to the charity plus an extra three pounds. The random variable $Y$ represents the total amount of money, in pounds, given by Ryan and Sasha.
Determine each of the following.
- E (Y)
- $\operatorname { Var } ( Y )$

OCR MEI Further Statistics A AS 2023 June Q2

2 A group of friends live by the sea. Each day they look out to sea in the hope of seeing a dolphin. The probability that they see a dolphin on any day is 0.15 . You should assume that this probability is not affected by whether or not they see a dolphin on any other day.

Explain why you can use a geometric distribution to model the number of days that it takes for them to first see a dolphin.
Find the probability that they see a dolphin for the first time on the fifth day.
Find the probability that they do not see a dolphin for at least 10 days.
Determine the mean and the variance of the number of days that it takes for them to see a dolphin.

OCR MEI Further Statistics A AS 2023 June Q3

3 At a pottery which manufactures mugs, it is known that $5 \%$ of mugs are faulty. The mugs are produced in batches of 20 . Faults are modelled as occurring randomly and independently. The number of faulty mugs in a batch is denoted by the random variable $X$.

Determine $\mathrm { P } ( X \geqslant 2 )$.
Find $\operatorname { Var } ( X )$. Independently of the mugs, the pottery also manufactures cups, and it is known that $7 \%$ of cups are faulty. The cups are produced in batches of 30 . Faults are modelled as occurring randomly and independently. The number of faulty cups in a batch is denoted by the random variable $Y$.
Determine the standard deviation of $X + Y$. When 10 batches of cups have been produced, a sample of 15 cups is tested to ensure that the handles of the cups are properly attached.
Explain why it might not be sensible to select a sample of 15 cups from the same batch.

OCR MEI Further Statistics A AS 2023 June Q4

4 At a parcel delivery company it is known that the probability that a parcel is delivered to the wrong address is 0.0005 . On a particular day, 15000 parcels are delivered. The number of parcels delivered to the wrong address is denoted by the random variable $X$.

Explain why the binomial distribution and the Poisson distribution could both be suitable models for the distribution of $X$.
Use a Poisson distribution to find each of the following.
- $\mathrm { P } ( X = 5 )$
- $\mathrm { P } ( X \geqslant 8 )$
You are given that 15000 parcels are delivered each day in a 5-day working week.
1. Determine the probability that at least 40 parcels are delivered to the wrong address during the week.
2. Determine the probability that at least 8 parcels are delivered to the wrong address on each of the 5 days in the week.

OCR MEI Further Statistics A AS 2023 June Q5

5 Two practice GCSE examinations in mathematics are given to all of the students in a large year group. A teacher wants to check whether there is a positive relationship between the marks obtained by the students in the two examinations. She selects a random sample of 20 students. Summary data for the marks obtained in the first and second practice examinations, $x$ and $y$ respectively, are as follows. $$\sum x = 565 \quad \sum y = 724 \quad \sum x ^ { 2 } = 17103 \quad \sum y ^ { 2 } = 29286 \quad \sum x y = 21635$$ The teacher decides to carry out a hypothesis test based on Pearson’s product moment correlation coefficient.

In this question you must show detailed reasoning. Calculate the value of Pearson's product moment correlation coefficient.
Carry out the test at the $5 \%$ significance level.
Given that the teacher did not draw a scatter diagram before carrying out the test, comment on the validity of the test.

OCR MEI Further Statistics A AS 2023 June Q6

6 An eight-sided dice has its faces numbered $1,2 , \ldots , 8$.

In this part of the question you should assume that the dice is fair.
1. State the probability that, when the dice is rolled once, the score is at least 6 .
2. Show that the probability that the score is within 2 standard deviations of its mean is 1 .
A student thinks that the dice may be biased. To investigate this, the student decides to roll the dice 80 times and then carry out a $\chi ^ { 2 }$ goodness of fit test of a uniform distribution. The spreadsheet below shows the data for the test, where some of the values have been deliberately omitted.
\multirow[b]{2}{*}{1} A B C D
Score Observed frequency Expected frequency Chi-squared contribution
2 1 14 10 1.6
3 2 4 10 3.6
4 3 10 10 0
5 4 15 10
6 5 6 10 1.6
7 6 11 10 0.1
8 7 7 10 0.9
9 8 10 0.9
1. Explain why all of the expected frequencies are equal to 10 .
2. Determine the missing values in each of the following cells.
  - B9
D5
(iii) In this question you must show detailed reasoning.

\(r\)	0	1	2	3	4	5
\(\mathrm { P } ( X = r )\)	\(\frac { 11 } { 30 }\)	\(\frac { 3 } { 8 }\)	\(\frac { 1 } { 6 }\)	\(\frac { 1 } { 12 }\)	0	\(\frac { 1 } { 120 }\)

\(r\)	1	2	3	4	5
\(\mathrm { P } ( X = r )\)	\(4 k\)	\(10 k\)

\(r\)	1	2	3	4	5
\(\mathrm { P } ( X = r )\)		\(7 k\)		\(3 k\)

\(x\)	45.9	48.3	52.2	64.6	66.6	67.6	69.3	75.0	77.4	82.8
\(y\)	25.4	23.9	26.6	18.8	18.9	19.0	16.8	16.3	17.8	17.2

	Neuroticism	Extroversion	Openness	Agreeableness	Conscientiousness
Neuroticism	1
Extroversion	-0.296*	1
Openness	-0.044	0.405*	1
Agreeableness	-0.190*	0.061	0.042	1
Conscientiousness	-0.485*	0.145	0.235*	0.112	1

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