Questions - OCR MEI - A-Level Maths

State whether either or both of the variables $d$ and $h$ are random variables. Summary data for the diameters and heights are as follows. $$\mathrm { n } = 15 \quad \sum \mathrm {~d} = 84.9 \quad \sum \mathrm {~h} = 124.7 \quad \sum \mathrm {~d} ^ { 2 } = 624.55 \quad \sum \mathrm {~h} ^ { 2 } = 1230.57 \quad \sum \mathrm { dh } = 866.63$$
Find the equation of the regression line of $h$ on $d$. Give your answer in the form $h = a d + b$, giving the values of $a$ and $b$ correct to $\mathbf { 2 }$ decimal places.
Use the regression line to predict the heights of beech trees with the following diameters.
- 7.5 cm
- 20.0 cm
- Comment on the reliability of your predictions.
- There are many mature beech trees with diameter of 60 cm or greater. However, there are no beech trees with a height of more than 50 m .
Comment on this in relation to your regression line.
State the coordinates of the point at which the regression line of $d$ on $h$ meets the line which you calculated in part (b).

OCR MEI Further Statistics Minor 2022 June Q3

3 Jane wonders whether the number of wasps entering a wasp's nest per 5 second interval can be modelled by a Poisson distribution with mean $\mu$. She counts the number of wasps entering the nest over 60 randomly selected 5 -second intervals. The results are shown in Fig. 3.1. \begin{table}[h]

Number of wasps	0	1	2	3	4	5	6	7	8	9	$\geqslant 10$
Frequency	0	2	5	5	12	10	10	11	1	4	0

\captionsetup{labelformat=empty} \caption{Fig. 3.1}

\end{table}

Show that a suitable estimate for the value of $\mu$ is 5.1. Fig. 3.2 shows part of a screenshot for a $\chi ^ { 2 }$ test to assess the goodness of fit of a Poisson model. The sample mean has been used as an estimate for the population mean. Some of the values in the spreadsheet have been deliberately omitted. \begin{table}[h]

	A	B	C	D	E
\includegraphics[max width=\textwidth, alt={}]{e8624e9b-5143-49d2-9683-cc3a1082694e-4_132_40_1069_273}	Number of wasps	Observed frequency	Poisson probability	Expected frequency	Chi-squared contribution
2	$\leqslant 2$	7	0.1165	6.9887	0.0000
3	3	5		8.0874	1.1786
4	4	12			0.2765
5	5	10			0.0255
6	6	10	0.1490	8.9400	0.1257
7	7	11	0.1086	6.5134	3.0904
8	$\geqslant 8$	5	0.1440	8.6414
9

\captionsetup{labelformat=empty} \caption{Fig. 3.2}

\end{table}

Determine the missing values in each of the following cells, giving your answers correct to 4 decimal places.
- C3
- D5
- E8
- Explain why some of the frequencies have been combined into the categories $\leqslant 2$ and $\geqslant 8$.
- In this question you must show detailed reasoning.
Carry out the hypothesis test at the 5\% significance level.
Jane also carries out a $\chi ^ { 2 }$ test for the number of wasps leaving another nest. As part of her calculations, she finds that the probability of no wasps leaving the nest in a 5 -second period is 0.0053 . She finds that a Poisson distribution is also an appropriate model in this case. Find a suitable estimate for the value of the mean number of wasps leaving the nest per 5-second period.

OCR MEI Further Statistics Minor 2022 June Q4

4 Alex is practising bowling at a cricket wicket. Every time she bowls a ball, she has a $30 \%$ chance of hitting the wicket.

Assuming that successive bowls are independent, determine the probability that Alex first hits the wicket on her third attempt.
Determine the probability that Alex hits the wicket for the fourth time on her tenth attempt.

OCR MEI Further Statistics Minor 2022 June Q5

5 A medical researcher is investigating whether there is any relationship between the age of a person and the level of a particular protein in the person’s blood. She measures the levels of the protein (measured in suitable units) in a random sample of 12 hospital patients of various ages (in years). The spreadsheet shows the values obtained, together with a scatter diagram which illustrates the data.
\includegraphics[max width=\textwidth, alt={}, center]{e8624e9b-5143-49d2-9683-cc3a1082694e-5_736_1470_1087_246}

The researcher 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 age and protein level.
Explain why the researcher chose a sample that was random.
The researcher had originally intended to use a sample size of 6 rather than the 12 that she actually used. Explain what advantage there is in using the larger sample size.

OCR MEI Further Statistics Minor 2022 June Q6

6 The random variable $X$ has a uniform distribution over the values $\{ 1,4,7 , \ldots , 3 n - 2 \}$, where $n$ is a positive integer.

Determine $\operatorname { Var } ( X )$ in terms of $n$.
Given that $n = 100$, find the probability that $X$ is within one standard deviation of the mean.

OCR MEI Further Statistics Minor 2023 June Q1

1 A fair spinner has ten sectors, labelled $1,2 , \ldots , 10$. In order to start a game, Kofi has to obtain an 8,9 or 10 on the spinner.

Find the probability that Kofi starts the game on the third spin.
Find the probability that Kofi takes at least 5 spins to start the game.
Determine the probability that the number of spins required to start the game is within 1 standard deviation of its mean.

OCR MEI Further Statistics Minor 2023 June Q2

2 A company manufactures batches of twenty thousand tins which are subsequently filled with fruit. The company tests tins from each batch to make sure that they are strong enough. The test is easy and cheap to carry out, but when a tin has been tested it is no longer suitable for filling with fruit.

1. Explain why a sample size of 5 tins per batch may not be appropriate in this case.
2. Explain why a sample size of 1000 tins per batch may not be appropriate in this case. The company tests a sample of 30 tins from each batch.
Explain why it would not be sensible for the sample to consist of the final 30 tins produced in a batch.
Give two features that the sample should have.

OCR MEI Further Statistics Minor 2023 June Q3

3 A fair four-sided dice has its faces numbered $0,1,2,3$. The dice is rolled three times. The discrete random variable $X$ is the sum of the lowest and highest scores obtained.

Show that $\mathrm { P } ( X = 1 ) = \frac { 3 } { 32 }$. The table below shows the probability distribution of $X$.
$r$ 0 1 2 3 4 5 6
$\mathrm { P } ( X = r )$ $\frac { 1 } { 64 }$ $\frac { 3 } { 32 }$ $\frac { 13 } { 64 }$ $\frac { 3 } { 8 }$ $\frac { 13 } { 64 }$ $\frac { 3 } { 32 }$ $\frac { 1 } { 64 }$
In this question you must show detailed reasoning. Find each of the following.
- $\mathrm { E } ( X )$
- $\operatorname { Var } ( X )$
- The random variable $Y$ represents the sum of 10 values of $X$.
  1. State a property of the 10 values of $X$ that would make it possible to deduce the standard deviation of $Y$.
  2. Given that this property holds, determine the standard deviation of $Y$.

OCR MEI Further Statistics Minor 2023 June Q4

4 Eve lives in a narrow lane in the country. She wonders whether the number of vehicles passing her house per minute can be modelled by a Poisson distribution with mean $\mu$. She counts the number of vehicles passing her house over 100 randomly selected one-minute intervals. The results are shown in Table 4.1. \begin{table}[h]

\captionsetup{labelformat=empty} \caption{Table 4.1}

Number of vehicles	0	1	2	3	4	5	6	7	8	9	10	$\geqslant 11$
Frequency	36	33	14	10	4	1	0	0	1	0	1	0

\end{table}

Use the results to find an estimate for $\mu$. The spreadsheet in Fig. 4.2 shows data for a $\chi ^ { 2 }$ test to assess the goodness of fit of a Poisson model. The sample mean from part (a) has been used as an estimate for the population mean. Some of the values in the spreadsheet have been deliberately omitted. \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Fig. 4.2}
\multirow[b]{2}{*}{1} A B C D E
Number of vehicles Observed frequency Poisson probability Expected frequency Chi-squared contribution
2 0 36 0.2725 27.2532 2.8073
3 1 33 0.3543 35.4291
4 2 14 3.5400
5 $\geqslant 3$ 17 0.5145
6
\end{table}
Calculate the missing values in each of the following cells, giving your answers correct to 4 decimal places.
- C4
- D5
- E3
- In this question you must show detailed reasoning.
Carry out the $\chi ^ { 2 }$ test at the 5\% significance level.
Eve checks her data and notices that the two largest numbers of vehicles per minute (8 and 10) occurred when some horses were being ridden along the lane, causing delays to the vehicles. She therefore repeats the analysis, missing out these two items of data. She finds that the value of the $\chi ^ { 2 }$ test statistic is now 4.748. The number of degrees of freedom of the test is unchanged. Make two comments about this revised test.

OCR MEI Further Statistics Minor 2023 June Q5

5 An ornithologist is investigating the link between the wing length and the mass of small birds, in order to try to predict the mass from the wing length without having to weigh birds. The ornithologist takes a random sample of 9 birds and measures their wing lengths $w \mathrm {~mm}$ and their masses $m g$. The spreadsheet below shows the data, together with a scatter diagram which illustrates the data.
\includegraphics[max width=\textwidth, alt={}, center]{72215d69-c3e6-492d-bb3e-bdc28aeb4613-5_719_1424_495_246}

Find the equation of the regression line of $m$ on $w$, giving the coefficients correct to $\mathbf { 3 }$ significant figures.
Use the equation which you found in part (a) to estimate the mass for each of the following wing lengths.
- 99 mm
- 110 mm
- Comment on the reliability of your estimates.
- The equation of the regression line of $w$ on $m$ is $w = 0.473 m + 87.5$. A friend of the ornithologist suggests that this equation could also be used to estimate the masses of birds from their wing lengths.
Comment on this suggestion.

OCR MEI Further Statistics Minor 2023 June Q6

6 Each competitor in a lumberjacking competition has to perform various disciplines for which they are timed. A spectator thinks that the times for two of the disciplines, chopping wood and sawing wood, are related. The table and the scatter diagram below show the times of a random sample of 8 competitors in these two disciplines.

Competitor	A	B	C	D	E	F	G	H
Sawing	17.1	16.7	14.3	14.0	12.8	21.5	15.3	14.4
Chopping	23.5	20.6	21.9	18.8	21.5	24.8	19.7	19.3

\includegraphics[max width=\textwidth, alt={}, center]{72215d69-c3e6-492d-bb3e-bdc28aeb4613-6_786_1130_708_239}

The spectator decides to carry out a hypothesis test to investigate whether there is any relationship. Explain why the spectator decides that a test based on Pearson's product moment correlation coefficient may not be valid.
Determine the value of Spearman's rank correlation coefficient.
Carry out a hypothesis test at the $5 \%$ significance level to investigate whether there is positive association between sawing and chopping times.

OCR MEI Further Statistics Minor 2023 June Q7

7 The discrete random variable $X$ has a uniform distribution over the set of all integers between 100 and $n$ inclusive, where $n$ is a positive integer with $n > 100$.

Given that $n$ is even, determine $\mathrm { P } \left( \mathrm { X } < \frac { 100 + \mathrm { n } } { 2 } \right)$.
Determine the variance of the sum of 50 independent values of $X$, giving your answer in the form $\mathrm { a } \left( \mathrm { n } ^ { 2 } + \mathrm { bn } + \mathrm { c } \right)$, where $a , b$ and $c$ are constants.

OCR MEI Further Statistics Minor 2024 June Q1

1 When a footballer takes a penalty kick the result is that either a goal is scored or a goal is not scored. It is known that, on average, a certain footballer scores a goal on $85 \%$ of penalty kicks. During one practice session, the footballer decides to take penalty kicks until a goal is not scored. You may assume that the outcome of each penalty kick that the footballer takes is independent of the outcome of each other penalty kick. The random variable representing the number of penalty kicks up to and including the first penalty kick that does not result in a goal is denoted by $X$.

State one further assumption that is necessary for $X$ to be modelled by a Geometric distribution. For the remainder of this question you may assume that this assumption is valid.
Find each of the following.
- $\mathrm { E } ( X )$
- $\operatorname { Var } ( X )$
- Find the probability that the footballer takes exactly 3 penalty kicks.
- Find the probability that the footballer takes at least 5 penalty kicks.

OCR MEI Further Statistics Minor 2024 June Q2

2 The sides of a fair 12 -sided spinner are labelled $1,2 , \ldots , 12$. The spinner is spun and $X$ is the random variable denoting the number on the side of the spinner that it lands on.

Suggest a suitable distribution to model $X$. You should state the value(s) of any parameter(s).
Find each of the following.
- $\mathrm { E } ( X )$
- $\operatorname { Var } ( X )$
You are given that $\mathrm { E } ( X )$ is denoted by $\mu$ and $\operatorname { Var } ( X )$ is denoted by $\sigma ^ { 2 }$.
Determine $\mathrm { P } \left( \left| \frac { 2 ( X - \mu ) } { \sigma } \right| > 1 \right)$.

OCR MEI Further Statistics Minor 2024 June Q3

3 The scatter diagram below illustrates data concerning average annual income per person, $
) x\(, and average life expectancy, $y$ years, for 45 randomly selected cities.
\includegraphics[max width=\textwidth, alt={}, center]{464c80be-007b-4d5a-9fe5-2f35100bdea6-3_860_1465_354_244}

State whether neither variable, one variable or both variables can be considered to be random in this situation. A student is researching possible positive association between average annual income and average life expectancy. The student decides that the data point labelled A on the scatter diagram is an outlier.
Describe the apparent relationship between average annual income and average life expectancy for this data point relative to the rest of the data. The data for point A is removed. The student now wishes to carry out a hypothesis test using the product moment correlation coefficient for the remaining 44 data points to investigate whether there is positive correlation between average annual income and average life expectancy.
Explain why this type of hypothesis test is appropriate in this situation. Justify your answer. The summary statistics for these 44 data points are as follows.
$\sum x = 751120 \sum y = 2397.1 \sum x ^ { 2 } = 14363849200 \sum y ^ { 2 } = 133014.63 \sum x y = 42465962$
Determine the value of the product moment correlation coefficient.
Carry out the test at the 1\% significance level.

OCR MEI Further Statistics Minor 2024 June Q4

4 A genetics researcher is investigating whether there is any association between natural hair colour and natural eye colour. A random sample of 800 adults is selected. Each adult can categorise their natural hair colour as blonde, brown, black or red and their natural eye colour as brown, blue or green.

Carry out the chi-squared test at the 10\% significance level. Table 4.2 shows the chi-squared contributions for some of the categories. The contributions for the categories relating to green eye colour have been deliberately omitted. \begin{table}[h]

\captionsetup{labelformat=empty} \caption{Table 4.2}

Calculate the chi-squared contribution for the green eye and blonde hair category.

With reference to the values in Table 4.2, discuss what the data suggest about brown eye colour and blue eye colour for people with blonde hair.

A different researcher, carrying out the same investigation, independently takes a different random sample of size 800 and performs the same hypothesis test, but at the 1\% significance level, reaching the same conclusion as the original test. By comparing only the significance level of the two tests, specify which test, the one at the 10\% significance level or the one at the 1\% significance level, provides stronger evidence for the conclusion. Justify your answer.

OCR MEI Further Statistics Minor 2024 June Q5

5 Over a long period of time, it is found that the mean number of mistakes made by a certain player when playing a particular piece of music is 5 . The number of mistakes that the player makes when playing the piece is denoted by the random variable $Y$.

State two assumptions necessary for $Y$ to be modelled by a Poisson distribution. For the remainder of this question you may assume that $Y$ can be modelled by a Poisson distribution.
1. Find the probability that the player makes exactly 3 mistakes when playing the piece.
2. Find the probability that the player makes fewer than 3 mistakes when playing the piece.
3. Find the probability that the player makes fewer than 6 mistakes in total when playing the piece twice, assuming that the performances are independent. In a recording studio, the player plays the piece once in the morning and once in the afternoon each day for one week (7 days). It can be assumed that all the performances are independent of each other. The performances are recorded onto two CDs, one for each of two critics, A and B, to review. The critics are interested in the total number of mistakes made by the player per day. Unfortunately, there is a recording error in one of the CDs; on this CD, every piece that is supposed to be an afternoon recording is in fact just a repeat of that morning’s recording. The random variables $M _ { 1 }$ and $M _ { 2 }$ represent the total number of mistakes per day for the correctly recorded CD and for the wrongly recorded CD respectively.
By considering the values of $\mathrm { E } \left( M _ { 1 } \right)$ and $\mathrm { E } \left( M _ { 2 } \right)$ explain why it is not possible to use the mean number of mistakes per day on the CDs to determine which critic received the wrongly recorded CD. Each critic counts the total number of mistakes made per day, for each of the 7 days of recordings on their CD. Summary data for this is given below. Critic A: $\quad n = 7 , \quad \sum x _ { A } = 70 , \quad \sum x _ { A } ^ { 2 } = 812$
Critic B: $\quad \mathrm { n } = 7 , \sum \mathrm { x } _ { \mathrm { B } } = 72 , \sum \mathrm { x } _ { \mathrm { B } } ^ { 2 } = 800$
By considering the values of $\operatorname { Var } \left( M _ { 1 } \right)$ and $\operatorname { Var } \left( M _ { 2 } \right)$ determine which critic is likely to have received the wrongly recorded CD.

OCR MEI Further Statistics Minor 2024 June Q6

6 The probability distribution of a discrete random variable, $X$, is shown in the table below.

$x$	0	1	2
$\mathrm { P } ( X = x )$	$1 - a - b$	$a$	$b$

Find $\mathrm { E } ( X )$ in terms of $a$ and $b$.
1. In the case where $\mathrm { E } ( \mathrm { X } ) = \mathrm { a } + 0.4$, find an expression for $\operatorname { Var } ( X )$ in terms of $a$.
2. In this case, show that the greatest possible value of $\operatorname { Var } ( X )$ is 0.65 . You must state the associated value of $a$.
You are now given instead that $\mathrm { E } ( X )$ is not known.
1. State the least possible value of $\operatorname { Var } ( X )$.
2. Give all possible pairs of values of $a$ and $b$ which give the least possible value of $\operatorname { Var } ( X )$ stated in part (c)(i).

OCR MEI Further Statistics Minor 2020 November Q1

1 A quiz team of 4 students is to be selected from a group of 7 girls and 5 boys. The team is selected at random from the students in the group. The number of girls in the team is denoted by the random variable $X$.

Show that $\mathrm { P } ( X = 4 ) = \frac { 7 } { 99 }$. Table 1 shows the probability distribution of $X$. \begin{table}[h]
$r$ 0 1 2 3 4
$\mathrm { P } ( X = r )$ $\frac { 1 } { 99 }$ $\frac { 14 } { 99 }$ $\frac { 42 } { 99 }$ $\frac { 35 } { 99 }$ $\frac { 7 } { 99 }$
\captionsetup{labelformat=empty} \caption{Table 1}
\end{table}
Find each of the following.
- $\mathrm { E } ( X )$
- $\operatorname { Var } ( X )$
It is decided that the quiz team must have at least 1 girl and at least 1 boy, but the team is still otherwise selected at random.
Explain whether $\mathrm { E } ( X )$ would be smaller than, equal to or larger than the value which you found in part (b).

OCR MEI Further Statistics Minor 2020 November Q2

2 On computer monitor screens there are often one or more tiny dots which are permanently dark and do not display any of the image. Such dots are known as 'dead pixels'. Dead pixels occur on screens randomly and independently of each other. A company manufactures three types of monitor, Types A, B and C. For a monitor of Type A, the screen has a total of 2304000 pixels. For this type of monitor, the probability of a randomly chosen pixel being dead is 1 in 500000 . Let $X$ represent the number of dead pixels on a monitor screen of this type.

Explain why you could use either a binomial distribution or a Poisson distribution to model the distribution of $X$.
Use a Poisson distribution to calculate estimates of each of the following probabilities.
- $\mathrm { P } ( X = 4 )$
- $\mathrm { P } ( X > 4 )$
- In this question you must show detailed reasoning.
For a monitor of Type B, the probability of a randomly chosen pixel being dead is also 1 in 500 000. The screen of a monitor of Type B has a total of $n$ pixels. Use a binomial distribution to find the least value of $n$ for which the probability of finding at least 1 dead pixel is greater than 0.99 . Give your answer in millions correct to 3 significant figures. For a monitor of Type C, the number of dead pixels on the screen is modelled by a Poisson distribution with mean $\lambda$.
Given that the probability of finding at least one dead pixel is 0.8 , find $\lambda$.

OCR MEI Further Statistics Minor 2020 November Q4

4 Cards are drawn at random from a standard pack of 52 cards, one at a time, until one of the 4 aces is drawn. After each card is drawn, it is replaced in the pack before the next one is drawn. The random variable $X$ represents the number of draws required to draw the first ace.

State fully the distribution of $X$.
Find $\mathrm { P } ( X = 10 )$.
Find each of the following.
- $\mathrm { E } ( X )$
- $\operatorname { Var } ( X )$
A further $k$ aces are added to the full pack and the process described above is repeated. The random variable $Y$ represents the number of draws required to draw the first ace.
In this question you must show detailed reasoning. Given that $\mathrm { P } ( Y = 2 ) = \frac { 8 } { 81 }$, find the two possible values of $k$.

OCR MEI Further Statistics Minor 2020 November Q5

5 A student is investigating immunisation. He wonders if there is any relationship between the percentage of young children who have been given measles vaccine and the percentage who have been given BCG vaccine in various countries. He takes a random sample of 8 countries and finds the data for the two variables. The spreadsheet in Fig. 5.1 shows the values obtained, together with a scatter diagram which illustrates the data. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{882f9f3c-40d8-4abb-822a-49bd505a33ea-5_910_1653_541_246} \captionsetup{labelformat=empty} \caption{Fig. 5.1}

\end{figure}

The student decides that a test based on Pearson's product moment correlation coefficient is not valid. Explain why he comes to this conclusion. The student carries out a test based on Spearman’s rank correlation coefficient.
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 measles and BCG vaccination levels. The student then decides to investigate the relationship between number of doctors per 1000 people in a country and unemployment rate in that country (unemployment rate is the percentage of the working age population who are not in paid work). He selects a random sample of 6 countries. The spreadsheet in Fig. 5.2 shows the values obtained, together with a scatter diagram which illustrates the data. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{882f9f3c-40d8-4abb-822a-49bd505a33ea-6_776_1649_495_248} \captionsetup{labelformat=empty} \caption{Fig. 5.2}
\end{figure}
Use your calculator to write down the equation of the regression line of unemployment rate on doctors per 1000.
Use the regression line to estimate the unemployment rate for a country with 2.00 doctors per 1000.
Comment briefly on the reliability of your answer to part (e). The student decides to add the data for another country with 3.99 doctors per 1000 and unemployment rate 11.42 to his diagram.
Add this point to the scatter diagram in the Printed Answer Booklet.
Without doing any further calculations, comment on what difference, if any, including this extra data point would make to the usefulness of a regression line of unemployment rate on doctors per 1000.

OCR MEI Further Statistics Minor 2020 November Q6

The random variable $X$ has a uniform distribution over the values $\{ 1,2 , \ldots , n \}$. Show that $\operatorname { Var } ( X )$ is given by $\frac { 1 } { 12 } \left( n ^ { 2 } - 1 \right)$.
The random variable $Y$ has a uniform distribution over the values $\{ 1,3,5 , \ldots , 2 n - 1 \}$. Using the result in part (a) or otherwise, show that $\operatorname { Var } ( Y )$ is given by $\frac { 1 } { 3 } \left( n ^ { 2 } - 1 \right)$.
Given that $n = 100$, find the least value of $k$ for which $\mathrm { P } ( \mu - k \sigma \leqslant Y \leqslant \mu + k \sigma ) = 1$, where the mean and standard deviation of $Y$ are represented by $\mu$ and $\sigma$ respectively.

OCR MEI Further Statistics Minor 2021 November Q1

1 The probability distribution of a discrete random variable $X$ is given by the formula $\mathrm { P } ( \mathrm { X } = \mathrm { r } ) = \mathrm { k } \left( ( \mathrm { r } - 1 ) ^ { 2 } + 1 \right)$ for $r = 1,2,3,4,5$.

Show that $k = \frac { 1 } { 35 }$. The distribution of $X$ is shown in the table.
$r$ 1 2 3 4 5
$\mathrm { P } ( \mathrm { X } = \mathrm { r } )$ $\frac { 1 } { 35 }$ $\frac { 2 } { 35 }$ $\frac { 1 } { 7 }$ $\frac { 2 } { 7 }$ $\frac { 17 } { 35 }$
Comment briefly on the shape of the distribution.
Find each of the following.
- $\mathrm { E } ( X )$
- $\operatorname { Var } ( X )$
The random variable $Y$ is given by $Y = 5 X - 10$.
Find each of the following.
- $\mathrm { E } ( Y )$
- $\operatorname { Var } ( Y )$

OCR MEI Further Statistics Minor 2021 November Q2

2 A road transport researcher is investigating the link between the age of a person, a years, and the distance, $d$ metres, at which the person can read a large road sign. The researcher selects 13 individuals of different ages between 20 and 80 and measures the value of $d$ for each of them. The spreadsheet below shows the data which the researcher obtained, together with a scatter diagram which illustrates the data.
\includegraphics[max width=\textwidth, alt={}, center]{691e8b55-e9a1-4fff-b9ee-a71ff1f73ead-3_725_1566_495_251}

Explain which of the two variables $a$ and $d$ is the independent variable.
Find the equation of the regression line of $d$ on $a$.
Use the regression line to predict the average distance at which a 60-year-old person can read the road sign.
Explain why it might not be sensible to use the regression line to predict the average distance at which a 5 -year-old child can read the road sign.
Determine the value of the residual for $a = 40$.
Explain why it would not be useful to find the equation of the regression line of $a$ on $d$.

\(r\)	0	1	2	3	4	5	6
\(\mathrm { P } ( X = r )\)	\(\frac { 1 } { 64 }\)	\(\frac { 3 } { 32 }\)	\(\frac { 13 } { 64 }\)	\(\frac { 3 } { 8 }\)	\(\frac { 13 } { 64 }\)	\(\frac { 3 } { 32 }\)	\(\frac { 1 } { 64 }\)

Number of wasps	0	1	2	3	4	5	6	7	8	9	\(\geqslant 10\)
Frequency	0	2	5	5	12	10	10	11	1	4	0

Number of vehicles	0	1	2	3	4	5	6	7	8	9	10	\(\geqslant 11\)
Frequency	36	33	14	10	4	1	0	0	1	0	1	0

\multirow[b]{2}{*}{1}	A	B	C	D	E
	Number of vehicles	Observed frequency	Poisson probability	Expected frequency	Chi-squared contribution
2	0	36	0.2725	27.2532	2.8073
3	1	33	0.3543	35.4291
4	2	14			3.5400
5	\(\geqslant 3\)	17			0.5145
6

\multirow{2}{*}{Observed frequency}		Hair Colour
		Blonde	Brown	Black	Red	Total
\multirow{3}{*}{Eye Colour}	Brown	47	153	196	36	432
	Blue	61	78	115	26	280
	Green	19	22	31	16	88
	Total	127	253	342	78	800

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OCR MEI Further Statistics Minor 2024 June Q5

OCR MEI Further Statistics Minor 2024 June Q6

OCR MEI Further Statistics Minor 2020 November Q1

OCR MEI Further Statistics Minor 2020 November Q2

OCR MEI Further Statistics Minor 2020 November Q4

OCR MEI Further Statistics Minor 2020 November Q5

OCR MEI Further Statistics Minor 2020 November Q6

OCR MEI Further Statistics Minor 2021 November Q1

OCR MEI Further Statistics Minor 2021 November Q2