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OCR MEI Further Statistics Major 2019 June Q9

15 marks Moderate -0.5

9 Every weekday Jonathan takes an underground train to work. On any weekday the time in minutes that he has to wait at the station for a train is modelled by the continuous uniform distribution over $[ 0,5 ]$.

Find the probability that Jonathan has to wait at least 3 minutes for a train. The total time that Jonathan has to wait on two days is modelled by the continuous random variable $X$ with probability density function given by $\mathrm { f } ( x ) = \begin{cases} \frac { 1 } { 25 } x & 0 \leqslant x \leqslant 5 , \\ \frac { 1 } { 25 } ( 10 - x ) & 5 < x \leqslant 10 , \\ 0 & \text { otherwise } . \end{cases}$

Find the probability that Jonathan has to wait a total of at most 6 minutes on two days. Jonathan's friend suggests that the total waiting time for 5 days, $T$ minutes, will almost certainly be less than 18 minutes. In order to investigate this suggestion, Jonathan constructs the simulation shown in Fig. 9. All of the numbers in the simulation have been rounded to 2 decimal places. \begin{table}[h]

	A	B	C	D	E	F
1	Mon	Tue	Wed	Thu	Fri	Total T
2	1.78	4.36	2.74	3.88	4.64	17.41
3	0.95	1.30	4.83	4.29	1.81	13.18
4	4.27	4.90	4.57	1.41	3.66	18.81
5	0.80	0.06	3.20	1.76	0.35	6.17
6	0.03	4.82	1.26	3.53	0.13	9.77
7	3.88	4.73	1.19	3.75	1.29	14.84
8	4.11	3.54	4.33	0.77	4.50	17.25
9	3.54	0.11	3.85	2.86	1.58	11.94
10	1.87	1.82	3.00	3.53	1.83	12.05
11	4.00	2.98	4.59	1.73	1.76	15.06
12	1.91	3.85	2.08	1.72	2.82	12.38
13	0.10	4.86	2.51	0.52	2.17	10.15
14	1.24	4.26	0.95	1.33	1.78	9.57
15	2.99	0.69	3.85	3.41	2.42	13.36
16	4.67	1.76	2.13	3.48	3.10	15.14
17	1.94	1.07	0.91	0.63	3.34	7.89
18	0.11	2.29	0.71	4.21	0.86	8.18
19	0.43	4.58	4.89	1.86	2.84	14.60
20	4.23	0.88	2.71	4.88	4.20	16.91
21	3.72	4.58	3.11	4.89	3.18	19.49

\captionsetup{labelformat=empty} \caption{Fig. 9}

\end{table}

Use the simulation to estimate $\mathrm { P } ( T > 18 )$.
Explain how Jonathan could obtain a better estimate. Jonathan thinks that he can use the Central Limit Theorem to provide a very good approximation to the distribution of $T$.
Find each of the following.
- $\mathrm { E } ( T )$
- $\operatorname { Var } ( T )$
- Use the Central Limit Theorem to estimate $\mathrm { P } ( T > 18 )$.
- Comment briefly on the use of the Central Limit Theorem in this case.
Jonathan travels to work on 200 days in a year.
Find the probability that the total waiting time for Jonathan in a year is more than 510 minutes.
[0pt] [3]

OCR MEI Further Statistics Major 2019 June Q10

14 marks Standard +0.8

10 The probability density function of the continuous random variable $X$ is given by $f ( x ) = \begin{cases} k x ^ { m } & 0 \leqslant x \leqslant a , \\ 0 & \text { otherwise, } \end{cases}$ where $a , k$ and $m$ are positive constants.

Show that $k = \frac { m + 1 } { a ^ { m + 1 } }$.
Find the cumulative distribution function of $X$ in terms of $x , a$ and $m$.
Given that $\mathrm { P } \left( \frac { 1 } { 4 } a < X < \frac { 1 } { 2 } a \right) = \frac { 1 } { 10 }$,
1. show that $2 p ^ { 2 } - 10 p + 5 = 0$, where $p = 2 ^ { m }$,
2. find the value of $m$. \section*{END OF QUESTION PAPER}

OCR MEI Further Statistics Major 2022 June Q1

7 marks Moderate -0.8

1 During a meteor shower, the number of meteors that can be seen at a particular location can be modelled by a Poisson distribution with mean 1.2 per minute.

Find the probability that exactly 2 meteors are seen in a period of 1 minute.
Find the probability that more than 3 meteors are seen in a period of 1 minute.
Find the probability that no more than 8 meteors are seen in a period of 10 minutes.
Explain what the fact that the number of meteors seen can be modelled by a Poisson distribution tells you about the occurrence of meteors.

OCR MEI Further Statistics Major 2022 June Q2

7 marks Standard +0.3

2 A manufacturer is testing how long coloured LED lights will last before the battery runs out, using two different battery types. The times in hours before the battery runs out are modelled by independent Normal distributions with means and standard deviations as shown in the table.

\cline { 2 - 3 } \multicolumn{1}{c|}{}

In a particular test, a battery of type A is used and the time taken for it to run out is recorded. This process is repeated until a total of 5 randomly selected batteries have been used. Determine the probability that the total time the 5 batteries last is at least 120 hours.
In a similar test, 3 randomly selected batteries of type A are used, one after the other. Then 2 randomly selected batteries of type B are used, one after the other. Determine the probability that the 3 type A batteries last longer in total than the 2 type B batteries.
Explain why it is necessary that the Normal distributions are independent in order to be able to find the probability in part (b).

OCR MEI Further Statistics Major 2022 June Q3

6 marks Standard +0.3

3 The table shows the probability distribution of the random variable $X$, where $a$ and $b$ are constants.

$r$	0	1	2	3	4
$\mathrm { P } ( X = r )$	$a$	$b$	0.24	0.32	$b ^ { 2 }$

Given that $\mathrm { E } ( X ) = 1.8$, determine the values of $a$ and $b$. The random variable $Y$ is given by $Y = 10 - 3 X$.
Using the values of $a$ and $b$ which you found in part (a), find each of the following.
- $\mathrm { E } ( Y )$
- $\operatorname { Var } ( Y )$

OCR MEI Further Statistics Major 2022 June Q4

5 marks Standard +0.8

4 A pack of $k$ cards is labelled $1,2 , \ldots , k$. A card is drawn at random from the pack. The random variable $X$ represents the number on the card.

Given that $k > 10$, find $\mathrm { P } ( X \geqslant 10 )$. You are now given that $k = 20$.
A card is drawn at random from the pack and the number on it is noted. The card is then returned to the pack. This process is repeated until the second occasion on which the number noted is less than 9 . Find the probability that no more than 4 cards have to be drawn. Answer all the questions. Section B (95 marks)

OCR MEI Further Statistics Major 2022 June Q5

11 marks Moderate -0.3

5 A motorist is investigating the relationship between tyre pressure and temperature. As the temperature increases during a hot day, she records the pressure (measured in bars) of one of her car tyres at specific temperatures of $20 ^ { \circ } \mathrm { C } , 22 ^ { \circ } \mathrm { C } , \ldots , 36 ^ { \circ } \mathrm { C }$. The results are shown in Table 5.1. \begin{table}[h]

Temperature $\left( t ^ { \circ } \mathrm { C } \right)$	20	22	24	26	28	30	32	34	36
Tyre pressure $( P$ bar $)$	2.012	2.036	2.065	2.074	2.114	2.140	2.149	2.176	2.192

\captionsetup{labelformat=empty} \caption{Table 5.1}

\end{table}

Calculate the equation of the regression line of pressure on temperature. Give your answer in the form $P = a t + b$, giving the values of $a$ and $b$ to $\mathbf { 4 }$ significant figures.
Table 5.2 shows the residuals for most of the data values. Complete the copy of the table in the Printed Answer Booklet. \begin{table}[h]
Temperature 20 22 24 26 28 30 32 34 36
Residual tyre
pressure
- 0.003 - 0.002 0.004 - 0.010 0.011 - 0.003 0.001
\captionsetup{labelformat=empty} \caption{Table 5.2}
\end{table}
With reference to the values of the residuals, comment on the goodness of fit of the regression line.
Use your answer to part (a) to calculate an estimate of the pressure in the tyre at each of the following temperatures, giving your answers to $\mathbf { 3 }$ decimal places.
- $25 ^ { \circ } \mathrm { C }$
- $10 ^ { \circ } \mathrm { C }$
- Comment on the reliability of each of your estimates.

OCR MEI Further Statistics Major 2022 June Q7

8 marks Standard +0.8

7 Amir is trying to thread a needle. On each attempt the probability that he is successful is 0.3 , independently of any other attempt. The random variable $X$ represents the number of attempts that he takes to thread the needle.

Find $\mathrm { P } ( X = 5 )$.
During the course of a day, Amir has to thread 6 needles. Determine the probability that it takes him more than 3 attempts to be successful for at least 4 of the 6 needles.
Amaya is also trying to thread a needle. On each attempt the probability that she is successful is $p$, independently of any other attempt. The probability that Amaya takes 2 attempts to thread a particular needle is $\frac { 28 } { 121 }$. Determine the possible values of $p$.

OCR MEI Further Statistics Major 2022 June Q8

14 marks Standard +0.3

8 A swimming coach is investigating whether there is correlation between the times taken by teenage swimmers to swim 50 m Butterfly and 50 m Freestyle. The coach selects a random sample of 11 teenage swimmers and records the times that each of them take for each event. The spreadsheet shows the data, together with a scatter diagram to illustrate the data. \includegraphics[max width=\textwidth, alt={}, center]{77eabbd6-a058-457f-9601-d66f3c2db005-06_712_1465_456_274}

In the scatter diagram, Butterfly times have been plotted on the horizontal axis and Freestyle times on the vertical axis. A student states that the variables should have been plotted the other way around. Explain whether the student is correct. The student decides to carry out a hypothesis test to investigate whether there is any correlation between the times taken for the two events.
Explain why the student decides to carry out a test based on Spearman's rank correlation coefficient.
In this question you must show detailed reasoning. Carry out the test at the 5\% significance level.
The student concludes that there is definitely no correlation between the times. Comment on the student's conclusion.

OCR MEI Further Statistics Major 2022 June Q9

11 marks Easy -1.2

9 The random variable $X$ has a discrete uniform distribution over the values $\{ 0,1,2 , \ldots , 20 \}$.

Find $\mathrm { P } ( X \leqslant 7 )$.

Find each of the following.

$\mathrm { E } ( X )$
$\operatorname { Var } ( X )$

The spreadsheet shows a simulation of the distribution of $X$. Each of the 25 rows of the spreadsheet below the heading row shows a simulation of 10 independent values of $X$ together with the value of the mean of the 10 values, denoted by $Y$.

\includegraphics[max width=\textwidth, alt={}]{77eabbd6-a058-457f-9601-d66f3c2db005-07_38_45_880_279}	A	B	C	D	E	F	G	H	I	J	K	L
1	$X _ { 1 }$	$X _ { 2 }$	$X _ { 3 }$	$X _ { 4 }$	$X _ { 5 }$	$X _ { 6 }$	$X _ { 7 }$	$X _ { 8 }$	$X _ { 9 }$	$X _ { 10 }$	$Y$
2	1	6	2	1	18	6	4	9	11	11	6.9
3	13	14	12	2	4	11	16	0	16	0	8.8
4	4	17	1	16	4	10	12	2	18	13	9.7
5	2	8	12	1	4	16	12	2	15	8	8.0
6	7	15	16	0	4	7	1	13	0	20	8.3
7	15	13	10	1	12	0	20	15	16	6	10.8
8	14	13	17	12	2	18	16	18	9	4	12.3
9	20	2	12	3	17	3	0	18	15	13	10.3
10	2	12	5	12	2	6	0	9	10	15	7.3
11	5	11	13	10	9	17	10	4	20	15	11.4
12	14	9	9	7	6	20	2	2	11	16	9.6
13	15	19	18	19	7	6	6	20	3	8	12.1
14	5	10	6	4	1	19	15	8	17	18	10.3
15	0	3	15	15	11	12	0	3	9	16	8.4
16	1	12	1	15	0	4	11	11	9	2	6.6
17	12	5	0	8	3	8	12	19	13	12	9.2
18	9	5	1	13	5	4	18	1	1	19	7.6
19	16	2	20	20	12	17	2	7	8	20	12.4
20	18	17	3	2	8	18	7	0	11	6	9.0
21	15	10	7	20	4	0	5	6	11	14	9.2
22	3	9	10	14	2	1	8	6	0	7	6.0
23	11	10	11	10	19	11	3	7	10	0	9.2
24	12	14	6	6	5	20	11	18	10	14	11.6
25	1	11	5	14	11	10	1	1	2	0	5.6
26	0	14	7	11	18	5	10	20	11	9	10.5
27

Use the spreadsheet to estimate $\mathrm { P } ( Y \leqslant 7 )$.
Explain why the true value of $\mathrm { P } ( Y \leqslant 7 )$ is less than $\mathrm { P } ( X \leqslant 7 )$, relating your answer to $\operatorname { Var } ( X )$ and $\operatorname { Var } ( Y )$.
The random variable $W$ is the mean of 30 independent values of $X$. Determine an estimate of $\mathrm { P } ( W \leqslant 7 )$.

OCR MEI Further Statistics Major 2022 June Q10

13 marks Standard +0.3

10 A scientist is researching dietary fat intake and cholesterol level. A random sample of 60 people is selected and their dietary fat intakes and cholesterol levels are measured. Dietary fat intakes are classified as low, medium and high, and cholesterol levels are classified as normal and high. The scientist decides to carry out a chi-squared test to investigate whether there is any association between dietary fat intake and cholesterol level. Tables $\mathbf { 1 0 . 1 }$ and $\mathbf { 1 0 . 2 }$ show the data and some of the expected frequencies for the test. \begin{table}[h]

\multirow{2}{*}{}		Dietary fat intake
		Low	Medium	High	Total
\multirow{2}{*}{Cholesterol level}	Normal	9	18	5	32
	High	3	13	12	28
	Total	12	31	17	60

\captionsetup{labelformat=empty} \caption{Table 10.1}

\end{table} \begin{table}[h]

\captionsetup{labelformat=empty} \caption{Table 10.2}

\end{table}

Complete the table of expected frequencies in the Printed Answer Booklet.

Determine the contribution to the chi-squared test statistic for people with normal cholesterol level and high dietary fat intake, giving your answer to $\mathbf { 4 }$ decimal places. The contributions to the chi-squared test statistic for the remaining categories are shown in Table 10.3. \begin{table}[h]

\captionsetup{labelformat=empty} \caption{Table 10.3} \end{table}

In this question you must show detailed reasoning. Carry out the test at the 5\% significance level.

For each level of dietary fat intake, give a brief interpretation of what the data suggest about the level of cholesterol.

OCR MEI Further Statistics Major 2022 June Q11

13 marks Standard +0.3

11 A particular dietary supplement, when taken for a period of 1 month, is claimed to increase lean body mass of adults by an average of 1 kg . A researcher believes that this claim overestimates the increase. She selects a random sample of 10 adults who then each take the supplement for a month. The increases in lean body masses in kg are as follows. $$\begin{array} { l l l l l l l l l l } - 0.84 & - 0.76 & - 0.16 & 0.43 & 1.31 & 1.32 & 1.47 & 1.64 & 1.93 & 2.14 \end{array}$$ A Normal probability plot and the $p$-value of the Kolmogorov-Smirnov test for these data are shown below. \includegraphics[max width=\textwidth, alt={}, center]{77eabbd6-a058-457f-9601-d66f3c2db005-09_575_1485_689_242}

The researcher decides to carry out a hypothesis test in order to investigate the claim. Comment on the type of hypothesis test that should be used. You should refer to
- The Normal probability plot
- The $p$-value of the Kolmogorov-Smirnov test
- Carry out a test at the $5 \%$ significance level to investigate whether the researcher's belief may be correct.
- If the Normal probability plot had been different, giving a $p$-value of 0.65 for the KolmogorovSmirnov test, a different procedure could have been used to investigate the researcher's belief.
- State what alternative test could have been used in this case.
- State what the hypotheses would have been.

OCR MEI Further Statistics Major 2022 June Q12

14 marks Challenging +1.2

12 The continuous random variable $X$ has cumulative distribution function given by $$F ( x ) = \begin{cases} 0 & x < 0 \\ k \left( a x - 0.5 x ^ { 2 } \right) & 0 \leqslant x \leqslant a \\ 1 & x > a \end{cases}$$ where $a$ and $k$ are positive constants.

Determine the median of $X$ in terms of $a$.
Given that $a = 10$, determine the probability that $X$ is within one standard deviation of its mean.

OCR MEI Further Statistics Major 2023 June Q1

10 marks Standard +0.3

1 A website simulates the outcome of throwing four fair dice. Ten thousand people take part in a challenge using the website in which they have one attempt at getting four sixes in the four throws of the dice. The number of people who succeed in getting four sixes is denoted by the random variable $X$.

Show that, for each person, the probability that the person gets four sixes is equal to $\frac { 1 } { 1296 }$.
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 each of the following probabilities.
- $\mathrm { P } ( X = 10 )$
- $\mathrm { P } ( X > 10 )$
- In another challenge on the website, 50 people are each given 20 independent attempts to try to get four sixes as often as they can.
Determine the probability that no more than 2 people succeed in getting four sixes at least once in their 20 attempts.

OCR MEI Further Statistics Major 2023 June Q2

5 marks Easy -1.2

2 A student is investigating the link between temperature and electricity consumption in the winter months. The student finds the average minimum temperature, $x ^ { \circ } \mathrm { C }$, from across the country on a day. The student then finds the total electricity consumption for that day, $y \mathrm { GWh }$. The scatter diagram below shows the values of $x$ and $y$ obtained from a random sample of 10 winter days. It also shows the equation of the regression line of $y$ on $x$ and the value of $r ^ { 2 }$, where $r$ is the product moment correlation coefficient. \includegraphics[max width=\textwidth, alt={}, center]{c692fb20-436f-4bc1-89bd-10fdba41ceba-03_776_1043_609_244}

Use the regression line to estimate the electricity consumption at each of the following average minimum temperatures.
- $5 ^ { \circ } \mathrm { C }$
- $- 4 ^ { \circ } \mathrm { C }$
- Comment on the reliability of your estimates.

OCR MEI Further Statistics Major 2023 June Q3

10 marks Moderate -0.3

3 A tennis player is practising her serve. Each time she serves, she has a $55 \%$ chance of being successful (getting the serve in the required area without hitting the net). You should assume that whether she is successful on any serve is independent of whether she is successful on any other serve.

Find the probability that the player is not successful in any of her first three serves.
Determine the probability that the player is successful at least 10 times in her first 20 serves.
Determine the probability that the player is successful for the first time on her fifth serve.
Determine the probability that the player is successful for the fifth time on her tenth serve. Another player is also practising his serve. Each time he serves, he has a probability $p$ of being successful. You should assume that whether he is successful on any serve is independent of whether he is successful on any other serve. The probability that he is successful for the first time on his second serve is 0.2496 and the probability that he is successful on both of his first two serves is less than 0.25 .
Determine the value of $p$.

OCR MEI Further Statistics Major 2023 June Q4

11 marks Standard +0.3

4 A machine manufactures batches of 100 titanium sheets. The thickness of every sheet in a batch is Normally distributed with mean $\mu \mathrm { mm }$ and standard deviation 0.03 mm . You should assume that each sheet is of uniform thickness and that the thicknesses of different sheets are independent of each other. The values of $\mu$ for three different batches, A, B and C, are 3.125, 3.117 and 3.109 respectively.

Determine the probability that the total thickness of 10 sheets from Batch A is less than 31.0 mm .
Determine the probability that, if a single sheet from Batch A is cut into pieces and 10 of the pieces are stacked together, the total thickness of the stack is less than 31.0 mm .
Determine the probability that, if one sheet from each of Batches A, B and C are stacked together, the total thickness of the stack is at least 9.4 mm .
Determine the probability that the total thickness of 10 sheets from Batch A is less than the total thickness of 10 sheets from Batch B.

OCR MEI Further Statistics Major 2023 June Q5

13 marks Standard +0.3

5 Amari is investigating how accurately people can estimate a short time period. He asks each of a random sample of 40 people to estimate a period of 20 seconds. For each person, he starts a stopwatch and then stops it when they tell him that they think that 20 s has elapsed. The times which he records are denoted by $x \mathrm {~s}$. You are given that $\sum x = 765 , \quad \sum x ^ { 2 } = 15065$.

Determine a 95\% confidence interval for the mean estimated time.
Amari says that the confidence interval supports the suggestion that people can estimate 20 s accurately. Make two comments about Amari's statement.
Discuss whether you could have constructed the confidence interval if there had only been 10 people involved in the experiment. Amari thinks that people would be able to estimate more accurately if he gave them a second attempt. He repeats the experiment with each person and again records the times. Software is used to produce a $95 \%$ confidence interval for the mean estimated time. The output from the software is shown below. Z Estimate of a Mean Confidence level 0.95 Sample
Mean 19.68
s 1.38
N 40
Result
Z Estimate of a Mean
Mean 19.68
s 1.38
SE 0.2182
N 40
Interval $19.68 \pm 0.4277$
State the confidence interval in the form $\mathrm { a } < \mu < \mathrm { b }$.
Make two comments based on this confidence interval about Amari's opinion that second attempts result in more accurate estimates.

OCR MEI Further Statistics Major 2023 June Q6

12 marks Standard +0.3

6 A student wonders if there is any correlation between download and upload speeds of data to and from the internet. The student decides to carry out a hypothesis test to investigate this and so measures the download speed $x$ and upload speed $y$ in suitable units on 20 randomly chosen occasions. The scatter diagram below illustrates the data which the student collected. \includegraphics[max width=\textwidth, alt={}, center]{c692fb20-436f-4bc1-89bd-10fdba41ceba-07_824_1411_440_246}

Explain why the student decides to carry out a test based on the product moment correlation coefficient. Summary statistics for the 20 occasions are as follows. $$\sum x = 342.10 \quad \sum y = 273.65 \quad \sum x ^ { 2 } = 5989.53 \quad \sum y ^ { 2 } = 3919.53 \quad \sum x y = 4713.62$$
In this question you must show detailed reasoning. Calculate the product moment correlation coefficient.
Carry out a hypothesis test at the $5 \%$ significance level to investigate whether there is any correlation between download speed and upload speed.
Both of the variables, download speed and upload speed, are random. Explain why, if download speed had been a non-random variable, the student could not have carried out the hypothesis test to investigate whether there was any correlation between download speed and upload speed.

OCR MEI Further Statistics Major 2023 June Q7

13 marks Standard +0.3

7 An analyst routinely examines bottles of hair shampoo in order to check that the average percentage of a particular chemical which the shampoo contains does not exceed the value of $1.0 \%$ specified by the manufacturer. The percentages of the chemical in a random sample of 12 bottles of the shampoo are as follows. $\begin{array} { l l l l l l l l l l l } 1.087 & 1.171 & 1.047 & 0.846 & 0.909 & 1.052 & 1.042 & 0.893 & 1.021 & 1.085 & 1.096 \end{array} 0.931$ The analyst uses software to draw a Normal probability plot for these data, and to carry out a Normality test as shown below. \includegraphics[max width=\textwidth, alt={}, center]{c692fb20-436f-4bc1-89bd-10fdba41ceba-08_524_1539_694_264}

The analyst is going to carry out a hypothesis test to check whether the average percentage exceeds 1.0\%. Explain which test the analyst should use, referring to each of the following.
- The Normal probability plot
- The $p$-value of the Kolmogorov-Smirnov test
- In this question you must show detailed reasoning.
Carry out the test at the 5\% significance level.

OCR MEI Further Statistics Major 2023 June Q8

12 marks Moderate -0.5

8 The random variable $X$ has a continuous uniform distribution over [0,10].

Find the probability that, if two independent values of $X$ are taken, one is less than 3 and the other is greater than 3 . The random variable $T$ denotes the sum of 5 independent values of $X$.

State the value of $\mathrm { P } ( T \leqslant 25 )$. The spreadsheet below shows the heading row and the first 20 data rows from a total of 100 data rows of a simulation of the distribution of $X$. Each of the 100 rows shows a simulation of 5 independent values of $X$, together with $T$, the sum of the 5 values. All of the values have been rounded to 2 decimal places. In column I the spreadsheet shows the number of values of $T$ that are less than or equal to the corresponding values in column H . For example, there are 75 simulated values of $T$ that are less than or equal to 30 .

	A	B	c	D	E	F	G	H	I
1	$\mathrm { X } _ { 1 }$	$\mathrm { X } _ { 2 }$	$\mathrm { X } _ { 3 }$	$\mathrm { X } _ { 4 }$	$\mathrm { X } _ { 5 }$	T		t	Number $\leqslant \mathrm { t }$
2	3.73	6.65	4.93	0.41	9.33	25.06		0	0
3	4.95	6.58	4.48	2.51	7.26	25.79		5	0
4	8.10	4.87	4.26	3.83	0.79	21.85		10	1
5	6.70	4.10	5.10	1.82	6.76	24.48		15	4
6	3.73	8.38	8.49	9.87	1.31	31.79		20	23
7	3.22	4.36	0.12	1.34	9.49	18.53		25	48
8	9.17	7.13	5.47	4.35	2.44	28.55		30	75
9	3.42	1.93	6.04	2.99	8.85	23.24		35	93
10	0.98	0.68	9.82	9.83	7.28	28.58		40	99
11	5.86	1.67	7.77	4.08	7.14	26.52		45	100
12	9.20	0.31	5.82	5.31	6.45	27.10		50	100
13	7.04	4.30	2.06	0.06	4.16	17.62
14	0.31	5.02	1.48	5.37	1.77	13.94
15	3.77	6.04	1.21	7.67	5.01	23.69
16	1.21	5.54	1.90	1.43	6.91	17.00
17	9.27	1.98	5.80	9.37	9.34	35.76
18	4.30	5.66	2.80	1.56	1.19	15.51
19	7.15	3.19	6.89	5.41	2.18	24.82
20	6.18	6.32	3.01	6.49	9.12	31.13
21	5.03	5.99	5.19	6.97	3.55	26.73

Use the spreadsheet output to estimate each of the following.
- $\mathrm { P } ( T \leqslant 25 )$
- $\mathrm { P } ( T > 35 )$
- In this question you must show detailed reasoning.
The random variable $Y$ is the mean of 100 independent values of $T$. Determine an estimate of $\mathrm { P } ( Y > 26 )$.

OCR MEI Further Statistics Major 2023 June Q9

10 marks Standard +0.3

9 A cyclist who lives on an island suspects that car drivers with locally registered number plates allow more space when passing her than those with non-locally registered number plates. She decides to carry out a hypothesis test and so over a period of time selects a random sample of 250 cars which pass her. For each car she estimates whether the car driver allows at least the recommended 1.5 metres when passing her. The table shows the data which she collected.

Where registered

\cline { 3 - 4 } \multicolumn{2}{|c|}{}

In this question you must show detailed reasoning. Carry out the test at the $5 \%$ significance level to examine whether there is any association between where the car is registered and passing distance.
A friend of the cyclist suggests that there may be a problem with the data, since the cyclist may have introduced some bias in estimating whether cars were allowing the recommended distance. Explain how any bias might have arisen.

OCR MEI Further Statistics Major 2023 June Q10

15 marks Challenging +1.2

10 The continuous random variable $X$ has probability density function given by $f ( x ) = \begin{cases} \frac { 4 } { 15 } \left( \frac { a } { x ^ { 2 } } + 3 x ^ { 2 } - \frac { 7 } { 2 } \right) & 1 \leqslant x \leqslant 2 , \\ 0 & \text { otherwise, } \end{cases}$ where $a$ is a positive constant.

Find the cumulative distribution function of $X$ in terms of $a$.
Hence or otherwise determine the value of $a$.
Show that the median value $m$ of $X$ satisfies the equation $$8 m ^ { 4 } - 28 m ^ { 2 } + 9 m - 4 = 0 .$$
Verify that the median value of $X$ is 1.74, correct to $\mathbf { 2 }$ decimal places.
Find $\mathrm { E } ( X )$.
Determine the mode of $X$.

OCR MEI Further Statistics Major 2023 June Q11

9 marks Moderate -0.5

11 The random variable $X$ takes the value 1 with probability $p$ and the value 0 with probability $1 - p$.

Find each of the following.
- $\mathrm { E } ( X )$
- $\operatorname { Var } ( X )$
- The random variable $Y \sim \mathrm {~B} ( 50,0.2 )$ has mean $\mu$ and variance $\sigma ^ { 2 }$.
Use the results of part (a) to prove that
- $\mu = 10$
- $\sigma ^ { 2 } = 8$.

OCR MEI Further Statistics Major 2024 June Q1

5 marks Easy -1.8

1 The number of insurance policy sales made per month by a salesperson is modelled by the random variable $X$, with probability distribution shown in the table.

$r$	0	1	2	3	4	5	6
$\mathrm { P } ( \mathrm { X } = \mathrm { r } )$	0.05	0.1	0.25	0.3	0.15	0.1	0.05

Find each of the following.
- $\mathrm { E } ( X )$
- $\operatorname { Var } ( X )$
The salesperson is paid a basic salary of $\pounds 1000$ per month plus $\pounds 500$ for each policy that is sold.
Find the mean and standard deviation of the salesperson's monthly salary.

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\(r\)	0	1	2	3	4
\(\mathrm { P } ( X = r )\)	\(a\)	\(b\)	0.24	0.32	\(b ^ { 2 }\)

Temperature \(\left( t ^ { \circ } \mathrm { C } \right)\)	20	22	24	26	28	30	32	34	36
Tyre pressure \(( P\) bar \()\)	2.012	2.036	2.065	2.074	2.114	2.140	2.149	2.176	2.192

\includegraphics[max width=\textwidth, alt={}]{77eabbd6-a058-457f-9601-d66f3c2db005-07_38_45_880_279}	A	B	C	D	E	F	G	H	I	J	K	L
1	\(X _ { 1 }\)	\(X _ { 2 }\)	\(X _ { 3 }\)	\(X _ { 4 }\)	\(X _ { 5 }\)	\(X _ { 6 }\)	\(X _ { 7 }\)	\(X _ { 8 }\)	\(X _ { 9 }\)	\(X _ { 10 }\)	\(Y\)
2	1	6	2	1	18	6	4	9	11	11	6.9
3	13	14	12	2	4	11	16	0	16	0	8.8
4	4	17	1	16	4	10	12	2	18	13	9.7
5	2	8	12	1	4	16	12	2	15	8	8.0
6	7	15	16	0	4	7	1	13	0	20	8.3
7	15	13	10	1	12	0	20	15	16	6	10.8
8	14	13	17	12	2	18	16	18	9	4	12.3
9	20	2	12	3	17	3	0	18	15	13	10.3
10	2	12	5	12	2	6	0	9	10	15	7.3
11	5	11	13	10	9	17	10	4	20	15	11.4
12	14	9	9	7	6	20	2	2	11	16	9.6
13	15	19	18	19	7	6	6	20	3	8	12.1
14	5	10	6	4	1	19	15	8	17	18	10.3
15	0	3	15	15	11	12	0	3	9	16	8.4
16	1	12	1	15	0	4	11	11	9	2	6.6
17	12	5	0	8	3	8	12	19	13	12	9.2
18	9	5	1	13	5	4	18	1	1	19	7.6
19	16	2	20	20	12	17	2	7	8	20	12.4
20	18	17	3	2	8	18	7	0	11	6	9.0
21	15	10	7	20	4	0	5	6	11	14	9.2
22	3	9	10	14	2	1	8	6	0	7	6.0
23	11	10	11	10	19	11	3	7	10	0	9.2
24	12	14	6	6	5	20	11	18	10	14	11.6
25	1	11	5	14	11	10	1	1	2	0	5.6
26	0	14	7	11	18	5	10	20	11	9	10.5
27

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