Questions - OCR MEI - A-Level Maths

Find $\mathrm { E } ( \mathrm { Z } )$.
Ben thinks that $Z$ will be well modelled by a continuous uniform distribution on $[ 0,10 ]$. By considering variances, show that he is not correct.

Ben's friend Jamila constructs the spreadsheet below, which shows a simulation of 20 values of $X , Y$ and $Z$. All of the values have been rounded to 2 decimal places.

\multirow[b]{3}{*}{

}

Write down an estimate of $\mathrm { P } ( Z > 6 )$.

Use a Normal approximation to determine the probability that Ben's total waiting time when travelling to and from work on 40 days is more than 210 minutes.

OCR MEI Further Statistics Major 2024 June Q11

11 The discrete random variable $X$ has a uniform distribution over the set of all integers between 25 and $n$ inclusive, where $n$ is a positive integer with $n > 25$.

Determine $\mathrm { P } \left( \mathrm { X } < \frac { \mathrm { n } + 25 } { 2 } \right)$ in each of the following cases.
- $n$ is even
- $n$ is odd
- Determine an expression in terms of $n$ for the variance of the mean of 100 independent values of $X$.
- Given that $n = 75$, calculate an estimate of the probability that the mean of 100 independent values of $X$ is less than 48 .

OCR MEI Further Statistics Major 2024 June Q12

12 The cumulative distribution function of the continuous random variable $X$ is given by
$F ( x ) = \begin{cases} 0 & x < 20 ,
a \left( x ^ { 2 } + b x + c \right) & 20 \leqslant x \leqslant 30 ,
1 & x > 30 , \end{cases}$
where $a$, $b$ and $c$ are constants.
You are given that $\mathrm { P } ( X < 25 ) = \frac { 11 } { 24 }$.

Find $\mathrm { P } ( X > 27 )$.
Find the 90th percentile of $X$.

OCR MEI Further Statistics Major 2020 November Q1

1 In a game at a fair, players choose 4 countries from a list of 10 countries. The names of all 10 countries are then put in a box and the player selects 4 of them at random. The random variable $X$ represents the number of countries that match those which the player originally chose.

Show that the probability that a randomly selected player matches all 4 countries is $\frac { 1 } { 210 }$. Table 1 shows the probability distribution of $X$. \begin{table}[h]
$r$ 0 1 2 3 4
$\mathrm { P } ( X = r )$ $\frac { 1 } { 14 }$ $\frac { 8 } { 21 }$ $\frac { 3 } { 7 }$ $\frac { 4 } { 35 }$ $\frac { 1 } { 210 }$
\captionsetup{labelformat=empty} \caption{Table 1}
\end{table}
Find each of the following.
- $\mathrm { E } ( X )$
- $\operatorname { Var } ( X )$
- A player has to pay $\pounds 1$ to play the game. The player gets 40 pence back for every country which is matched.
Find the mean and standard deviation of the player's loss per game.
In order to try to attract more customers, the rules will be changed as follows. The game will still cost $\pounds 1$ to play. The player will get 25 pence back for every country which is matched, plus an additional bonus of $\pounds 100$ if all four countries are matched. Find the player's mean gain or loss per game with these new rules.

OCR MEI Further Statistics Major 2020 November Q2

2 On average 1 in 4000 people have a particular antigen in their blood (an antigen is a molecule which may cause an adverse reaction).

1. A random sample of 1200 people is selected. The random variable $X$ represents the number of people in the sample who have this antigen in their blood. Explain why you could use either a binomial distribution or a Poisson distribution to model the distribution of $X$.
2. Use either a binomial or a Poisson distribution to calculate each of the following probabilities.
  - $\mathrm { P } ( X = 3 )$
$\mathrm { P } ( X > 3 )$
A researcher needs to find 2 people with the antigen. Find the probability that at most 5000 people have to be tested in order to achieve this.

OCR MEI Further Statistics Major 2020 November Q3

3 A supermarket sells cashew nuts in three different sizes of bag: small, medium and large. The weights in grams of the nuts in each type of bag are modelled by independent Normal distributions as shown in Table 3. \begin{table}[h]

Bag size	Mean	Standard deviation
Small	51.5	1.1
Medium	100.7	1.6
Large	201.3	1.7

\captionsetup{labelformat=empty} \caption{Table 3}

\end{table}

Find the probability that the mean weight of two randomly selected large bags is at least 200 g .
Find the probability that the total weight of eight randomly selected small bags is greater than the total weight of two randomly selected medium bags and one randomly selected large bag.

OCR MEI Further Statistics Major 2020 November Q4

4 An amateur meteorologist records the total rainfall at her home each day using a traditional rain gauge. This means that she has to go out each day at 9 am to read the rain gauge and then to empty it. She wants to save time by using a digital rain gauge, but she also wants to ensure that the readings from the digital gauge are similar to those of her traditional gauge. Over a period of 100 days, she uses both gauges to measure the rainfall. The meteorologist uses software to produce a 95\% confidence interval for the difference between the two readings (the traditional gauge reading minus the digital gauge reading). The output from the software is shown in Fig. 4. Although rainfall was measured over a period of 100 days, there was no rain on 40 of those days and so the sample size in the software output is 60 rather than 100. \begin{table}[h]

\captionsetup{labelformat=empty} \caption{Fig. 4}

\end{table}

Explain why this confidence interval can be calculated even though nothing is known about the distribution of the population of differences.
State the confidence interval which the software gives in the form $a < \mu < b$.
Show how the value 0.07444 (labelled SE) was calculated.
Comment on whether you think that the confidence interval suggests that the two different methods of measurement are broadly in agreement.

OCR MEI Further Statistics Major 2020 November Q5

5 A hearing expert is investigating whether web-based hearing tests can be used instead of hearing tests in a hearing laboratory. The expert selects a random sample of 16 people with normal hearing. Each of them is given two hearing tests, one in the laboratory and one web-based. The scores in the laboratory-based test, $x$, and the web-based test, $y$, are both measured in the same suitable units.

Half of the participants do the laboratory-based test first and the other half do the web-based test first. Explain why the expert adopts this approach. The scatter diagram in Fig. 5 shows the data that the expert collected. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8d36bc92-07ac-40c3-9e75-26f2bc9d2fcc-05_785_1360_1009_242} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure} Summary statistics for these data are as follows. $$\Sigma x = 198.0 \quad \Sigma x ^ { 2 } = 2936.92 \quad \Sigma y = 188.7 \quad \Sigma y ^ { 2 } = 2605.35 \quad \Sigma x y = 2554.87$$
Calculate the equation of the regression line suitable for estimating web-based scores from laboratory-based scores.
Estimate the web-based scores of people whose laboratory-based scores were as follows.
- 12
- 25
- Comment on the reliability of each of your estimates.
- A colleague of the expert suggests that the regression line is not valid because one of the data values is an outlier.
Stating the approximate coordinates of the outlier, suggest what the expert should do.

OCR MEI Further Statistics Major 2020 November Q6

6 A pollution control officer is investigating a possible link between the levels of various pollutants in the air and the speed of the wind at various sites. A random sample of 60 values of the windspeed together with the levels of a variety of pollutants is taken at a particular site. The product moment correlation coefficient between wind-speed and nitrogen dioxide level is 0.3231 .

Carry out a hypothesis test at the $10 \%$ significance level to investigate whether there is any correlation between wind-speed and nitrogen dioxide level.

State the condition required for the test carried out in part (a) to be valid. Table 6.1 shows the values of the product moment correlation coefficient between 5 different measures of pollution and also wind-speed for a very large random sample of values at another site. Those correlations that are significant at the $10 \%$ level are denoted by a * after the value of the correlation. \begin{table}[h]

Correlations	PM10	SPEED	$\mathrm { NO } _ { 2 }$	$\mathrm { O } _ { 3 }$	PM25	$\mathrm { SO } _ { 2 }$
PM10	1.00
SPEED	0.08*	1.00
$\mathrm { NO } _ { 2 }$	0.59*	0.25*	1.00
$\mathbf { O } _ { \mathbf { 3 } }$	-0.05*	-0.04*	-0.30*	1.00
PM25	0.85*	-0.01	0.56*	-0.02	1.00
$\mathrm { SO } _ { 2 }$	0.42*	0.15*	0.73*	-0.63*	0.40*	1.00

\captionsetup{labelformat=empty} \caption{Table 6.1}

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

\captionsetup{labelformat=empty} \caption{Table 6.2 shows standard guidelines for effect sizes.}

Product moment

correlation coefficient

\end{table} Table 6.2 The officer analyses these data for effect size.

Explain how the very large sample size relates to the interpretation of the correlation coefficients shown in Table 6.1.
Comment briefly on what the pollution control officer might conclude from these tables, relevant to her investigation into wind-speed and pollutant levels.

OCR MEI Further Statistics Major 2020 November Q7

10 marks

7 The lengths in mm of a random sample of 6 one-year-old fish of a particular species are as follows.
$\begin{array} { l l l l l l } 271 & 293 & 306 & 287 & 264 & 290 \end{array}$

State an assumption required in order to find a confidence interval for the mean length of one-year-old fish of this species. Fig. 7 shows a Normal probability plot for these data. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8d36bc92-07ac-40c3-9e75-26f2bc9d2fcc-07_599_753_646_246} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
Explain why the Normal probability plot suggests that the assumption in part (a) may be valid.
In this question you must show detailed reasoning. Assuming that this assumption is true, find a 95\% confidence interval for the mean length of one-year-old fish of this species.

OCR MEI Further Statistics Major 2020 November Q9

9 A supermarket sells trays of peaches. Each tray contains 10 peaches. Often some of the peaches in a tray are rotten. The numbers of rotten peaches in a random sample of 150 trays are shown in Table 9.1. \begin{table}[h]

Number of rotten peaches	0	1	2	3	4	5	6	$\geqslant 7$
Frequency	39	39	33	19	8	8	4	0

\captionsetup{labelformat=empty} \caption{Table 9.1}

\end{table} A manager at the supermarket thinks that the number of rotten peaches in a tray may be modelled by a binomial distribution.

Use these data to estimate the value of the parameter $p$ for the binomial model $\mathrm { B } ( 10 , p )$. The manager decides to carry out a goodness of fit test to investigate further. The screenshot in Fig. 9.2 shows part of a spreadsheet to assess the goodness of fit of the distribution $\mathrm { B } ( 10 , p )$, using the value of $p$ estimated from the data. \begin{table}[h]

-	A	B	C	D	E
1	Number of rotten peaches	Observed frequency	Binomial probability	Expected frequency	Chi-squared contribution
2	0	39
3	1	39			1.4229
4	2	33	0.2941	44.1167	2.8012
5	3	19	0.1629	24.4383	1.2102
6	$\geqslant 4$	20	0.0769	11.5311	6.2199
7

\captionsetup{labelformat=empty} \caption{Fig. 9.2}

\end{table}

Calculate the missing values in each of the following cells.
- C2
- D2
- E2
- Explain why the numbers for 4, 5, 6 and at least 7 rotten peaches have been combined into the single category of at least 4 rotten peaches, as shown in the spreadsheet.
- Carry out the test at the $1 \%$ significance level.
- Using the values of the contributions, comment on the results of the test.

OCR MEI Further Statistics Major 2020 November Q10

10 The discrete random variables $X$ and $Y$ have distributions as follows: $X \sim \mathrm {~B} ( 20,0.3 )$ and $Y \sim \operatorname { Po } ( 3 )$. The spreadsheet in Fig. 10 shows a simulation of the distributions of $X$ and $Y$. Each of the 20 rows below the heading row consists of a value of $X$, a value of $Y$, and the value of $X - 2 Y$. \begin{table}[h]

1	A	B	C
1	X	Y	$X - 2 Y$
2	6	6	-6
3	5	4	-3
4	8	1	6
5	6	5	-4
6	6	3	0
7	8	1	6
8	6	4	-2
9	5	4	-3
10	7	4	-1
11	8	3	2
12	6	2	2
13	5	1	3
14	6	1	4
15	5	4	-3
16	7	2	3
17	5	2	1
18	4	4	-4
19	5	0	5
20	5	1	3
21	4	2	0
nn

\captionsetup{labelformat=empty} \caption{Fig. 10}

\end{table}

Use the spreadsheet to estimate each of the following.
- $\mathrm { P } ( X - 2 Y > 0 )$
- $\mathrm { P } ( X - 2 Y > 1 )$
- How could the estimates in part (a) be improved?
The mean of 50 values of $X - 2 Y$ is denoted by the random variable $W$.
Calculate an estimate of $\mathrm { P } ( W > 1 )$.

OCR MEI Further Statistics Major 2020 November Q11

11 The length of time in minutes for which a particular geyser erupts is modelled by the continuous random variable $T$ with cumulative distribution function given by
$\mathrm { F } ( t ) = \begin{cases} 0 & t \leqslant 2 ,
k \left( 8 t ^ { 2 } - t ^ { 3 } - 24 \right) & 2 < t < 4 ,
1 & t \geqslant 4 , \end{cases}$
where $k$ is a positive constant.

Show that $k = \frac { 1 } { 40 }$.
Find the probability that a randomly selected eruption time lies between 2.5 and 3.5 minutes.
Show that the median $m$ of the distribution satisfies the equation $m ^ { 3 } - 8 m ^ { 2 } + 44 = 0$.
Verify that the median eruption time is 2.95 minutes, correct to 2 decimal places. The mean and standard deviation of $T$ are denoted by $\mu$ and $\sigma$ respectively.
Find $\mathrm { P } ( \mu - \sigma < T < \mu + \sigma )$.
Sketch the graph of the probability density function of $T$.
A Normally distributed random variable $X$ has the same mean and standard deviation as $T$. By considering the shape of the Normal distribution, and without doing any calculations, explain whether $\mathrm { P } ( \mu - \sigma < X < \mu + \sigma )$ will be greater than, equal to or less than the probability that you calculated in part (e).

OCR MEI Further Statistics Major 2021 November Q1

1 When babies are born, their head circumferences are measured. A random sample of 50 newborn female babies is selected. The sample mean head circumference is 34.711 cm . The sample standard deviation head circumference is 1.530 cm .

Determine a 95\% confidence interval for the population mean head circumference of newborn female babies.
Explain why you can calculate this interval even though the distribution of the population of head circumferences of newborn female babies is unknown.

OCR MEI Further Statistics Major 2021 November Q2

2 In a game at a charity fair, a player rolls 3 unbiased six-sided dice. The random variable $X$ represents the difference between the highest and lowest scores.

Show that $\mathrm { P } ( X = 0 ) = \frac { 1 } { 36 }$. The table shows the probability distribution of $X$.
$r$ 0 1 2 3 4 5
$\mathrm { P } ( \mathrm { X } = \mathrm { r } )$ $\frac { 1 } { 36 }$ $\frac { 5 } { 36 }$ $\frac { 2 } { 9 }$ $\frac { 1 } { 4 }$ $\frac { 2 } { 9 }$ $\frac { 5 } { 36 }$
Draw a graph to illustrate the distribution.
Describe the shape of the distribution.
In this question you must show detailed reasoning. Find each of the following.
- $\mathrm { E } ( X )$
- $\operatorname { Var } ( X )$
As a result of playing the game, the player receives $30 X$ pence from the organiser of the game.
Find the variance of the amount that the player receives.
The player pays $k$ pence to play the game. Given that the average profit made by the organiser is 12.5 pence per game, determine the value of $k$.

OCR MEI Further Statistics Major 2021 November Q3

3 In air traffic management, air traffic controllers send radio messages to pilots. On receiving a message, the pilot repeats it back to the controller to check that it has been understood correctly. At a particular site, on average $4 \%$ of messages sent by controllers are not repeated back correctly and so have been misunderstood. You should assume that instances of messages being misunderstood occur randomly and independently.

Find the probability that exactly 2 messages are misunderstood in a sequence of 50 messages.
Find the probability that in a sequence of messages, the 10th message is the first one which is misunderstood.
Find the probability that in a sequence of 20 messages, there are no misunderstood messages.
Determine the expected number of messages required for 3 of them to be misunderstood.
Determine the probability that in a sequence of messages, the 3rd misunderstood message is the 60th message in the sequence.

OCR MEI Further Statistics Major 2021 November Q4

4 A radioactive source contains 1000000 nuclei of a particular radioisotope. On average 1 in 200000 of these nuclei will decay in a period of 1 second. The random variable $X$ represents the number of nuclei which decay in a period of 1 second. You should assume that nuclei decay randomly and independently of each other.

Explain why you could use either a binomial distribution or a Poisson distribution to model the distribution of $X$. Use a Poisson distribution to answer parts (b) and (c).
Calculate each of the following probabilities.
- $\mathrm { P } ( X = 6 )$
- $\mathrm { P } ( X > 6 )$
- Determine an estimate of the probability that at least 60 nuclei decay in a period of 10 seconds.

OCR MEI Further Statistics Major 2021 November Q5

5 A manufacturer uses three types of capacitor in a particular electronic device. The capacitances, measured in suitable units, are modelled by independent Normal distributions with means and standard deviations as shown in the table.

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

Determine the probability that the total capacitance of a randomly chosen capacitor of Type B and two randomly chosen capacitors of Type A is at least 16 units.
Determine the probability that the capacitance of a randomly chosen capacitor of Type C is within 1 unit of the total capacitance of four randomly chosen capacitors of Type B. When the manufacturer gets a new batch of 1000 capacitors from the supplier, a random sample of 10 of them is tested to check the capacitances. For a new batch of Type C capacitors, summary statistics for the capacitances, $x$ units, of the random sample are as follows.
$n = 10$ $$\sum x = 299.6 \quad \sum x ^ { 2 } = 8981.0$$ You should assume that the capacitances of the sample come from a Normally distributed population, but you should not assume that the standard deviation is 0.64 as for previous Type C capacitors.
In this question you must show detailed reasoning. Carry out a hypothesis test at the $5 \%$ significance level to check whether it is reasonable to assume that the capacitors in this batch have the specified mean capacitance for Type C of 30.2 units.

OCR MEI Further Statistics Major 2021 November Q6

6 Cosmic rays passing through the upper atmosphere cause muons, and other types of particle, to be formed. Muons can be detected when they reach the surface of the earth. It is known that the mean number of muons reaching a particular detector is 1.7 per second. The numbers of muons reaching this detector in 200 randomly selected periods of 1 second are shown in Fig. 6.1. \begin{table}[h]

Number of muons	0	1	2	3	4	5	6	$\geqslant 7$
Frequency	34	65	55	24	14	6	2	0

\captionsetup{labelformat=empty} \caption{Fig. 6.1}

\end{table}

Use the values of the sample mean and sample variance to discuss the suitability of a Poisson distribution as a model. The screenshot in Fig. 6.2 shows part of a spreadsheet to assess the goodness of fit of the distribution Po(1.7). \begin{table}[h]

	A	B	C	D	E
1	Number of muons	Observed frequency	Poisson probability	Expected frequency	Chi-squared contribution
2	0	34	0.1827	36.5367	0.1761
3	1	65
4	2	55	0.2640	52.7955	0.0920
5	3	24	0.1496	29.9175	1.1704
6	4	14			0.1299
7	$\geqslant 5$	8	0.0296	5.9230	0.7284

\captionsetup{labelformat=empty} \caption{Fig. 6.2}

\end{table}

Calculate the missing values in each of the following cells.
- C3
- D3
- E3
- Explain why the numbers for 5, 6 and at least 7 muons have been combined into the single category of at least 5 muons, as shown in Fig. 6.2.
- In this question you must show detailed reasoning.
Carry out the test at the 5\% significance level.

OCR MEI Further Statistics Major 2021 November Q7

7 A physiotherapist is investigating hand grip strength in adult women under 30 years old. She thinks that the grip strength of the dominant hand will be on average 2 kg higher than the grip strength of the non-dominant hand. The physiotherapist selects a random sample of 12 adult women under 30 years old and measures the grip strength of each of their hands. She then uses software to produce a $95 \%$ confidence interval for the mean difference in grip strength between the two hands (dominant minus nondominant), as shown in Fig. 7. \begin{table}[h]

}


Result
T Estimate of a Mean
Mean	2.79
s	3.92
SE	1.13161
N	12
df	11
Lower Limit	0.29935
Upper Limit	5.28065
Interval	$2.79 \pm 2.49065$

\captionsetup{labelformat=empty} \caption{Fig. 7} \end{table}

Explain why the physiotherapist used the same people for testing their dominant and nondominant grip strengths.
State any assumptions necessary in order to construct the confidence interval shown in Fig. 7.
Explain whether the confidence interval supports the physiotherapist's belief.
The physiotherapist then finds some data which have previously been collected on grip strength using a sample of 100 adult women. A 95\% confidence interval, based on this sample and calculated using a Normal distribution, for the mean difference in grip strength between the two hands (dominant minus non-dominant) is (1.94, 2.84).
1. For this sample, find
  - the mean difference
the standard deviation of the differences.
(ii) Explain what you would need to know about the nature of this sample if you wanted to draw conclusions about the mean difference in grip strength in the population of adult women.

OCR MEI Further Statistics Major 2021 November Q8

$\mathrm { VO } _ { 2 \max }$ is a measure of athletic fitness. Since $\mathrm { VO } _ { 2 \max }$ is fairly time-consuming and expensive to measure, an exercise scientist wants to predict $\mathrm { VO } _ { 2 _ { \text {max } } }$ from data such as times for running different distances. The scientist uses these data for a random sample of 15 athletes to predict their $\mathrm { V } \mathrm { O } _ { 2 \text { max } }$ values, denoted by $y$, in suitable units. She also obtains accurate measurements of the $\mathrm { V } \mathrm { O } _ { 2 \text { max } }$ values, denoted by $x$, in the same units. The scatter diagram in Fig. 8.1 shows the values of $x$ and $y$ obtained, together with the equation of the regression line of $y$ on $x$ and the value of $r ^ { 2 }$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ce557137-f9eb-4c09-a7e3-e4ec626109dc-08_750_1324_660_317} \captionsetup{labelformat=empty} \caption{Fig. 8.1}
\end{figure}
1. Use the regression line to estimate the predicted $\mathrm { VO } _ { 2 \text { max } }$ of an athlete whose accurately measured $\mathrm { VO } _ { 2 \text { max } }$ is 50 .
2. Comment on the reliability of your estimate.
3. The equation of the regression line of $x$ on $y$ is $x = 0.7565 y + 10.493$. Find the coordinates of the point at which the two regression lines meet.
4. State what the point you found in part (iii) represents.
It is known that there is negative correlation between $\mathrm { VO } _ { 2 \text { max } }$ and marathon times in very good runners (those whose best marathon times are under 3 hours). The exercise scientist wishes to know whether the same applies to runners who take longer to run a marathon. She selects a random sample of 20 runners whose best marathon times are between $3 \frac { 1 } { 2 }$ hours and $4 \frac { 1 } { 2 }$ hours and accurately measures their $\mathrm { VO } _ { 2 \text { max } }$. Fig. 8.2 is a scatter diagram of accurately measured $\mathrm { VO } _ { \text {2max } }$, $v$ units, against best marathon time, $t$ hours, for these runners. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ce557137-f9eb-4c09-a7e3-e4ec626109dc-09_671_1064_648_319} \captionsetup{labelformat=empty} \caption{Fig. 8.2}
\end{figure}
1. Explain why the exercise scientist comes to the conclusion that a test based on Pearson's product moment correlation coefficient may be valid. Summary statistics for the 20 runners are as follows. $$\sum t = 80.37 \quad \sum v = 970.86 \quad \sum t ^ { 2 } = 324.71 \quad \sum v ^ { 2 } = 47829.24 \quad \sum t v = 3886.53$$
2. Find the value of Pearson's product moment correlation coefficient.
3. Carry out a test at the $5 \%$ significance level to investigate whether there is negative correlation between accurately measured $\mathrm { VO } _ { 2 _ { \text {max } } }$ and best marathon time for runners whose best marathon times are between $3 \frac { 1 } { 2 }$ hours and $4 \frac { 1 } { 2 }$ hours.

OCR MEI Further Statistics Major 2021 November Q9

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

Given that $n$ is odd, determine $\mathrm { P } \left( \mathrm { X } > \frac { 1 } { 2 } \mathrm { n } \right)$, giving your answer as a single fraction in terms of $n$.
Determine the variance of the sum of 10 independent values of $X$, giving your answer in the form $\mathrm { an } ^ { 2 } + \mathrm { bn }$, where $a$ and $b$ are constants.

OCR MEI Further Statistics Major 2021 November Q10

10 Sarah takes a bus to work each weekday morning and returns each evening. The times in minutes that she has to wait for the bus in the morning and evening are modelled by uniform distributions over the intervals $[ 0,10 ]$ and $[ 0,6 ]$ respectively. The times in minutes for the bus journeys in the morning and evening are modelled by $\mathrm { N } ( 25,4 )$ and $\mathrm { N } ( 28,16 )$ respectively. You should assume that all of the times are independent. The total time in minutes that she takes for her two journeys, including the waiting times, is denoted by the random variable $T$. The spreadsheet below shows the first 20 rows of a simulation of 500 return journeys. It also shows in column H the numbers of values of $T$ that are less than or equal to the corresponding values in column G. For example, there are 156 out of the 500 simulated values of $T$ which are less than or equal to 58 minutes. All of the times have been rounded to 2 decimal places.

	A	B	C	D	E	F	G	H
1	Waiting time morning	Journey time morning	Waiting time evening	Journey time evening	Total time		Total time t	Number $\leqslant \mathbf { t }$
2	0.89	20.78	1.88	26.30	49.86		46	0
3	3.55	21.24	1.04	29.61	55.44		48	4
4	2.13	21.83	2.40	28.64	55.00		50	13
5	5.12	25.04	3.13	24.30	57.60		52	35
6	4.03	27.49	2.19	30.81	64.52		54	57
7	2.47	20.54	4.32	34.61	61.93		56	104
8	3.21	26.93	3.78	27.66	61.58		58	156
9	9.72	24.15	0.63	27.53	62.03		60	218
10	1.59	28.45	0.08	35.87	65.99		62	288
11	7.34	23.04	4.02	24.77	59.17		64	357
12	1.04	24.69	1.66	31.95	59.33		66	408
13	7.17	22.16	2.55	25.39	57.28		68	441
14	5.20	26.97	2.41	30.05	64.62		70	475
15	5.01	26.84	1.88	36.21	69.93		72	490
16	3.76	26.03	2.21	30.96	62.96		74	496
17	0.96	23.72	2.55	29.36	56.59		76	500
18	8.64	24.97	2.82	26.39	62.82
19	0.59	20.82	4.57	31.41	57.38
20	9.85	23.68	5.54	29.92	68.99
01

Use the spreadsheet output to estimate each of the following.
- $\mathrm { P } ( T \leqslant 56 )$
- $\mathrm { P } ( T > 61 )$
- The random variable $W$ is Normally distributed with the same mean and variance as $T$. Find each of the following.
- $\mathrm { P } ( W \leqslant 56 )$
- $\mathrm { P } ( W > 61 )$
- Explain why, if many more journeys were used in the simulation, you would expect $\mathrm { P } ( T > 61 )$ to be extremely close to $\mathrm { P } ( W > 61 )$.

OCR MEI Further Statistics Major 2021 November Q11

11 The continuous random variable $X$ has probability density function given by
$f ( x ) = \begin{cases} a x ^ { 2 } & 0 \leqslant x < 2 ,
b ( 3 - x ) ^ { 2 } & 2 \leqslant x \leqslant 3 ,
0 & \text { otherwise } \end{cases}$
where $a$ and $b$ are positive constants.

Given that $\mathrm { E } ( X ) = 2$, determine the values of $a$ and $b$.
Determine the median value of $X$.
A random sample of 50 observations of $X$ is selected. Given that $\operatorname { Var } ( X ) = 0.2$, determine an estimate of the probability that the mean value of the 50 observations is less than 1.9.

OCR MEI Further Numerical Methods 2019 June Q1

1 Fig. 1 shows some spreadsheet output concerning the values of a function, $\mathrm { f } ( x )$. \begin{table}[h]

	A	B	C
1	$x$	$\mathrm { f } ( x )$
2	1	0.367879441	0.367879441
3	2	0.018315639	0.38619508
4	3	0.00012341	0.38631849
5	4	$1.12535 \mathrm { E } - 07$	0.386318602
6	5	$1.38879 \mathrm { E } - 11$	0.386318602

\captionsetup{labelformat=empty} \caption{Fig. 1}

\end{table} The formula in cell B2 is ==EXP(-(A2\^{}2)) and equivalent formulae are in cells B3 to B6. The formula in cell C 2 is $= \mathrm { B } 2$.
The formula in cell C3 is $\quad = \mathrm { C } 2 + \mathrm { B } 3$. Equivalent formulae are in cells C4 to C6.

Use sigma notation to express the formula in cell C5 in standard mathematical notation.
Explain why the same value is displayed in cells C 5 and C 6. Now suppose that the value in cell C2 is chopped to 3 decimal places and used to approximate the value in cell C2.
Calculate the relative error when this approximation is used. Suppose that the values in cells B4, B5 and B6 are chopped to 3 decimal places and used as approximations to the original values in cells B4, B5 and B6 respectively.
Explain why the relative errors in these approximations are all the same.

\(r\)	0	1	2	3	4
\(\mathrm { P } ( X = r )\)	\(\frac { 1 } { 14 }\)	\(\frac { 8 } { 21 }\)	\(\frac { 3 } { 7 }\)	\(\frac { 4 } { 35 }\)	\(\frac { 1 } { 210 }\)

\(r\)	0	1	2	3	4	5
\(\mathrm { P } ( \mathrm { X } = \mathrm { r } )\)	\(\frac { 1 } { 36 }\)	\(\frac { 5 } { 36 }\)	\(\frac { 2 } { 9 }\)	\(\frac { 1 } { 4 }\)	\(\frac { 2 } { 9 }\)	\(\frac { 5 } { 36 }\)

Number of rotten peaches	0	1	2	3	4	5	6	\(\geqslant 7\)
Frequency	39	39	33	19	8	8	4	0

1	A	B	C
1	X	Y	\(X - 2 Y\)
2	6	6	-6
3	5	4	-3
4	8	1	6
5	6	5	-4
6	6	3	0
7	8	1	6
8	6	4	-2
9	5	4	-3
10	7	4	-1
11	8	3	2
12	6	2	2
13	5	1	3
14	6	1	4
15	5	4	-3
16	7	2	3
17	5	2	1
18	4	4	-4
19	5	0	5
20	5	1	3
21	4	2	0
nn

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