Questions - OCR MEI - A-Level Maths

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

OCR MEI Further Statistics Major 2024 June Q2

2 The number of cars arriving per minute to queue at a drive-through fast-food restaurant is modelled by the random variable $X$. The standard deviation of $X$ is 0.6 . You should assume that arrivals are random and independent and occur at a constant average rate.

Find the mean of $X$.
1. Calculate $\mathrm { P } ( X = 1 )$.
2. Calculate $\mathrm { P } ( X > 1 )$.
Find the probability that fewer than 5 cars arrive in a randomly chosen 20 -minute period.

OCR MEI Further Statistics Major 2024 June Q3

3 At a launderette the process of cleaning a load of clothes consists of three stages: washing, drying and folding. The times in minutes for each process are modelled by independent Normal distributions with means and standard deviations as shown in the table.

\cline { 2 - 3 } \multicolumn{1}{c\|}{}	Mean	Standard deviation
Washing	35	2.4
Drying	46	3.1
Folding	12	2.2

Find the probability that drying a randomly chosen load of clothes takes more than 50 minutes.
It is given that for $99 \%$ of loads of clothes the washing time is less than $k$ minutes. Find the value of $k$.
Determine the probability that the drying time for a randomly chosen load of clothes is less than the total of the washing and folding times.
Determine the probability that the mean time for cleaning 5 randomly chosen loads of clothes is less than 90 minutes. You should assume that the time for cleaning any load is independent of the time for cleaning any other load.

OCR MEI Further Statistics Major 2024 June Q4

4 An archer fires arrows at a circular target of radius 50 cm . The distance in cm that an arrow lands from the centre of the target is modelled by the random variable $X$, with probability density function given by
$f ( x ) = \begin{cases} a x & 0 \leqslant x \leqslant 50 ,
0 & \text { otherwise, } \end{cases}$
where $a$ is a constant.

Determine the value of $a$.
Determine the probability that an arrow will land within 5 cm of the centre of the target.
Determine the median distance from the centre of the target that an arrow will land.

OCR MEI Further Statistics Major 2024 June Q5

5 A researcher is investigating whether doing yoga has any effect on quality of sleep in older people. The researcher selects a random sample of 40 older people, who then complete a yoga course. Before they start the course and again at the end, the 40 people fill in a questionnaire which measures their perceived sleep quality. The higher the score, the better is the perceived quality of sleep. The researcher uses software to produce a 90\% confidence interval for the difference in mean sleep quality (sleep quality after the course minus sleep quality before the course). The output from the software is shown below. Z Estimate of a Mean Confidence level □ 0.9 Sample

Mean	0.586
$s$	2.14
	40

Result
Z Estimate of a Mean

Mean	0.586
s	2.14
SE	0.3384
N	40
Lower limit	0.029
Upper limit	1.143
Interval	$0.586 \pm 0.557$

Explain why the confidence interval is based on the Normal distribution even though the distribution of the population of differences is not known.
Explain whether the confidence interval suggests that the mean sleep qualities before and after completing a yoga course are different.
In the output from the software, SE stands for 'standard error'.
1. Explain what standard error is.
2. Show how the standard error was calculated in this case.
A colleague of the researcher suggests that the confidence level should have been $95 \%$ rather than $90 \%$. Determine whether this would have made a difference to your answer to part (b).

OCR MEI Further Statistics Major 2024 June Q6

6 A student is investigating the relationship between age and grip strength in adults. The student selects 10 people and records their ages in years and the grip strengths of their dominant hand, measured in kg. The data are shown in the table below, together with a scatter diagram to illustrate the data.

Age	22	29	36	39	53	57	60	71	76	82
Grip strength	38	46	42	49	37	47	36	33	34	24

\includegraphics[max width=\textwidth, alt={}]{bab116b3-6e5f-44db-ac86-670e4040d649-05_634_1107_641_239}

The student decides to carry out a hypothesis test to investigate whether there is negative association between age and grip strength.

Explain why the student decides to carry out a test based on Spearman's rank correlation coefficient.
State what property of the sample is required in order for it to be valid to carry out a hypothesis test.
In this question you must show detailed reasoning. Assuming that the property in part (b) holds, carry out the test at the $5 \%$ significance level.

OCR MEI Further Statistics Major 2024 June Q7

7 An environmental investigator wants to check whether the level of selenium in carrots in fields near a mine is different from the usual level in the country, which is $9.4 \mathrm { ng } / \mathrm { g }$ (nanograms per gram). She takes a random sample of 10 carrots from fields near the mine and measures the selenium level of each of them in $\mathrm { ng } / \mathrm { g }$, with results as follows.
$\begin{array} { l l l l l l l l l l } 6.20 & 10.72 & 11.42 & 16.32 & 15.33 & 10.56 & 8.83 & 9.21 & 7.78 & 14.32 \end{array}$

Find estimates of each of the following.
- The population mean
- The population standard deviation
The investigator produces a Normal probability plot and carries out a Kolmogorov-Smirnov test for these data as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{bab116b3-6e5f-44db-ac86-670e4040d649-06_583_1499_959_242}
Comment on what the Normal probability plot and the $p$-value of the test suggest about the data.
State the null hypothesis for the Kolmogorov-Smirnov test for Normality.
In this question you must show detailed reasoning. Carry out a test at the $5 \%$ significance level to investigate whether the mean selenium level in carrots from fields near the mine is different from $9.4 \mathrm { ng } / \mathrm { g }$.
If the $p$-value of the Kolmogorov-Smirnov test for Normality had been 0.007, explain what procedure you could have used to investigate the selenium level in carrots from fields near the mine.

OCR MEI Further Statistics Major 2024 June Q8

8 An estate agent collects data for a random selection of 13 flats in order to investigate the link between the floor areas of flats and their price. The scatter diagram shows the floor areas, $x \mathrm {~m} ^ { 2 }$, and prices, $\pounds y$ thousand, of the 13 flats.
\includegraphics[max width=\textwidth, alt={}, center]{bab116b3-6e5f-44db-ac86-670e4040d649-07_613_1246_386_242}

The estate agent notes that two of the data points are outliers. One is Flat A which has a large floor area but is in poor condition. The other is Flat B which has a balcony with a desirable view overlooking the sea. Label these two data points on the copy of the scatter diagram in the Printed Answer Booklet. The estate agent decides to remove these two data points from the analysis. Summary statistics for the remaining 11 flats are as follows. $$\sum x = 652.5 \quad \sum y = 5067 \quad \sum x ^ { 2 } = 41987.35 \quad \sum y ^ { 2 } = 2456813 \quad \sum x y = 315928.2$$
In this question you must show detailed reasoning. Calculate the equation of a regression line which is suitable for estimating the price of a flat from its floor area.
Use the regression line to estimate the price for the following floor areas.
- $40 \mathrm {~m} ^ { 2 }$
- $110 \mathrm {~m} ^ { 2 }$
- Given that the value of the product moment correlation coefficient for these 11 data items is 0.765 , comment on the reliability of your estimates.
- The estate agent thinks that he can predict the floor area of a flat from its price, using the equation of the regression line found in part (b).
Comment briefly on the estate agent's idea.

OCR MEI Further Statistics Major 2024 June Q9

9 A cyclist has 3 bicycles, a road bike, a gravel bike and an electric bike. She wishes to know if the bicycle which she is riding makes any difference to whether she reaches a speed of 25 mph or greater on a journey. She selects a random sample of 120 journeys and notes the bicycle and whether or not her maximum speed was 25 mph or greater. She decides to carry out a chisquared test to investigate whether there is any association between bicycle type and whether her maximum speed is 25 mph or greater. Tables 9.1 and 9.2 show the data and some of the expected frequencies for the test. \begin{table}[h]

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

\multirow{2}{*}{}		Bicycle
		Road	Gravel	Electric	Total
\multirow{2}{*}{Maximum speed}	Less than 25 mph	2	21	19	42
	25 mph or greater	13	47	18	78
	Total	15	68	37	120

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

\captionsetup{labelformat=empty} \caption{Table 9.2}

\multirow{2}{*}{Expected frequency}		Bicycle
		Road	Gravel	Electric
\multirow{2}{*}{Maximum speed}	Less than 25 mph			12.95
	25 mph or greater			24.05

\end{table}

Complete the table of expected frequencies in the Printed Answer Booklet.
Determine the contribution to the chi-squared test statistic for the Electric bicycle and maximum speed 25 mph or greater. Give your answer correct to 4 decimal places. The contributions to the chi-squared test statistic for the remaining categories are shown in Table 9.3. \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Table 9.3}
\multirow{2}{*}{Contribution to the test statistic} Bicycle
Road Gravel Electric
\multirow{2}{*}{Maximum speed} Less than 25 mph 2.0119 0.3294 2.8264
25 mph or greater 1.0833 0.1774
\end{table}
In this question you must show detailed reasoning. Carry out the test at the 5\% significance level.
For each type of bicycle, give a brief interpretation of what the data suggest about maximum speed.

\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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