Questions S3 - A-Level Maths

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Edexcel S3 2022 January Q1

8 marks Easy -1.2

The Headteacher of a school is thinking about making changes to the school day. She wants to take a sample of 60 students so that she can find out what the students think about the proposed changes.

The names of the 1200 students of the school are listed alphabetically.

Explain how the Headteacher could take a systematic sample of 60 students.
1. Explain why systematic sampling is likely to be quicker than simple random sampling in this situation.
2. With reference to this situation,
  - explain why systematic sampling may introduce bias compared to simple random sampling
give an example of the bias that may occur when using this alphabetical list
Explain why this suggests that a stratified sample of size 60 may be better than the systematic sample taken by the Headteacher.

Edexcel S3 2022 January Q2

8 marks Standard +0.3

2. Krishi owns a farm on which he keeps chickens. He selects, at random, 10 of the eggs produced and weighs each of them.
You may assume that these weights are a random sample from a normal distribution with standard deviation 1.9 g The total weight of these 10 eggs is 537.2 g

Find a $95 \%$ confidence interval for the mean weight of the eggs produced by Krishi's chickens. Krishi was hoping to obtain a $99 \%$ confidence interval of width at most 1.5 g
Calculate the minimum sample size necessary to achieve this. \includegraphics[max width=\textwidth, alt={}, center]{fc43aabf-ad04-4852-8539-981cef608f31-04_2662_95_107_1962}

Edexcel S3 2022 January Q3

8 marks Standard +0.3

3. The table shows the time, in seconds, of the fastest qualifying lap for 10 different Formula One racing drivers and their finishing position in the actual race.

Calculate the value of Spearman's rank correlation coefficient for these data.
Stating your hypotheses clearly, test at the $1 \%$ level of significance, whether or not there is evidence of a positive correlation between the fastest qualifying lap time and finishing position for these Formula One racing drivers.

Edexcel S3 2022 January Q4

10 marks Standard +0.3

4. A manager at a large estate agency believes that the type of property affects the time taken to sell it. A random sample of 125 properties sold is shown in the table.

\multirow{2}{*}{}	Type of property
	Bungalow	Flat	House	Total
Sold within three months	7	29	46	82
Sold in more than three months	9	19	15	43
Total	16	48	61	125

Test, at the $5 \%$ level of significance, whether there is evidence for an association between the type of property and the time taken to sell it. You should state your hypotheses, expected frequencies, test statistic and the critical value used for this test.

Edexcel S3 2022 January Q5

15 marks Standard +0.3

A dog breeder claims that the mean weight of male Great Dane dogs is 20 kg more than the mean weight of female Great Dane dogs.

Tammy believes that the mean weight of male Great Dane dogs is more than 20 kg more than the mean weight of female Great Dane dogs. She takes random samples of 50 male and 50 female Great Dane dogs and records their weights. The results are summarised below, where $x$ denotes the weight, in kg , of a male Great Dane dog and $y$ denotes the weight, in kg, of a female Great Dane dog. $$\sum x = 3610 \quad \sum x ^ { 2 } = 260955.6 \quad \sum y = 2585 \quad \sum y ^ { 2 } = 133757.2$$

Find unbiased estimates for the mean and variance of the weights of
1. the male Great Dane dogs,
2. the female Great Dane dogs.
Stating your hypotheses clearly, carry out a suitable test to assess Tammy's belief. Use a $5 \%$ level of significance and state your critical value.
For the test in part (b), state whether or not it is necessary to assume that the weights of the Great Dane dogs are normally distributed. Give a reason for your answer.
State an assumption you have made in carrying out the test in part (b).

Edexcel S3 2022 January Q6

12 marks Standard +0.8

The number of emails per hour received by a helpdesk were recorded. The results for a random sample of 80 one-hour periods are shown in the table.

Number of emails per hour	0	1	2	3	4	5	6
Frequencies	1	10	23	15	19	9	3

Show that the mean number of emails per hour in the sample is 3 The manager believes that the number of emails per hour received could be modelled by a Poisson distribution. The following table shows some of the expected frequencies.
Number of emails per hour Expected Frequencies
0 $r$
1 11.949
2 17.923
3 17.923
4 13.443
5 $s$
$\geqslant 6$ $t$
Find the values of $r , s$ and $t$, giving your answers to 3 decimal places.
Using a 10\% significance level, test whether or not a Poisson model is reasonable. You should clearly state your hypotheses, test statistic and the critical value used.

Edexcel S3 2022 January Q7

14 marks Standard +0.8

A market stall sells vegetables. Two of the vegetables sold are broccoli heads and cabbages.

The weights of these broccoli heads, $B$ kilograms, follow a normal distribution $$B \sim \mathrm {~N} \left( 0.588,0.084 ^ { 2 } \right)$$ The weights of these cabbages, $C$ kilograms, follow a normal distribution $$C \sim \mathrm {~N} \left( 0.908,0.039 ^ { 2 } \right)$$

Find the probability that the total weight of two randomly chosen broccoli heads is less than the weight of a randomly chosen cabbage. Broccoli heads cost $\pounds 2.50$ per kg and cabbages cost $\pounds 3.00$ per kg. Jaymini buys 1 broccoli head and 2 cabbages, chosen randomly.
Find the probability that she pays more than £7 The market stall offers a discount for buying 5 or more broccoli heads. The price with the discount is $\pounds w$ per kg. Let $\pounds D$ be the price with the discount of 5 broccoli heads.
Find, in terms of $w$, the mean and standard deviation of $D$ Given that $\mathrm { P } ( D < 6 ) < 0.1$
find the smallest possible value of $w$, giving your answer to 2 decimal places.

Edexcel S3 2022 January Q1

10 marks Moderate -0.3

The weights, $x \mathrm {~kg}$, of each of 10 watermelons selected at random from Priya's shop were recorded. The results are summarised as follows

$$\sum x = 114.2 \quad \sum x ^ { 2 } = 1310.464$$

Calculate unbiased estimates of the mean and the variance of the weights of the watermelons in Priya's shop. Priya researches the weight of watermelons, for the variety she has in her shop, and discovers that the weights of these watermelons are normally distributed with a standard deviation of 0.8 kg
Calculate a $95 \%$ confidence interval for the mean weight of watermelons in Priya's shop. Give the limits of your confidence interval to 2 decimal places. Priya claims that the confidence interval in part (b) suggests that nearly all of the watermelons in her shop weigh more than 10.5 kg
Use your answer to part (b) to estimate the smallest proportion of watermelons in her shop that weigh less than 10.5 kg

Edexcel S3 2022 January Q2

8 marks Standard +0.3

Secondary schools in a region conduct ability testing at the start of Year 7 and the start of Year 8. Each year a regional education officer randomly selects 240 Year 7 students and 240 Year 8 students from across the region. The results for last year are summarised in the table below.

\cline { 2 - 3 } \multicolumn{1}{c\|}{}	Mean score	Variance of scores
Year 7	101	38
Year 8	103	42

The regional education officer claims that there is no difference between the mean scores of these two year groups.

Test the regional education officer's claim at the $1 \%$ significance level. You should state your hypotheses, test statistic and critical value clearly.
Explain the significance of the Central Limit Theorem in part (a).

Edexcel S3 2022 January Q3

12 marks Moderate -0.3

A medical research team carried out an investigation into the metabolic rate, MR, of men aged between 30 years and 60 years.

A random sample of 10 men was taken from this age group.
The table below shows for each man his MR and his body mass index, BMI. The table also shows the rank for the level of daily physical activity, DPA, which was assessed by the medical research team. Rank 1 was assigned to the man with the highest level of daily physical activity.

Man	$A$	$B$	$C$	$D$	$E$	$F$	$G$	$H$	$I$	$J$
MR ( $\boldsymbol { x }$ )	6.24	5.94	6.83	6.53	6.31	7.44	7.32	8.70	7.88	7.78
BMI ( $\boldsymbol { y }$ )	19.6	19.2	23.6	21.4	20.2	20.8	22.9	25.5	23.3	25.1
DPA rank	10	7	9	8	6	3	1	4	5	2

$$\text { [You may use } \quad \mathrm { S } _ { x y } = 15.1608 \quad \mathrm {~S} _ { x x } = 6.90181 \quad \mathrm {~S} _ { y y } = 45.304 \text { ] }$$

Calculate the value of the product moment correlation coefficient between MR and BMI for these 10 men.
Use your value of the product moment correlation coefficient to test, at the 5\% significance level, whether or not there is evidence of a positive correlation between MR and BMI.
State your hypotheses clearly.
State an assumption that must be made to carry out the test in part (b).
Calculate the value of Spearman's rank correlation coefficient between MR and DPA for these 10 men.
Use a two-tailed test and a $5 \%$ level of significance to assess whether or not there is evidence of a correlation between MR and DPA.

Edexcel S3 2022 January Q4

14 marks Standard +0.3

A survey was carried out with students that had studied Maths, Physics and Chemistry at a college between 2016 and 2020. The students were divided into two groups $A$ and $B$.
1. Explain how a sample could be obtained from this population using quota sampling.
The students were asked which of the three subjects they enjoyed the most. The results of the survey are shown in the table.
\multirow{2}{*}{} Subject enjoyed the most
Maths Physics Chemistry Total
Group A 16 10 13 39
Group B 38 13 10 61
Total 54 23 23 100
Test, at the $5 \%$ level of significance, whether the subject enjoyed the most is independent of group. You should state your hypotheses, expected frequencies, test statistic and the critical value used for this test. The Headteacher discovered later that the results were actually based on a random sample of 200 students but had been recorded in the table as percentages.
For the test in part (b), state with reasons the effect, if any, that this information would have on
1. the null and alternative hypotheses,
2. the critical value,
3. the value of the test statistic,
4. the conclusion of the test.

Edexcel S3 2022 January Q5

17 marks Standard +0.3

Charlie is training for three events: a 1500 m swim, a 40 km bike ride and a 10 km run.

From past experience his times, in minutes, for each of the three events independently have the following distributions. $$\begin{aligned} & S \sim \mathrm {~N} \left( 41,5.2 ^ { 2 } \right) \text { represents the time for the swim } \\ & B \sim \mathrm {~N} \left( 81,4.2 ^ { 2 } \right) \text { represents the time for the bike ride } \\ & R \sim \mathrm {~N} \left( 57,6.6 ^ { 2 } \right) \text { represents the time for the run } \end{aligned}$$

Find the probability that Charlie's total time for a randomly selected swim, bike ride and run exceeds 3 hours.
Find the probability that the time for a randomly selected swim will be at least 20 minutes quicker than the time for a randomly selected run. Given that $\mathrm { P } ( S + B + R > t ) = 0.95$
find the value of $t$ A triathlon consists of a 1500 m swim, immediately followed by a 40 km bike ride, immediately followed by a 10 km run. Charlie uses the answer to part (a) to find the probability that, in 6 successive independent triathlons, his time will exceed 3 hours on at least one occasion.
Find the answer Charlie should obtain. Jane says that Charlie should not have used the answer to part (a) for the calculation in part (d).
Explain whether or not Jane is correct.

Edexcel S3 2022 January Q6

14 marks Standard +0.3

A farmer sells strawberries in baskets. The contents of each of 100 randomly selected baskets were weighed and the results, given to the nearest gram, are shown below.

Weight of strawberries (grams)	Number of baskets
302-303	5
304-305	13
306-307	10
308-309	18
310-311	25
312-313	20
314-315	5
316-317	4

The farmer proposes that the weight of strawberries per basket, in grams, should be modelled by a normal distribution with a mean of 310 g and standard deviation 4 g . Using his model, the farmer obtains the following expected frequencies.

Weight of strawberries (s, grams)	Expected frequency
$s \leqslant 303.5$	$a$
$303.5 < s \leqslant 305.5$	7.8
$305.5 < s \leqslant 307.5$	13.6
$307.5 < s \leqslant 309.5$	18.4
$309.5 < s \leqslant 311.5$	19.6
$311.5 < s \leqslant 313.5$	16.3
$313.5 < s \leqslant 315.5$	10.6
$s > 315.5$	$b$

Find the value of $a$ and the value of $b$. Give your answers correct to one decimal place. Before $s \leqslant 303.5$ and $s > 315.5$ are included, for the remaining cells, $$\sum \frac { ( O - E ) ^ { 2 } } { E } = 9.71$$
Using a 5\% significance level, test whether the data are consistent with the model. You should state your hypotheses, the test statistic and the critical value used. An alternative model uses estimates for the population mean and standard deviation from the data given. Using these estimated values no expected frequency is below 5
Another test is to be carried out, using a $5 \%$ significance level, to assess whether the data are consistent with this alternative model.
State the effect, if any, on the critical value for this test. Give a reason for your answer.

Edexcel S3 2023 January Q1

12 marks Standard +0.3

1 A machine fills bottles with mineral water.
The machine is checked every day to ensure that it is working correctly. On a particular day a random sample of 100 bottles is taken. The volume of water, $x$ millilitres, for each bottle is measured and each measurement is coded using $$y = x - 1000$$ The results are summarised below $$\sum y = 847 \quad \sum y ^ { 2 } = 13510.09$$

1. Show that the value of the unbiased estimate of the mean of $x$ is 1008.47
2. Calculate the unbiased estimate of the variance of $x$ The machine was initially set so that the volume of water in a bottle had a mean value of 1010 millilitres. Later, a test at the $5 \%$ significance level is used to determine whether or not the mean volume of water in a bottle has changed. If it has changed then the machine is stopped and reset.
Write down suitable null and alternative hypotheses for a 2-tailed test.
Find the critical region for $\bar { X }$ in the above test.
Using your answer to part (a) and your critical region found in part (c), comment on whether or not the machine needs to be stopped and reset.
Give a reason for your answer.
Explain why the use of $\sigma ^ { 2 } = s ^ { 2 }$ is reasonable in this situation.

Edexcel S3 2023 January Q2

12 marks Standard +0.3

2 The table shows the season's best times, $x$ seconds, for the 8 athletes who took part in the 200 m final in the 2021 Tokyo Olympics. It also shows their finishing position in the race.

Athlete	$A$	$B$	$C$	$D$	$E$	$F$	$G$	$H$
Season's best time	19.89	19.83	19.74	19.84	19.91	19.99	20.13	20.10
Finishing position	1	2	3	4	5	6	7	8

Given that the fastest season's best time is ranked number 1

calculate the value of the Spearman's rank correlation coefficient for these data.
Stating your hypotheses clearly, test, at the $1 \%$ level of significance, whether or not there is evidence of a positive correlation between the rank of the season's best time and the finishing position for these athletes. Chris suggests that it would be better to use the actual finishing time, $y$ seconds, of these athletes rather than their finishing position. Given that $$S _ { x x } = 0.1286875 \quad S _ { y y } = 0.55275 \quad S _ { x y } = 0.225175$$
calculate the product moment correlation coefficient between the season's best time and the finishing time for these athletes.
Give your answer correct to 3 decimal places.
Use your value of the product moment correlation coefficient to test, at the $1 \%$ level of significance, whether or not there is evidence of a positive correlation between the season's best time and the finishing time for these athletes.

Edexcel S3 2023 January Q3

9 marks Moderate -0.8

3 A mobile phone company offers an insurance policy to its customers when they purchase a mobile phone. The company conducted a survey on the age of the customers and whether or not claims were made. A random sample of 1200 customers from this company was investigated for 2020 and the results are shown in the table below.

		Claim made in 2020	No claim made in 2020	Total
\multirow{3}{*}{Age}	17-20 years	24	176	200
	21-50 years	48	652	700
	51 years and over	14	286	300
	Total	86	1114	1200

The data are to be used to determine whether or not making a claim is independent of age.

Calculate the expected frequencies for the age group 51 years and over that
1. made a claim in 2020
2. did not make a claim in 2020 The 4 classes of customers aged between 17 and 50 give a value of $\sum \frac { ( O - E ) ^ { 2 } } { E } = 7.123$ correct to 3 decimal places.
Test, at the $1 \%$ level of significance, whether or not making a claim is independent of age. Show your working clearly, stating your hypotheses, the degrees of freedom, the test statistic and the critical value used.

Edexcel S3 2023 January Q4

14 marks Standard +0.3

4 A research student is investigating the number of children who are girls in families with 4 children. The table below shows her results for 200 such families.

Number of girls	0	1	2	3	4
Frequency	15	68	69	38	10

The research student suggests that a binomial distribution with $p = \frac { 1 } { 2 }$ could be a suitable model for the number of children who are girls in a family of 4 children.

Using her results and a $5 \%$ significance level, test the research student's claim. You should state your hypotheses, expected frequencies, test statistic and the critical value used. The research student decides to refine the model and retains the idea of using a binomial distribution but does not specify the probability that the child is a girl.
Use the data in the table to show that the probability that a child is a girl is 0.45 The research student uses the probability from part (b) to calculate a new set of expected frequencies, none of which are less than 5
The statistic $\sum \frac { ( O - E ) ^ { 2 } } { E }$ is evaluated and found to be 2.47
Test, at the $5 \%$ significance level, whether using a binomial distribution is suitable to model the number of children who are girls in a family of 4 children. You should state your hypotheses and the critical value used.

Edexcel S3 2023 January Q5

10 marks Standard +0.3

5 Claire grows strawberries on her farm. She wants to compare two brands of fertiliser, brand $A$ and brand $B$. She grows two sets of plants of the same variety of strawberries under the same conditions, fertilising one set with brand $A$ and the other with brand $B$. The yields per plant, in grams, from each set of plants are summarised below.

	Mean	Standard deviation	Number of plants
Fertiliser A	1377	17.8	50
Fertiliser B	1368	18.4	40

Stating your hypotheses clearly, carry out a suitable test to assess whether the mean yield from plants using fertiliser $A$ is greater than the mean yield from plants using fertiliser $B$.
Use a 1\% level of significance and state your test statistic and critical value. The total cost of fertiliser $A$ for Claire's 50 plants was $\pounds 75$ The total cost of fertiliser $B$ for Claire's 40 plants was $\pounds 50$ Claire sells all her strawberries at $\pounds 3$ per kilogram.
Use this information, together with your answer in part (a), to advise Claire on which of the two brands of fertiliser she should use next year in order to maximise her expected profit per plant, giving a reason for your answer.

Edexcel S3 2023 January Q6

10 marks Moderate -0.3

6 A garden centre sells bags of stones and large bags of gravel.
The weight, $X$ kilograms, of stones in a bag can be modelled by a normal distribution with unknown mean $\mu$ and known standard deviation 0.4 The stones in each of a random sample of 36 bags from a large batch is weighed. The total weight of stones in these 36 bags is found to be 806.4 kg

Find a 98\% confidence interval for the mean weight of stones in the batch.
Explain why the use of the Central Limit theorem is not required to answer part (a) The manufacturer of these bags of stones claims that bags in this batch have a mean weight of 22.5 kg
Using your answer to part (a), comment on the claim made by the manufacturer. The weight, $Y$ kilograms, of gravel in a large bag can be modelled by a normal distribution with mean 850 kg and standard deviation 5 kg A builder purchases 10 large bags of gravel.
Find the probability that the mean weight of gravel in the 10 large bags is less than 848 kg

Edexcel S3 2023 January Q7

8 marks Standard +0.3

7 At a particular supermarket, the times taken to serve each customer in a queue at a standard checkout may be modelled by a normal distribution with mean 240 seconds and standard deviation 20 seconds. There is a queue of 3 customers at a standard checkout.
Making a reasonable assumption about the times taken to serve these customers,

find the probability that the total time taken to serve the 3 customers will be less than 11 minutes.
State the assumption you have made in part (a) In the supermarket there is also an express checkout, which is reserved for customers buying 10 or fewer items. The time taken to serve a customer at this express checkout may be modelled by a normal distribution with mean 100 seconds and standard deviation 8 seconds. On a particular day Jiang has 8 items to pay for and has to choose whether to join a queue of 3 customers waiting at a standard checkout or a queue of 7 customers waiting at the express checkout. Using a similar assumption to that made in part (a),
find the probability that the total time taken to serve the 3 customers at the standard checkout will exceed the total time taken to serve the 7 customers at the express checkout.

Edexcel S3 2024 January Q1

8 marks Standard +0.3

Chen is treating vines to prevent fungus appearing. One month after the treatment, Chen monitors the vines to see if fungus is present.

The contingency table shows information about the type of treatment for a sample of 150 vines and whether or not fungus is present.

\multirow{2}{*}{}	Type of treatment
	None	Sulphur	Copper sulphate
No fungus present	20	55	48
Fungus present	10	8	9

Test, at the $5 \%$ level of significance, whether or not there is any association between the type of treatment and the presence of fungus.
Show your working clearly, stating your hypotheses, expected frequencies, test statistic and critical value.

Edexcel S3 2024 January Q2

7 marks Easy -1.8

A company has 800 employees.

The manager of the company is going to take a sample of 80 employees.

Explain how this sample can be taken using systematic sampling. The company has offices in London, Edinburgh and Cardiff. The table shows the number of employees in each city.
City London Edinburgh Cardiff
Number of employees 430 250 120
The president of the company is going to take a sample of 100 employees to determine the average time employees spend in front of a computer each week.
Explain how this sample can be taken using stratified sampling.
Explain an advantage of using stratified sampling rather than simple random sampling.

Edexcel S3 2024 January Q3

12 marks Standard +0.3

The table shows the annual tea consumption, $t$ (kg/person), and population, $p$ (millions), for a random sample of 7 European countries.

Country	A	B	C	D	E	F	G
Annual tea consumption, $\boldsymbol { t }$ (kg/person)	0.27	0.15	0.42	0.06	1.94	0.78	0.44
Population, $\boldsymbol { p }$ (millions)	5.4	5.8	9	10.2	67.9	17.1	8.7

$$\text { (You may use } \mathrm { S } _ { t t } = 2.486 \quad \mathrm {~S} _ { p p } = 3026.234 \quad \mathrm {~S} _ { p t } = 83.634 \text { ) }$$ Angela suggests using the product moment correlation coefficient to calculate the correlation between annual tea consumption and population.

Use Angela's suggestion to test, at the $5 \%$ level of significance, whether or not there is evidence of any correlation between annual tea consumption and population. State your hypotheses clearly and the critical value used. Johan suggests using Spearman's rank correlation coefficient to calculate the correlation between the rank of annual tea consumption and the rank of population.
Calculate Spearman's rank correlation coefficient between the rank of annual tea consumption and the rank of population.
Use Johan's suggestion to test, at the $5 \%$ level of significance, whether or not there is evidence of a positive correlation between annual tea consumption and population.
State your hypotheses clearly and the critical value used.

Edexcel S3 2024 January Q4

10 marks Standard +0.3

The number of jobs sent to a printer per hour in a small office is recorded for 120 hours. The results are summarised in the following table.

Number of jobs	0	1	2	3	4	5
Frequency	24	34	28	21	8	5

Show that the mean number of jobs sent to the printer per hour for these data is 1.75 The office manager believes that the number of jobs sent to the printer per hour can be modelled using a Poisson distribution. The office manager uses the mean given in part (a) to calculate the expected frequencies for this model. Some of the results are given in the following table.
Number of jobs 0 1 2 3 4 5 or more
Expected frequency 20.85 36.49 31.93 $r$ $s$ 3.95
Show that the value of $s$ is 8.15 to 2 decimal places.
Find the value of $r$ to 2 decimal places. The value of $\sum \frac { \left( O _ { i } - E _ { i } \right) ^ { 2 } } { E _ { i } }$ for the first four frequencies in the table is 1.43
Test, at the $5 \%$ level of significance, whether or not the number of jobs sent to the printer per hour can be modelled using a Poisson distribution. Show your working clearly, stating your hypotheses, test statistic and critical value.

Edexcel S3 2024 January Q5

9 marks Standard +0.3

A professor claims that undergraduates studying History have a typing speed of more than 15 words per minute faster than undergraduates studying Maths.

A sample is taken of 38 undergraduates studying History and 45 undergraduates studying Maths. The typing speed, $x$ words per minute, of each undergraduate is recorded. The results are summarised in the table below.

	$n$	$\bar { x }$	$s ^ { 2 }$
Undergraduates studying History	38	56.3	27.2
Undergraduates studying Maths	45	39.8	18.5

Use a suitable test, at the $5 \%$ level of significance, to investigate the professor's claim.
State clearly your hypotheses, test statistic and critical value.
State two assumptions you have made in carrying out the test in part (a).

Weight of strawberries (s, grams)	Expected frequency
\(s \leqslant 303.5\)	\(a\)
\(303.5 < s \leqslant 305.5\)	7.8
\(305.5 < s \leqslant 307.5\)	13.6
\(307.5 < s \leqslant 309.5\)	18.4
\(309.5 < s \leqslant 311.5\)	19.6
\(311.5 < s \leqslant 313.5\)	16.3
\(313.5 < s \leqslant 315.5\)	10.6
\(s > 315.5\)	\(b\)

Number of emails per hour	Expected Frequencies
0	\(r\)
1	11.949
2	17.923
3	17.923
4	13.443
5	\(s\)
\(\geqslant 6\)	\(t\)

Man	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)	\(I\)	\(J\)
MR ( \(\boldsymbol { x }\) )	6.24	5.94	6.83	6.53	6.31	7.44	7.32	8.70	7.88	7.78
BMI ( \(\boldsymbol { y }\) )	19.6	19.2	23.6	21.4	20.2	20.8	22.9	25.5	23.3	25.1
DPA rank	10	7	9	8	6	3	1	4	5	2

\multirow{2}{*}{}	Subject enjoyed the most
	Maths	Physics	Chemistry	Total
Group A	16	10	13	39
Group B	38	13	10	61
Total	54	23	23	100

Country	A	B	C	D	E	F	G
Annual tea consumption, \(\boldsymbol { t }\) (kg/person)	0.27	0.15	0.42	0.06	1.94	0.78	0.44
Population, \(\boldsymbol { p }\) (millions)	5.4	5.8	9	10.2	67.9	17.1	8.7

Number of jobs	0	1	2	3	4	5 or more
Expected frequency	20.85	36.49	31.93	\(r\)	\(s\)	3.95

	\(n\)	\(\bar { x }\)	\(s ^ { 2 }\)
Undergraduates studying History	38	56.3	27.2
Undergraduates studying Maths	45	39.8	18.5

City	London	Edinburgh	Cardiff
Number of employees	430	250	120

Questions S3 (621 questions)

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