Questions S3 - A-Level Maths

Type of round	Mean	Standard deviation
Starter	200	15
Middle	220	25
Final	250	20

The game consists of 4 starter, 12 middle and 4 final rounds. Find the probability that

the mean time per round for the 4 final rounds will exceed 260 seconds,
all 20 rounds will be completed in a total time of 75 minutes or less,
the 12 middle rounds will take at least 3.5 times as long in total as the 4 starter rounds,
the mean time per round for the 12 middle rounds will be at least 25 seconds less than the mean time per round for the 4 final rounds.

OCR MEI S3 2016 June Q2

A genetic model involving body colour and eye colour of fruit flies predicts that offspring will consist of four phenotypes in the ratio $9 : 3 : 3 : 1$. A random sample of 200 such offspring is taken. Their phenotypes are found to be as follows.
Phenotype Brown body Red eye Brown body Brown eye Black body Red eye Black body Brown eye
Frequency 125 37 32 6
Relative proportion from model 9 3 3 1
Carry out a test, using a $2.5 \%$ level of significance, of the goodness of fit of the genetic model to these data.
The median length of European fruit flies is 2.5 mm . South American fruit flies are believed to be larger than European fruit flies. A random sample of 12 South American fruit flies is taken. The flies are found to have the following lengths (in mm).
$1.7 \quad 1.4$
$3.1 \quad 3.5$
3.8
4.2
2.2
2.9
4.4
2.6
$3.9 \quad 3.2$ Carry out a Wilcoxon signed rank test, using a $5 \%$ level of significance, to test this belief.

OCR MEI S3 2016 June Q3

3 The random variable $X$ has the following probability density function: $$\mathrm { f } ( x ) = \begin{cases} k \left( 1 - x ^ { 2 } \right) & - 1 \leqslant x \leqslant 1
0 & \text { elsewhere } \end{cases}$$ where $k$ is a positive constant.

Calculate the value of $k$.
Sketch the probability density function.
Calculate $\operatorname { Var } ( X )$.
Find a cubic equation satisfied by the upper quartile $q$, and hence verify that $q = 0.35$ to 2 decimal places.
A random sample of 40 values of $X$ is taken. Using a suitable approximating distribution, calculate the probability that the mean of these values is greater than 0.125 . Justify your choice of distribution.

OCR MEI S3 2016 June Q4

4 An insurance company is investigating a new system designed to reduce the average time taken to process claim forms. The company has decided to use 10 experienced employees to process claims using the old system and the new system. Two procedures for comparing the systems are proposed.
Procedure $A$ There are two sets of claim forms, set 1 and set 2. Each contains the same number of forms. Each employee processes set 1 on the old system and set 2 on the new system. The times taken are compared. Procedure $B$ There is just one set of claim forms which each employee processes firstly on the old system and then on the new system. The times taken are compared.

State one weakness of each of these procedures. In fact a third procedure which avoids these two weaknesses is adopted. In this procedure each employee is given a randomly selected set of claim forms. Each set contains the same number of forms. The employees each process their set of claim forms on both systems. The times taken, in minutes, are shown in the table.
Employee 1 2 3 4 5 6 7 8 9 10
Old system 40.5 42.9 52.8 51.7 77.2 66.7 65.2 49.2 55.6 58.3
New system 39.2 40.7 50.6 50.7 71.4 70.5 71.1 47.7 52.1 55.5
Carry out a paired $t$ test at the $5 \%$ level of significance to investigate whether the mean length of time taken to process a set of forms has reduced using the new system.
State fully the usual conditions for a paired $t$ test.
Construct a $99 \%$ confidence interval for the mean reduction in time taken to process a set of forms using the new system.

OCR MEI S3 2008 January Q1

The time (in milliseconds) taken by my computer to perform a particular task is modelled by the random variable $T$. The probability that it takes more than $t$ milliseconds to perform this task is given by the expression $\mathrm { P } ( T > t ) = \frac { k } { t ^ { 2 } }$ for $t \geqslant 1$, where $k$ is a constant.
1. Write down the cumulative distribution function of $T$ and hence show that $k = 1$.
2. Find the probability density function of $T$.
3. Find the mean time for the task.
For a different task, the times (in milliseconds) taken by my computer on 10 randomly chosen occasions were as follows. $$\begin{array} { c c c c c c c c c c } 6.4 & 5.9 & 5.0 & 6.2 & 6.8 & 6.0 & 5.2 & 6.5 & 5.7 & 5.3 \end{array}$$ From past experience it is thought that the median time for this task is 5.4 milliseconds. Carry out a test at the $5 \%$ level of significance to investigate this, stating your hypotheses carefully.

OCR MEI S3 2008 January Q2

2 In the vegetable section of a local supermarket, leeks are on sale either loose (and unprepared) or prepared in packs of 4 . The weights of unprepared leeks are modelled by the random variable $X$ which has the Normal distribution with mean 260 grams and standard deviation 24 grams. The prepared leeks have had $40 \%$ of their weight removed, so that their weights, $Y$, are modelled by $Y = 0.6 X$.

Find the probability that a randomly chosen unprepared leek weighs less than 300 grams.
Find the probability that a randomly chosen prepared leek weighs more than 175 grams.
Find the probability that the total weight of 4 randomly chosen prepared leeks in a pack is less than 600 grams.
What total weight of prepared leeks in a randomly chosen pack of 4 is exceeded with probability 0.975 ?
Sandie is making soup. She uses 3 unprepared leeks and 2 onions. The weights of onions are modelled by the Normal distribution with mean 150 grams and standard deviation 18 grams. Find the probability that the total weight of her ingredients is more than 1000 grams.
A large consignment of unprepared leeks is delivered to the supermarket. A random sample of 100 of them is taken. Their weights have sample mean 252.4 grams and sample standard deviation 24.6 grams. Find a $99 \%$ confidence interval for the true mean weight of the leeks in this consignment.

OCR MEI S3 2008 January Q3

3 Engineers in charge of a chemical plant need to monitor the temperature inside a reaction chamber. Past experience has shown that when functioning correctly the temperature inside the chamber can be modelled by a Normal distribution with mean $380 ^ { \circ } \mathrm { C }$. The engineers are concerned that the mean operating temperature may have fallen. They decide to test the mean using the following random sample of 12 recent temperature readings.

374.0	378.1	363.0	357.0	377.9	388.4
379.6	372.4	362.4	377.3	385.2	370.6

Give three reasons why a $t$ test would be appropriate.
Carry out the test using a $5 \%$ significance level. State your hypotheses and conclusion carefully.
Find a 95\% confidence interval for the true mean temperature in the reaction chamber.
Describe briefly one advantage and one disadvantage of having a 99\% confidence interval instead of a 95\% confidence interval.

OCR MEI S3 2008 January Q4

In Germany, towards the end of the nineteenth century, a study was undertaken into the distribution of the sexes in families of various sizes. The table shows some data about the numbers of girls in 500 families, each with 5 children. It is thought that the binomial distribution $\mathrm { B } ( 5 , p )$ should model these data.
Number of girls Number of families
0 32
1 110
2 154
3 125
4 63
5 16
1. Use this information to calculate an estimate for the mean number of girls per family of 5 children. Hence show that 0.45 can be taken as an estimate of $p$.
2. Investigate at a $5 \%$ significance level whether the binomial model with $p$ estimated as 0.45 fits the data. Comment on your findings and also on the extent to which the conditions for a binomial model are likely to be met.
A researcher wishes to select 50 families from the 500 in part (a) for further study. Suggest what sort of sample she might choose and describe how she should go about choosing it.

Edexcel S3 2021 January Q1

A journalist is going to interview a sample of 10 players from the 60 players in a local football club. The journalist uses the random numbers on page 27 of the formula booklet and starts at the top of the 10th column, where the first number is 96

The journalist worked down the 10th column to select 10 numbers. The first 3 numbers selected were: 33, 15 and 23

Find the other 7 numbers to complete the sample of ten. There are 24 girls and 36 boys who play football for the club.
The journalist labels the girls from 1 to 24 and the boys from 25 to 60
Show how the journalist can use her 10 random numbers to select a stratified sample of 10 players from the club to interview. The club provided the journalist with a list of the players in ascending order of ages, numbered 1 to 60. The journalist uses the 10 random numbers to select a simple random sample of the players.
State, giving a reason, a group of players who may not be represented in this sample.

Edexcel S3 2021 January Q2

2. A teacher believes that those of her students with strong mathematical ability may also have enhanced short-term memory. She shows a random sample of 11 students a tray of different objects for eight seconds and then asks them to write down as many of the objects as they can remember. The results, along with their percentage score in a recent mathematics test, are given in the table below.

Student	$A$	$B$	$C$	$D$	$E$	$F$	$G$	$H$	$I$	$J$	$K$
No. of objects	8	11	9	15	17	6	10	14	12	13	5
\% in maths test	30	62	57	80	75	43	65	51	48	55	32

Calculate Spearman's rank correlation coefficient for these data. Show your working clearly.
Stating your hypotheses clearly, carry out a suitable test to assess the teacher's belief. Use a $5 \%$ level of significance and state your critical value. The teacher shows these results to her class and argues that spending more time trying to improve their short-term memory would improve their mathematical ability.
Explain whether or not you agree with the teacher's argument.

Edexcel S3 2021 January Q3

3. The students in a group of schools can choose a club to join. There are 4 clubs available: Music, Art, Sports and Computers. The director collected information about the number of students in each club, using a random sample of 88 students from across the schools. The results are given in Table 1 below. \begin{table}[h]

\cline { 2 - 5 } \multicolumn{1}{c\|}{}	Music	Art	Sports	Computers
No. of students	14	28	27	19

\captionsetup{labelformat=empty} \caption{Table 1}

\end{table} The director uses a chi-squared test to determine whether or not the students are uniformly distributed across the 4 clubs.

1. Find the expected frequencies he should use. Given that the test statistic he calculated was 6.09 (to 3 significant figures)
2. use a $5 \%$ level of significance to complete the test. You should state the degrees of freedom and the critical value used. The director wishes to examine the situation in more detail and takes a second random sample of 88 students. The director assumes that within each school, students select their clubs independently. The students come from 3 schools and the distribution of the students from each school amongst the clubs is given in Table 2 below. \begin{table}[h]
  School Club Music Art Sports Computers
  School $\boldsymbol { A }$ 3 10 9 8
  School $\boldsymbol { B }$ 1 11 13 5
  School $\boldsymbol { C }$ 11 6 7 4
  \captionsetup{labelformat=empty} \caption{Table 2}
  \end{table} The director wishes to test for an association between a student's school and the club they choose.
State hypotheses suitable for such a test.
Calculate the expected frequency for School $C$ and the Computers club. The director calculates the test statistic to be 7.29 (to 3 significant figures) with 4 degrees of freedom.
Explain clearly why his test has 4 degrees of freedom.
Complete the test using a $5 \%$ level of significance and stating clearly your critical value.

Edexcel S3 2021 January Q4

4. The scores in a national test of seven-year-old children are normally distributed with a standard deviation of 18
A random sample of 25 seven-year-old children from town $A$ had a mean score of 52.4

Calculate a 98\% confidence interval for the mean score of the seven-year-old children from town $A$.
(4) An independent random sample of 30 seven-year-old children from town $B$ had a mean score of 57.8
A local newspaper claimed that the mean score of seven-year-old children from town $B$ was greater than the mean score of seven-year-old children from town $A$.
Stating your hypotheses clearly, use a $5 \%$ significance level to test the newspaper's claim. You should show your working clearly. The mean score for the national test of seven-year-old children is $\mu$. Considering the two samples of seven-year-old children separately, at the $5 \%$ level of significance, there is insufficient evidence that the mean score for town $A$ is less than $\mu$, and insufficient evidence that the mean score for town $B$ is less than $\mu$.
Find the largest possible value for $\mu$. \includegraphics[max width=\textwidth, alt={}, center]{ba3f3f9c-53d2-4e95-b2f3-3f617f1821ed-11_2255_50_314_34}

VIXV SIHIANI III IM IONOO VIAV SIHI NI JYHAM ION OO VI4V SIHI NI JLIYM ION OO

Edexcel S3 2021 January Q5

5. Chrystal is studying the lengths of pine cones that have fallen from a tree. She believes that the length, $X \mathrm {~cm}$, of the pine cones can be modelled by a normal distribution with mean 6 cm and standard deviation 0.75 cm . She collects a random sample of 80 pine cones and their lengths are recorded in the table below.

Length, $x$ cm	$x < 5$	$5 \leqslant x < 5.5$	$5.5 \leqslant x < 6$	$6 \leqslant x < 6.5$	$x \geqslant 6.5$
Frequency	6	14	24	26	10

Stating your hypotheses clearly and using a $10 \%$ level of significance, test Chrystal's belief. Show your working clearly and state the expected frequencies, the test statistic and the critical value used.
(10) Chrystal's friend David asked for more information about the lengths of the 80 pine cones. Chrystal told him that $$\sum x = 464 \quad \text { and } \quad \sum x ^ { 2 } = 2722.59$$
Calculate unbiased estimates of the mean and variance of the lengths of the pine cones. David used the calculations from part (b) to test whether or not the lengths of the pine cones are normally distributed using Chrystal's sample. His test statistic was 3.50 (to 3 significant figures) and he did not pool any classes.
Using a $10 \%$ level of significance, complete David's test stating the critical value and the degrees of freedom used.
Estimate, to 2 significant figures, the proportion of pine cones from the tree that are longer than 7 cm . \includegraphics[max width=\textwidth, alt={}, center]{ba3f3f9c-53d2-4e95-b2f3-3f617f1821ed-15_2255_50_314_34}

Edexcel S3 2021 January Q6

6. A potter makes decorative tiles in two colours, red and yellow. The length, $R \mathrm {~cm}$, of the red tiles has a normal distribution with mean 15 cm and standard deviation 1.5 cm . The length, $Y \mathrm {~cm}$, of the yellow tiles has the normal distribution $\mathrm { N } \left( 12,0.8 ^ { 2 } \right)$. The random variables $R$ and $Y$ are independent. A red tile and a yellow tile are chosen at random.

Find the probability that the yellow tile is longer than the red tile. Taruni buys 3 red tiles and 1 yellow tile.
Find the probability that the total length of the 3 red tiles is less than 4 times the length of the yellow tile. Stefan defines the random variable $X = a R + b Y$, where $a$ and $b$ are constants. He wants to use values of $a$ and $b$ such that $X$ has a mean of 780 and minimum variance.
Find the value of $a$ and the value of $b$ that Stefan should use.
\includegraphics[max width=\textwidth, alt={}, center]{ba3f3f9c-53d2-4e95-b2f3-3f617f1821ed-19_2255_50_314_34}

Edexcel S3 2022 January Q1

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

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

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

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

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

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

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

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

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

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

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.

Length, \(x\) cm	\(x < 5\)	\(5 \leqslant x < 5.5\)	\(5.5 \leqslant x < 6\)	\(6 \leqslant x < 6.5\)	\(x \geqslant 6.5\)
Frequency	6	14	24	26	10

Phenotype	Brown body Red eye	Brown body Brown eye	Black body Red eye	Black body Brown eye
Frequency	125	37	32	6
Relative proportion from model	9	3	3	1

Employee	1	2	3	4	5	6	7	8	9	10
Old system	40.5	42.9	52.8	51.7	77.2	66.7	65.2	49.2	55.6	58.3
New system	39.2	40.7	50.6	50.7	71.4	70.5	71.1	47.7	52.1	55.5

Student	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)	\(I\)	\(J\)	\(K\)
No. of objects	8	11	9	15	17	6	10	14	12	13	5
\% in maths test	30	62	57	80	75	43	65	51	48	55	32

School Club	Music	Art	Sports	Computers
School \(\boldsymbol { A }\)	3	10	9	8
School \(\boldsymbol { B }\)	1	11	13	5
School \(\boldsymbol { C }\)	11	6	7	4

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

\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

Number of girls	Number of families
0	32
1	110
2	154
3	125
4	63
5	16

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