Questions - Edexcel S3 - A-Level Maths

Direction of flight	Frequency
$0 \leqslant x < 72$	78
$72 \leqslant x < 140$	69
$140 \leqslant x < 190$	51
$190 \leqslant x < 260$	108
$260 \leqslant x < 360$	144

Kylie believes that a continuous uniform distribution over the interval [0,360] is a suitable model for the direction of flight. Stating your hypotheses clearly, use a 1\% level of significance to test Kylie's belief. Show your working clearly.

Edexcel S3 2016 June Q6

6. The random variable $W$ is defined as $$W = 3 X - 4 Y$$ where $X \sim \mathrm {~N} \left( 21,2 ^ { 2 } \right)$ and $Y \sim \mathrm {~N} \left( 8.5 , \sigma ^ { 2 } \right)$ and $X$ and $Y$ are independent.
Given that $\mathrm { P } ( W < 44 ) = 0.9$

find the value of $\sigma$, giving your answer to 2 decimal places. The random variables $A _ { 1 } , A _ { 2 }$ and $A _ { 3 }$ each have the same distribution as $A$, where $A \sim \mathrm {~N} \left( 28,5 ^ { 2 } \right)$ The random variable $B$ is defined as $$B = 2 X + \sum _ { i = 1 } ^ { 3 } A _ { i }$$ where $X , A _ { 1 } , A _ { 2 }$ and $A _ { 3 }$ are independent.
Find $\mathrm { P } ( B \leqslant 145 \mid B > 120 )$

Edexcel S3 2016 June Q7

7. A random sample of 8 apples is taken from an orchard and the weight, in grams, of each apple is measured. The results are given below. $$\begin{array} { l l l l l l l l } 143 & 131 & 165 & 122 & 137 & 155 & 148 & 151 \end{array}$$

Calculate unbiased estimates for the mean and the variance of the weights of apples. A population has an unknown mean $\mu$ and an unknown variance $\sigma ^ { 2 }$
A random sample represented by $X _ { 1 } , X _ { 2 } , X _ { 3 } , \ldots , X _ { 8 }$ is taken from this population.
Explain why $\sum _ { i = 1 } ^ { 8 } \left( X _ { i } - \mu \right) ^ { 2 }$ is not a statistic. Given that $\mathrm { E } \left( S ^ { 2 } \right) = \sigma ^ { 2 }$, where $S ^ { 2 }$ is an unbiased estimator of $\sigma ^ { 2 }$ and the statistic $$Y = \frac { 1 } { 8 } \left( \sum _ { i = 1 } ^ { 8 } X _ { i } ^ { 2 } - 8 \bar { X } ^ { 2 } \right)$$
find $\mathrm { E } ( Y )$ in terms of $\sigma ^ { 2 }$
Hence find the bias, in terms of $\sigma ^ { 2 }$, when $Y$ is used as an estimator of $\sigma ^ { 2 }$

Edexcel S3 2016 June Q8

8. A six-sided die is labelled with the numbers $1,2,3,4,5$ and 6 A group of 50 students want to test whether or not the die is fair for the number six.
The 50 students each roll the die 30 times and record the number of sixes they each obtain.
Given that $\bar { X }$ denotes the mean number of sixes obtained by the 50 students, and using $$\mathrm { H } _ { 0 } : p = \frac { 1 } { 6 } \text { and } \mathrm { H } _ { 1 } : p \neq \frac { 1 } { 6 }$$ where $p$ is the probability of rolling a 6 ,

use the Central Limit Theorem to find an approximate distribution for $\bar { X }$, if $\mathrm { H } _ { 0 }$ is true.
Hence find, in terms of $\bar { X }$, the critical region for this test. Use a $5 \%$ level of significance.

Edexcel S3 2017 June Q1

The ages, in years, of a random sample of 8 parrots are shown in the table below.

Parrot	$A$	$B$	$C$	$D$	$E$	$F$	$G$	$H$
Age	10	4	13	15	2	1	8	6

A parrot breeder does not know the ages of these 8 parrots. She examines each of these 8 parrots and is asked to put them in order of decreasing age. She puts them in the order $$\begin{array} { l l l l l l l l } D & G & H & C & A & B & F & E \end{array}$$

Find, to 3 decimal places, Spearman's rank correlation coefficient between the breeder's order and the actual order.
(5)
Use your value of Spearman's rank correlation coefficient to test for evidence of the breeder's ability to order parrots correctly, by their age, after examining them. Use a $1 \%$ significance level and state your hypotheses clearly.

Edexcel S3 2017 June Q2

2. A school uses online report cards to promote both hard work and good behaviour of its pupils. Each card details a pupil's recent achievement and contains exactly one of three inspirational messages $A , B$ or $C$, chosen by the pupil's teacher. The headteacher believes that there is an association between the pupil's gender and the inspirational message chosen. He takes a random sample of 225 pupils and examines the card for each pupil. His results are shown in Table 1. \begin{table}[h]

\cline { 2 - 5 } \multicolumn{2}{c|}{}

Inspirational message

\multirow{2}{*}{Total}

\cline { 3 - 5 } \multicolumn{2}{c|}{}

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

\end{table} Stating your hypotheses clearly, test, at the $10 \%$ level of significance, whether or not there is evidence to support the headteacher's belief. Show your working clearly. You should state your expected frequencies correct to 2 decimal places.

Edexcel S3 2017 June Q3

3. The manager of a gym claimed that the mean age of its customers is 30 years. A random sample of 75 customers is taken and their ages have a mean of 28.2 years and a standard deviation, $s$, of 8.5 years.

Stating your hypotheses clearly and using a 10\% level of significance, test whether or not the manager's claim is supported by the data.
Explain the relevance of the Central Limit Theorem to your calculation in part (a).
State an additional assumption needed to carry out the test in part (a).

Edexcel S3 2017 June Q4

4. The number of emergency plumbing calls received per day by a local council was recorded over a period of 80 days. The results are summarised in the table below.

Number of calls, $\boldsymbol { x }$	0	1	2	3	4	5	6	7	8
Frequency	3	13	14	15	10	8	8	6	3

Show that the mean number of emergency plumbing calls received per day is 3.5 A council officer suggests that a Poisson distribution can be used to model the number of emergency plumbing calls received per day. He uses the mean from the sample above and calculates the expected frequencies shown in the table below.
$\boldsymbol { x }$ 0 1 2 3 4 5 6 7
8 or
more
Expected
frequency
2.42 8.46 14.80 $r$ 15.10 10.57 6.17 3.08 $s$
Calculate the value of $r$ and the value of $s$, giving your answers correct to 2 decimal places.
Test, at the $5 \%$ level of significance, whether or not the Poisson distribution is a suitable model for the number of emergency plumbing calls received per day. State your hypotheses clearly.

Edexcel S3 2017 June Q5

5. A dance studio has 800 dancers of which \begin{displayquote} 452 are beginners
251 are intermediates
97 are professionals

Explain in detail how a stratified sample of size 50 could be taken.
State an advantage of stratified sampling rather than simple random sampling in this situation. \end{displayquote} Independent random samples of 80 beginners and 60 intermediates are chosen. Each of these dancers is given an assessment score, $x$, based on the quality of their dancing. The results are summarised in the table below.
$\bar { x }$ $s ^ { 2 }$ $n$
Beginners 31.7 57.3 80
Intermediates 36.9 38.1 60
The studio manager believes that the mean score of intermediates is more than 3 points greater than the mean score of beginners.
Stating your hypotheses clearly and using a $5 \%$ level of significance, test whether or not these data support the studio manager's belief.

Edexcel S3 2017 June Q6

6. A company produces a certain type of mug. The masses of these mugs are normally distributed with mean $\mu$ and standard deviation 1.2 grams. A random sample of 5 mugs is taken and the mass, in grams, of each mug is measured. The results are given below. \section*{$\begin{array} { l l l l l } 229.1 & 229.6 & 230.9 & 231.2 & 231.7 \end{array}$}

Find a $95 \%$ confidence interval for $\mu$, giving your limits correct to 1 decimal place. Sonia plans to take 20 random samples, each of 5 mugs. A 95\% confidence interval for $\mu$ is to be determined for each sample.
Find the probability that more than 3 of these intervals will not contain $\mu$.

Edexcel S3 2017 June Q7

7. The independent random variables $X$ and $Y$ are such that $$X \sim \mathrm {~N} \left( 30,4.5 ^ { 2 } \right) \text { and } Y \sim \mathrm {~N} \left( 20,3.5 ^ { 2 } \right)$$ The random variables $X _ { 1 } , X _ { 2 }$ and $X _ { 3 }$ are independent and each has the same distribution as $X$. The random variables $Y _ { 1 }$ and $Y _ { 2 }$ are independent and each has the same distribution as $Y$. Given that the random variable $A$ is defined as $$A = \frac { X _ { 1 } + X _ { 2 } + X _ { 3 } + Y _ { 1 } + Y _ { 2 } } { 5 }$$

find $\mathrm { P } ( A < 24 )$ The random variable $W$ is such that $W \sim \mathrm {~N} \left( \mu , 2.8 ^ { 2 } \right)$ Given that $\mathrm { P } ( W - X < 4 ) = 0.1$ and that $W$ and $X$ are independent,
find the value of $\mu$, giving your answer to 3 significant figures.

Edexcel S3 2017 June Q8

8. The random variable $X$ has a continuous uniform distribution over the interval $[ \alpha + 3,2 \alpha + 9 ]$ where $\alpha$ is a constant. The mean of a random sample of size $n$, taken from this distribution, is denoted by $\bar { X }$

Show that $\bar { X }$ is a biased estimator of $\alpha$
Hence find the bias, in terms of $\alpha$, when $\bar { X }$ is used as an estimator of $\alpha$ Given that $Y = \frac { 2 \bar { X } } { 3 } + k$ is an unbiased estimator of $\alpha$
find the value of the constant $k$ A random sample of 8 values of $X$ is taken and the results are as follows
4.8
5.8
6.5
7.1
8.2
9.5
9.9
10.6
Use the sample to estimate the maximum value that $X$ can take.
\includegraphics[max width=\textwidth, alt={}]{3b6aaa8a-aeac-4a44-820b-e82f317d0c85-28_2633_1828_119_121}

Edexcel S3 2018 June Q1

A random sample of 9 footballers is chosen to participate in an obstacle course. The time taken, $y$ seconds, for each footballer to complete the obstacle course is recorded, together with the footballer's Body Mass Index, $x$. The results are shown in the table below.

Footballer	Body Mass Index, $\boldsymbol { x }$	Time taken to complete the obstacle course, $y$ seconds
A	18.7	690
B	19.5	801
C	20.2	723
D	20.4	633
E	20.8	660
F	21.9	655
G	23.2	711
H	24.3	642
I	24.8	607

Russell claims, that for footballers, as Body Mass Index increases the time taken to complete the obstacle course tends to decrease.

Find, to 3 decimal places, Spearman's rank correlation coefficient between $x$ and $y$.
Use your value of Spearman's rank correlation coefficient to test Russell's claim. Use a 5\% significance level and state your hypotheses clearly. The product moment correlation coefficient for these data is - 0.5594
Use the value of the product moment correlation coefficient to test for evidence of a negative correlation between Body Mass Index and the time taken to complete the obstacle course. Use a 5\% significance level.
Using your conclusions to part (b) and part (c), describe the relationship between Body Mass Index and the time taken to complete the obstacle course.

Edexcel S3 2018 June Q2

A random sample of 75 packets of seeds is selected from a production line. Each packet contains 12 seeds. The seeds are planted and the number of seeds that germinate from each packet is recorded. The results are as follows.

Number of seeds that

germinate from each packet

6 or

fewer

Number of packets

Show that the probability of a randomly selected seed from this sample germinating is 0.82 A gardener suggests that a binomial distribution can be used to model the number of seeds that germinate from a packet of 12 seeds. She uses a binomial distribution with the estimated probability 0.82 of a seed germinating. Some of the calculated expected frequencies are shown in the table below.
Number of seeds that
germinate from each packet
6 or
fewer
7 8 9 10 11 12
Expected frequency $s$ 2.80 7.97 $r$ 22.04 18.26 6.93
Calculate the value of $r$ and the value of $s$, giving your answers correct to 2 decimal places.
Test, at the $10 \%$ level of significance, whether or not these data suggest that the binomial distribution is a suitable model for the number of seeds that germinate from a packet of 12 seeds. State your hypotheses clearly and show your working.

Edexcel S3 2018 June Q3

3. Star Farm produces duck eggs. Xander takes a random sample of 20 duck eggs from Star Farm and their widths, $x \mathrm {~cm}$, are recorded. Xander's results are summarised as follows. $$\sum x = 92.0 \quad \sum x ^ { 2 } = 433.4974$$

Calculate unbiased estimates of the mean and the variance of the width of duck eggs produced by Star Farm. Yinka takes an independent random sample of 30 duck eggs from Star Farm and their widths, $y \mathrm {~cm}$, are recorded. Yinka's results are summarised as follows. $$\sum y = 142.5 \quad \sum y ^ { 2 } = 689.5078$$
Treating the combined sample of 50 duck eggs as a single sample, estimate the standard error of the mean.
(5) Research shows that the population of duck egg widths is normally distributed with standard deviation 0.71 cm . The farmer claims that the mean width of duck eggs produced by Star Farm is greater than 4.5 cm .
Using your combined mean, test, at the $5 \%$ level of significance, the farmer's claim. State your hypotheses clearly.

Edexcel S3 2018 June Q4

4. A company selects a random sample of five of its warehouses. The table below summarises the number of employees, in thousands, at each warehouse and the number of reported first aid incidents at each warehouse during 2017

Warehouse	A	$B$	C	D	E
Number of employees, (in thousands)	2	1	3.8	3	2.2
Number of reported first aid incidents	15	10	40	26	23

The personnel manager claims that the mean number of reported first aid incidents per 1000 employees is the same at each of the company's warehouses.

Stating your hypotheses clearly, use a $5 \%$ level of significance to test the manager's claim. Jean, the safety officer at warehouse $C$, kept a record of each reported first aid incident at warehouse $C$ in 2017. Jean wishes to select a systematic sample of 10 records from warehouse $C$.
Explain, in detail, how Jean should obtain such a sample.

Edexcel S3 2018 June Q5

5. A factory produces steel sheets whose weights, $X \mathrm {~kg}$, have a normal distribution with an unknown mean $\mu \mathrm { kg }$ and known standard deviation $\sigma \mathrm { kg }$. A random sample of 25 sheets gave both a

$95 \%$ confidence interval for $\mu$ of $( 30.612,31.788 )$
$c \%$ confidence interval for $\mu$ of $( 30.66,31.74 )$
1. Find the value of $\sigma$
2. Find the value of $c$, giving your answer correct to 3 significant figures.

Edexcel S3 2018 June Q6

6. The continuous random variable $Y$ is uniformly distributed over the interval $$[ a - 3 , a + 6 ]$$ where $a$ is a constant. A random sample of 60 observations of $Y$ is taken.
Given that $\bar { Y } = \frac { \sum _ { i = 1 } ^ { 60 } Y _ { i } } { 60 }$

use the Central Limit Theorem to find an approximate distribution for $\bar { Y }$ Given that the 60 observations of $Y$ have a sample mean of 13.4
find a $98 \%$ confidence interval for the maximum value that $Y$ can take.

Edexcel S3 2018 June Q7

7．（i）As part of a recruitment exercise candidates are required to complete three separate tasks．The times taken，$A , B$ and $C$ ，in minutes，for candidates to complete the three tasks are such that $$A \sim \mathrm {~N} \left( 21,2 ^ { 2 } \right) , B \sim \mathrm {~N} \left( 32,7 ^ { 2 } \right) \text { and } C \sim \mathrm {~N} \left( 45,9 ^ { 2 } \right)$$ The time taken by an individual candidate to complete each task is assumed to be independent of the time taken to complete each of the other tasks． A candidate is selected at random．
（a）Find the probability that the candidate takes a total time of more than 90 minutes to complete all three tasks．
（b）Find $\mathrm { P } ( A > B )$
（ii）A simple random sample，$X _ { 1 } , X _ { 2 } , X _ { 3 } , X _ { 4 }$ ，is taken from a normal population with mean $\mu$ and standard deviation $\sigma$ Given that $$\bar { X } = \frac { X _ { 1 } + X _ { 2 } + X _ { 3 } + X _ { 4 } } { 4 }$$ and that $$\mathrm { P } \left( X _ { 1 } > \bar { X } + k \sigma \right) = 0.1$$ where $k$ is a constant，
find the value of $k$ ，giving your answer correct to 3 significant figures．

END

Edexcel S3 2021 June Q1

A plant biologist claims that as the percentage moisture content of the soil in a field increases, so does the percentage plant coverage. He splits the field into equal areas labelled $A , B , C , D$ and $E$ and measures the percentage plant coverage and the percentage moisture content for each area. The results are shown in the table below.

\cline { 2 - 6 } \multicolumn{1}{c\|}{}	$A$	$B$	$C$	$D$	$E$
Coverage \%	10	12	25	0	6
Moisture \%	30	20	40	10	25

Calculate Spearman's rank correlation coefficient for these data.
Stating your hypotheses clearly, test at the $5 \%$ level of significance, whether or not these data provide support for the plant biologist's claim.

Edexcel S3 2021 June Q2

A doctor believes that the diet of her patients and their health are not independent.

She takes a random sample of 200 patients and records whether they are in good health or poor health and whether they have a good diet or a poor diet. The results are summarised in the table below.

\cline { 2 - 3 } \multicolumn{1}{c\|}{}	Good health	Poor health
Good diet	86	8
Poor diet	91	15

Stating your hypotheses clearly, test the doctor's belief using a $5 \%$ level of significance. Show your working for your test statistic and state your critical value clearly.

Edexcel S3 2021 June Q3

Components are manufactured such that their length in mm is normally distributed with mean $\mu$ and variance $\sigma ^ { 2 }$. Below is a 95\% confidence interval for $\mu$ calculated from a random sample of components.
(11.52, 13.75)

Using the same random sample,

find a $90 \%$ confidence interval for $\mu$. Four 90\% confidence intervals are found from independent random samples.
Calculate the probability that only 3 of these 4 intervals will contain $\mu$.

Edexcel S3 2021 June Q4

A college runs academic and vocational courses. The college has 1680 academic students and 2520 vocational students.
1. Describe how a stratified sample of 70 students at the college could be taken.
All students at the college take a basic skills test. A random sample of 50 academic students has a mean score of 57 and a variance of 60. An independent random sample of 80 vocational students has a mean score of 62 with a variance of 70
Stating your hypotheses clearly, test at the $5 \%$ level of significance, whether or not the mean basic skills score for vocational students is greater than the mean basic skills score for academic students.
Explain the importance of the Central Limit Theorem to the test in part (b).
State an assumption that is required to carry out the test in part (b). All the academic students at the college take a basic skills course. Another random sample of 50 academic students and another independent random sample of 80 vocational students retake the basic skills test. The hypotheses used in part (b) are then tested again at the same level of significance. The value of the test statistic $z$ is now 1.54
Comment on the mean basic skills scores of academic and vocational students after taking this course.
Considering the outcomes of the tests in part (b) and part (e), comment on the effectiveness of the basic skills course.

Edexcel S3 2021 June Q5

A researcher is looking into the effectiveness of a new medicine for the relief of symptoms. He collects random samples of 8 people who are taking the medicine from each of 50 different medical practices. The number of people who say that the medicine is a success, in each sample, is recorded. The results are summarised in the table below.

Number of successes	0	1	2	3	4	5	6	7	8
Number of practices	4	6	3	12	10	7	4	2	2

The researcher decides to model this data using a binomial distribution.

State two necessary assumptions that the researcher made in order to use this model.
Show that the mean number of successes per sample is 3.54 He decides to use this mean to calculate expected frequencies. The results are shown in the table below.
Number of successes 0 1 2 3 4 5 6 7 8
Expected frequency 0.47 2.96 8.23 13.07 $f$ 8.23 3.27 0.74 $g$
Calculate the value of $f$ and the value of $g$. Give your answers to 2 decimal places.
Stating your hypotheses clearly, test at the $10 \%$ level of significance, whether or not the binomial distribution is a suitable model for the number of successes in samples of 8 people.

Edexcel S3 2021 June Q6

A baker produces bread buns and bread rolls. The weights of buns, $B$ grams, and the weights of rolls, $R$ grams, are such that $B \sim \mathrm {~N} \left( 55,1.3 ^ { 2 } \right)$ and $R \sim \mathrm {~N} \left( 51,1.2 ^ { 2 } \right)$

A bun and a roll are selected at random.

Find the probability that the bun weighs less than $110 \%$ of the weight of the roll. Two buns are chosen at random.
Find the probability that their weights differ by more than 1 gram. The baker sells bread in bags. Each bag contains either 10 buns or 11 rolls. The weight of an empty bag, $S$ grams, is such that $S \sim \mathrm {~N} \left( 3,0.2 ^ { 2 } \right)$
Find the probability that a bag of buns weighs less than a bag of rolls.

Parrot	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)
Age	10	4	13	15	2	1	8	6

Direction of flight	Frequency
\(0 \leqslant x < 72\)	78
\(72 \leqslant x < 140\)	69
\(140 \leqslant x < 190\)	51
\(190 \leqslant x < 260\)	108
\(260 \leqslant x < 360\)	144

Number of calls, \(\boldsymbol { x }\)	0	1	2	3	4	5	6	7	8
Frequency	3	13	14	15	10	8	8	6	3

	\(\bar { x }\)	\(s ^ { 2 }\)	\(n\)
Beginners	31.7	57.3	80
Intermediates	36.9	38.1	60

Footballer	Body Mass Index, \(\boldsymbol { x }\)	Time taken to complete the obstacle course, \(y\) seconds
A	18.7	690
B	19.5	801
C	20.2	723
D	20.4	633
E	20.8	660
F	21.9	655
G	23.2	711
H	24.3	642
I	24.8	607

Number of successes	0	1	2	3	4	5	6	7	8
Expected frequency	0.47	2.96	8.23	13.07	\(f\)	8.23	3.27	0.74	\(g\)

Questions — Edexcel S3 (313 questions)

Browse by module

Browse by board

Edexcel S3 2016 June Q5

Edexcel S3 2016 June Q6

Edexcel S3 2016 June Q7

Edexcel S3 2016 June Q8

Edexcel S3 2017 June Q1

Edexcel S3 2017 June Q2

Edexcel S3 2017 June Q3

Edexcel S3 2017 June Q4

Edexcel S3 2017 June Q5

Edexcel S3 2017 June Q6

Edexcel S3 2017 June Q7

Edexcel S3 2017 June Q8

Edexcel S3 2018 June Q1

Edexcel S3 2018 June Q2

Edexcel S3 2018 June Q3

Edexcel S3 2018 June Q4

Edexcel S3 2018 June Q5

Edexcel S3 2018 June Q6

Edexcel S3 2018 June Q7

Edexcel S3 2021 June Q1

Edexcel S3 2021 June Q2

Edexcel S3 2021 June Q3

Edexcel S3 2021 June Q4

Edexcel S3 2021 June Q5

Edexcel S3 2021 June Q6