Questions - Edexcel S3 - A-Level Maths

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

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

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

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

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

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).

Edexcel S3 2024 January Q6

A random sample of 8 three-month-old golden retriever dogs is taken.

The heights of the golden retrievers are recorded.
Using this sample, a 95\% confidence interval for the mean height, in cm, of three-month-old golden retrievers is found to be $( 45.72,53.88 )$

Find a 99\% confidence interval for the mean height. You may assume that the heights are normally distributed with known population standard deviation. Some summary statistics for the weights, $x \mathrm {~kg}$, of this sample are given below. $$\sum x = 91.2 \quad \sum x ^ { 2 } = 1145.16 \quad n = 8$$
Calculate unbiased estimates of the mean and the variance of the weights of three-month-old golden retrievers. A further random sample of 24 three-month-old golden retrievers is taken. The unbiased estimates of the mean and the variance of the weights, in kg , from this sample are found to be 10.8 and 17.64 respectively.
Estimate the standard error of the mean weight for the combined sample of 32 three-month-old golden retrievers.

Edexcel S3 2024 January Q7

Small containers and large containers are independently filled with fruit juice.

The amounts of fruit juice in small containers are normally distributed with mean 180 ml and standard deviation 4.5 ml The amounts of fruit juice in large containers are normally distributed with mean 330 ml and standard deviation 6.7 ml The random variable $W$ represents the total amount of fruit juice in a random sample of 2 small containers minus the amount of fruit juice in 1 randomly selected large container.
$W \sim \mathrm {~N} ( a , b )$ where $a$ and $b$ are positive constants.

Find the value of $a$ and the value of $b$
Find the probability that a randomly chosen large container of fruit juice contains more than 1.8 times the amount of fruit juice in a randomly chosen small container. A random sample of 3 small containers of fruit juice is taken.
Find the probability that the first container of fruit juice in this sample contains at least 5 ml more than the mean amount of fruit juice in all 3 small containers.

Edexcel S3 2014 June Q1

A tennis club's committee wishes to select a sample of 50 members to fill in a questionnaire about the club's facilities. The 300 members, of whom 180 are males, are listed in alphabetical order and numbered $1 - 300$ in the club’s membership book.

The club's committee decides to use a random number table to obtain its sample.
The first three lines of the random number table used are given below.

319	952	241	343	278	811	394	165	008	413	063	179	749
722	962	334	461	267	114	806	992	414	837	837	657	339
470	684	554	127	067	459	142	920	144	575	311	605	412

Starting with the top left-hand corner (319) and working across, the committee selects 50 random numbers. The first 2 suitable numbers are 241 and 278. Numbers greater than 300 are ignored.

Find the next two suitable numbers. When the club's committee looks at the members corresponding to their random numbers they find that only 1 female has been selected.
The committee does not want to be accused of being biased towards males so considers using a systematic sample instead.
1. Explain clearly how the committee could take a systematic sample.
2. Explain why a systematic sample may not give a sample that represents the proportion of males and females in the club. The committee decides to use a stratified sample instead.
Describe how to choose members for the stratified sample.
Explain an advantage of using a stratified sample rather than a quota sample.

Edexcel S3 2014 June Q2

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

Show that $\bar { X }$ is a biased estimator of $\alpha$, and state the bias. Given that $Y = k \bar { X }$ is an unbiased estimator for $\alpha$
find the value of $k$. A random sample of 10 values of $X$ is taken and the results are as follows $$\begin{array} { l l l l l l l l l l } 3 & 5 & 8 & 12 & 4 & 13 & 10 & 8 & 5 & 12 \end{array}$$
Hence estimate the maximum value of $X$

Edexcel S3 2014 June Q3

3. A grocer believes that the average weight of a grapefruit from farm $A$ is greater than the average weight of a grapefruit from farm $B$. The weights, in grams, of 80 grapefruit selected at random from farm $A$ have a mean value of 532 g and a standard deviation, $s _ { A }$, of 35 g . A random sample of 100 grapefruit from farm $B$ have a mean weight of 520 g and a standard deviation, $s _ { B }$, of 28 g . Stating your hypotheses clearly and using a 1\% level of significance, test whether or not the grocer's belief is supported by the data.

Edexcel S3 2014 June Q4

4. In a survey 10 randomly selected men had their systolic blood pressure, $x$, and weight, $w$, measured. Their results are as follows

Man	$\boldsymbol { A }$	$\boldsymbol { B }$	$\boldsymbol { C }$	$\boldsymbol { D }$	$\boldsymbol { E }$	$\boldsymbol { F }$	$\boldsymbol { G }$	$\boldsymbol { H }$	$\boldsymbol { I }$	$\boldsymbol { J }$
$x$	123	128	137	143	149	153	154	159	162	168
$w$	78	93	85	83	75	98	88	87	95	99

Calculate the value of Spearman's rank correlation coefficient between $x$ and $w$.
Stating your hypotheses clearly, test at the $5 \%$ level of significance, whether or not there is evidence of a positive correlation between systolic blood pressure and weight. The product moment correlation coefficient for these data is 0.5114
Use the value of the product moment correlation coefficient to test, at the $5 \%$ level of significance, whether or not there is evidence of a positive correlation between systolic blood pressure and weight.
Using your conclusions to part (b) and part (c), describe the relationship between systolic blood pressure and weight.

Edexcel S3 2014 June Q5

A random sample of 200 people were asked which hot drink they preferred from tea, coffee and hot chocolate. The results are given below.

\cline { 3 - 6 } \multicolumn{2}{|c|}{}

	\multirow{2}{*}{Total}
\cline { 3 - 5 } \multicolumn{2}{\|c\|}{}	Tea	Coffee	Hot Chocolate
\multirow{2}{*}{Gender}	Males	57	26	11	94
\cline { 2 - 6 }	Females	42	47	17	106
Total		99	73	28	200

Test, at the $5 \%$ significance level, whether or not there is an association between type of drink preferred and gender. State your hypotheses and show your working clearly. You should state your expected frequencies to 2 decimal places.
State what difference using a $0.5 \%$ significance level would make to your conclusion. Give a reason for your answer.

Edexcel S3 2014 June Q6

6. Eight tasks were given to each of 125 randomly selected job applicants. The number of tasks failed by each applicant is recorded. The results are as follows

Number of tasks failed by an applicant	0	1	2	3	4	5	6 or more
Frequency	2	21	45	42	12	3	0

Show that the probability of a randomly selected task, from this sample, being failed is 0.3 An employer believes that a binomial distribution might provide a good model for the number of tasks, out of 8, that an applicant fails. He uses a binomial distribution, with the estimated probability 0.3 of a task being failed. The calculated expected frequencies are as follows
Number of tasks failed by an applicant 0 1 2 3 4 5 6 or more
Expected frequency 7.21 24.71 37.06 $r$ 17.02 5.83 $s$
Find the value of $r$ and the value of $s$ giving your answers to 2 decimal places.
Test, at the $5 \%$ level of significance, whether or not a binomial distribution is a suitable model for these data. State your hypotheses and show your working clearly. The employer believes that all applicants have the same probability of failing each task.
Use your result from part(c) to comment on this belief.

Edexcel S3 2014 June Q7

7. The random variable $X$ is defined as $$X = 4 Y - 3 W$$ where $Y \sim \mathrm {~N} \left( 40,3 ^ { 2 } \right) , W \sim \mathrm {~N} \left( 50,2 ^ { 2 } \right)$ and $Y$ and $W$ are independent.

Find $\mathrm { P } ( X > 25 )$ The random variables $Y _ { 1 } , Y _ { 2 }$ and $Y _ { 3 }$ are independent and each has the same distribution as $Y$. The random variable $A$ is defined as $$A = \sum _ { i = 1 } ^ { 3 } Y _ { i }$$ The random variable $C$ is such that $C \sim \mathrm {~N} \left( 115 , \sigma ^ { 2 } \right)$ Given that $\mathrm { P } ( A - C < 0 ) = 0.2$ and that $A$ and $C$ are independent,
find the variance of $C$.

Edexcel S3 2015 June Q1

The names of the 720 members of a swimming club are listed alphabetically in the club's membership book. The chairman of the swimming club wishes to select a systematic sample of 40 names. The names are numbered from 001 to 720 and a number between 001 and $w$ is selected at random. The corresponding name and every $x$ th name thereafter are included in the sample.
1. Find the value of $w$.
2. Find the value of $x$.
3. Write down the probability that the sample includes both the first name and the second name in the club's membership book.
4. State one advantage and one disadvantage of systematic sampling in this case.
5. Nine dancers, Adilzhan $( A )$, Bianca $( B )$, Chantelle $( C )$, Lee $( L )$, Nikki $( N )$, Ranjit $( R )$, Sergei $( S )$, Thuy $( T )$ and Yana $( Y )$, perform in a dancing competition.
Two judges rank each dancer according to how well they perform. The table below shows the rankings of each judge starting from the dancer with the strongest performance.
Rank 1 2 3 4 5 6 7 8 9
Judge 1 $S$ $N$ $B$ $C$ $T$ $A$ $Y$ $R$ $L$
Judge 2 $S$ $T$ $N$ $B$ $C$ $Y$ $L$ $A$ $R$
Calculate Spearman's rank correlation coefficient for these data.
Stating your hypotheses clearly, test at the $1 \%$ level of significance, whether or not the two judges are generally in agreement.

Edexcel S3 2015 June Q3

The number of accidents on a particular stretch of motorway was recorded each day for 200 consecutive days. The results are summarised in the following table.

Number of accidents	0	1	2	3	4	5
Frequency	47	57	46	35	9	6

Show that the mean number of accidents per day for these data is 1.6 A motorway supervisor believes that the number of accidents per day on this stretch of motorway can be modelled by a Poisson distribution. She uses the mean found in part (a) to calculate the expected frequencies for this model. Her results are given in the following table.
Number of accidents 0 1 2 3 4 5 or more
Frequency 40.38 64.61 $r$ 27.57 11.03 $s$
Find the value of $r$ and the value of $s$, giving your answers to 2 decimal places.
Stating your hypotheses clearly, use a $10 \%$ level of significance to test the motorway supervisor's belief. Show your working clearly.

Edexcel S3 2015 June Q4

A farm produces potatoes. The potatoes are packed into sacks.

The weight of a sack of potatoes is modelled by a normal distribution with mean 25.6 kg and standard deviation 0.24 kg

Find the probability that two randomly chosen sacks of potatoes differ in weight by more than 0.5 kg
(6) Sacks of potatoes are randomly selected and packed onto pallets.
The weight of an empty pallet is modelled by a normal distribution with mean 20.0 kg and standard deviation 0.32 kg Each full pallet of potatoes holds 30 sacks of potatoes.
Find the probability that the total weight of a randomly chosen full pallet of potatoes is greater than 785 kg

Edexcel S3 2015 June Q5

A Head of Department at a large university believes that gender is independent of the grade obtained by students on a Business Foundation course. A random sample was taken of 200 male students and 160 female students who had studied the course.

The results are summarised below.

\cline { 3 - 4 } \multicolumn{2}{c\|}{}	Male	Female
\multirow{3}{*}{Grade}	Distinction	$18.5 \%$	$27.5 \%$
\cline { 2 - 4 }	Merit	$63.5 \%$	$60.0 \%$
\cline { 2 - 4 }	Unsatisfactory	$18.0 \%$	$12.5 \%$

Stating your hypotheses clearly, test the Head of Department's belief using a 5\% level of significance. Show your working clearly.

Edexcel S3 2015 June Q6

As part of an investigation, a random sample was taken of 50 footballers who had completed an obstacle course in the early morning. The time taken by each of these footballers to complete the obstacle course, $x$ minutes, was recorded and the results are summarised by

$$\sum x = 1570 \quad \text { and } \quad \sum x ^ { 2 } = 49467.58$$

Find unbiased estimates for the mean and variance of the time taken by footballers to complete the obstacle course in the early morning. An independent random sample was taken of 50 footballers who had completed the same obstacle course in the late afternoon. The time taken by each of these footballers to complete the obstacle course, $y$ minutes, was recorded and the results are summarised as $$\bar { y } = 30.9 \quad \text { and } \quad s _ { y } ^ { 2 } = 3.03$$
Test, at the $5 \%$ level of significance, whether or not the mean time taken by footballers to complete the obstacle course in the early morning, is greater than the mean time taken by footballers to complete the obstacle course in the late afternoon. State your hypotheses clearly.
Explain the relevance of the Central Limit Theorem to the test in part (b).
State an assumption you have made in carrying out the test in part (b).

Edexcel S3 2015 June Q7

A fair six-sided die is labelled with the numbers $1,2,3,4,5$ and 6 The die is rolled 40 times and the score, $S$, for each roll is recorded.
1. Find the mean and the variance of $S$.
2. Find an approximation for the probability that the mean of the 40 scores is less than 3 (3)
3. A factory produces steel sheets whose weights $X \mathrm {~kg}$, are such that $X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)$
A random sample of these sheets is taken and a $95 \%$ confidence interval for $\mu$ is found to be (29.74, 31.86)
Find, to 2 decimal places, the standard error of the mean.
Hence, or otherwise, find a $90 \%$ confidence interval for $\mu$ based on the same sample of sheets. Using four different random samples, four $90 \%$ confidence intervals for $\mu$ are to be found.
Calculate the probability that at least 3 of these intervals will contain $\mu$.

Edexcel S3 2016 June Q1

The table below shows the distance travelled by car and the amount of commission earned by each of 8 salespersons in 2015

Salesperson	Distance travelled (in 1000's of km)	Commission earned (in \\(1000's)
A	20.4	17.7
B	22.2	24.1
C	29.9	20.3
D	37.8	28.3
E	25.5	34.9
\)F$	30.2	29.3
G	35.3	23.6
H	16.5	26.8

Find Spearman's rank correlation coefficient for these data.
Stating your hypotheses clearly, test, at the $5 \%$ level of significance, whether or not there is evidence of a positive correlation between the distance travelled by car and the amount of commission earned.

Edexcel S3 2016 June Q2

2. A researcher investigates the results of candidates who took their driving test at one of three driving test centres. A random sample of 620 candidates gave the following results.

\multirow{2}{*}{}		Driving test centre			\multirow{2}{*}{Total}
		$\boldsymbol { A }$	B	C
\multirow{2}{*}{Result}	Pass	99	110	68	277
	Fail	108	116	119	343
Total		207	226	187	620

Test, at the $5 \%$ level of significance, whether there is an association between the results of candidates' driving tests and the driving test centre. State your hypotheses and show your working clearly. You should state your expected frequencies correct to 2 decimal places. The researcher decides to conduct a further investigation into the results of candidates' driving tests.
State which driving test centre you would recommend for further investigation. Give a reason for your answer.

Edexcel S3 2016 June Q3

3. A company wants to survey its employees' attitudes to work. The company's workforce is located at three offices. The number of employees at each location is summarised in the table below.

Office location	Number of employees
Bristol	856
Dudley	429
Glasgow	1215

Each employee is located at only one office. A personnel assistant plans to survey the first 50 employees who arrive for work at the Bristol office on a Monday morning.

Give two reasons why this survey is likely to lead to a biased response. A personnel manager has access to the company's information system that holds details of each employee including their place of work. The manager decides to take a stratified sample of 150 employees.
Describe how to choose employees for this stratified sample.
Explain an advantage of using a stratified sample rather than a quota sample.

Edexcel S3 2016 June Q4

4. A random sample of 60 children and a random sample of 50 adults were taken and each person was given the same task to complete. The table below summarises the times taken, $t$ seconds, to complete the task.

	Mean, $\overline { \boldsymbol { t } }$	Standard deviation, $\boldsymbol { s }$	$\boldsymbol { n }$
Children	61.2	5.9	60
Adults	59.1	5.2	50

Stating your hypotheses clearly, test, at the $5 \%$ level of significance, whether or not there is evidence that the mean time taken to complete the task by children is greater than the mean time taken by adults.
(6)
Explain the relevance of the Central Limit Theorem to your calculation in part (a).
State an assumption you have made to carry out the test in part (a).

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

319	952	241	343	278	811	394	165	008	413	063	179	749
722	962	334	461	267	114	806	992	414	837	837	657	339
470	684	554	127	067	459	142	920	144	575	311	605	412

Man	\(\boldsymbol { A }\)	\(\boldsymbol { B }\)	\(\boldsymbol { C }\)	\(\boldsymbol { D }\)	\(\boldsymbol { E }\)	\(\boldsymbol { F }\)	\(\boldsymbol { G }\)	\(\boldsymbol { H }\)	\(\boldsymbol { I }\)	\(\boldsymbol { J }\)
\(x\)	123	128	137	143	149	153	154	159	162	168
\(w\)	78	93	85	83	75	98	88	87	95	99

City	London	Edinburgh	Cardiff
Number of employees	430	250	120

319	952	241	343	278	811	394	165	008	413	063	179	749
722	962	334	461	267	114	806	992	414	837	837	657	339
470	684	554	127	067	459	142	920	144	575	311	605	412

Questions — Edexcel S3 (313 questions)

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319	952	241	343	278	811	394	165	008	413	063	179	749
722	962	334	461	267	114	806	992	414	837	837	657	339
470	684	554	127	067	459	142	920	144	575	311	605	412