Questions - Edexcel S3 - A-Level Maths

	$n$	$\bar { x }$	$\sum x ^ { 2 }$
Type $A$ oranges	40	140.4	790258
Type $B$ oranges	32	134.7	581430

Calculate unbiased estimates for the variance of the weights of the population of type $A$ oranges and the variance of the weights of the population of type $B$ oranges.
Test, at the $5 \%$ level of significance, the greengrocer's belief. You should state the hypotheses and the critical value used for this test.
Explain how you have used the fact that the sample sizes are large in your answer to part (b).

Edexcel S3 2020 October Q6

6. The number of toasters sold by a shop each week may be modelled by a Poisson distribution with mean 4 A random sample of 35 weeks is taken and the mean number of toasters sold per week is found.

Write down the approximate distribution for the mean number of toasters sold per week from a random sample of 35 weeks. The number of kettles sold by the shop each week may be modelled by a Poisson distribution with mean $\lambda$ A random sample of 40 weeks is taken and the mean number of kettles sold per week is found. The width of the $99 \%$ confidence interval for $\lambda$ is 2.6
Find an estimate for $\lambda$ A second, independent random sample of 40 weeks is taken and a second $99 \%$ confidence interval for $\lambda$ is found.
Find the probability that only one of these two confidence intervals contains $\lambda$

Edexcel S3 2020 October Q7

7. A company makes cricket balls and tennis balls. The weights of cricket balls, $C$ grams, follow a normal distribution $$C \sim \mathrm {~N} \left( 160,1.25 ^ { 2 } \right)$$ Three cricket balls are selected at random.

Find the probability that their total weight is more than 475.8 grams. The weights of tennis balls, $T$ grams, follow a normal distribution $$T \sim \mathrm {~N} \left( 60,2 ^ { 2 } \right)$$ Five tennis balls and two cricket balls are selected at random.
Find the probability that the total weight of the five tennis balls and the two cricket balls is more than 625 grams. A random sample of $n$ tennis balls $T _ { 1 } , T _ { 2 } , \ldots , T _ { n }$ is taken.
The random variable $Y = ( n - 1 ) T _ { 1 } - \sum _ { r = 2 } ^ { n } T _ { r }$
Given that $\mathrm { P } ( Y > 40 ) = 0.0838$ correct to 4 decimal places,
find $n$.

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Edexcel S3 2021 October Q1

A machine makes screws with a mean length of 30 mm and a standard deviation of 2.5 mm .

A manager claims that, following some repairs, the machine is now making screws with a mean length of less than 30 mm . The manager takes a random sample of 80 screws and finds that they have a mean length of 29.5 mm . Use a suitable test, at the $5 \%$ level of significance, to determine whether there is evidence to support the manager's claim. State your hypotheses clearly.

Edexcel S3 2021 October Q2

2. Andy has some apple trees. Over many years she has graded each apple from her trees as $A , B , C , D$ or $E$ according to the quality of the apple, with $A$ being the highest quality and $E$ being the lowest quality. She knows that the proportion of apples in each grade produced by her trees is as follows.

Grade	$A$	$B$	$C$	$D$	$E$
Proportion	$4 \%$	$28 \%$	$52 \%$	$10 \%$	$6 \%$

Raj advises Andy to add potassium to the soil around her apple trees. Andy believes that adding potassium will not affect the distribution of grades for the quality of the apples. To test her belief Andy adds potassium to the soil around her apple trees. The following year she counts the number of apples in each grade. The number of apples in each grade is shown in the table below.

Grade	$A$	$B$	$C$	$D$	$E$
Frequency	9	71	136	21	3

Test Andy's belief using a $5 \%$ level of significance. Show your working clearly, stating your hypotheses, expected frequencies and degrees of freedom. 2 continued

Edexcel S3 2021 October Q3

3. A cafe owner wishes to know whether the price of strawberry jam is related to the taste of the jam. He finds a website that lists the price per 100 grams and a mark for the taste, out of 100, awarded by a judge, for 9 different strawberry jams $A , B , C , D , E , F , G , H$ and $I$. He then ranks the marks for taste and the prices. The ranks are shown in the table below.

Rank	1	2	3	4	5	6	7	8	9
Price	$A$	$B$	$E$	$C$	$D$	$F$	$G$	$H$	$I$
Taste	$A$	$B$	$F$	$E$	$H$	$G$	$I$	$C$	$D$

Calculate Spearman's rank correlation coefficient for these data.
Test, at the $5 \%$ level of significance, whether or not there is a relationship between the price and the taste of these strawberry jams. State your hypotheses clearly. A friend suggests that it would be better to use the price per 100 grams, $c$, and the mark for the taste, $m$, for each strawberry jam rather than rank them. Given that $$\mathrm { S } _ { c c } = 2.0455 \quad \mathrm {~S} _ { m m } = 243.5556 \quad \mathrm {~S} _ { c m } = 16.4943$$
calculate the product moment correlation coefficient between the price and the mark for taste of these strawberry jams, giving your answer correct to 3 decimal places.
Use your 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 the price and the mark for taste of these 9 strawberry jams. State your hypotheses clearly.
State which of the tests in parts (b) and (d) is more appropriate for the cafe owner to use. Give a reason for your answer.

Edexcel S3 2021 October Q4

A local village radio station, LSB, decides to survey adults in its broadcasting area about the programmes it produces.
$L S B$ broadcasts to 4 villages $\mathrm { A } , \mathrm { B } , \mathrm { C }$ and D .
The number of households in each of the villages is given below.

Village	Number of households
A	41
B	164
C	123
D	82

LSB decides to take a stratified sample of 200 households.

Explain how to select the households for this stratified sample.
(3) One of the questions in the survey related to the age group of each member of the household and whether they listen to $L S B$. The data received are shown below.
\multirow{2}{*}{} Age group
18-49 50-69 Older than 69
Listen to LSB 130 162 65
Do not listen to LSB 78 98 62
The data are to be used to determine whether or not there is an association between the age group and whether they listen to $L S B$.
Calculate the expected frequencies for the age group 50-69 that
1. listen to $L S B$
2. do not listen to $L S B$
  (2) Given that for the other 4 classes $\sum \frac { ( O - E ) ^ { 2 } } { E } = 4.657$ to 3 decimal places,
test at the $5 \%$ level of significance, whether or not there is evidence of an association between age and listening to $L S B$. Show your working clearly, stating the degrees of freedom and the critical value.

Edexcel S3 2021 October Q5

Assam produces bags of flour. The stated weight printed on the bags of flour is 3 kg . The weights of the bags of flour are normally distributed with standard deviation 0.015 kg .

Assam weighs a random sample of 9 bags of flour and finds their mean weight is 2.977 kg .

Calculate the $99 \%$ confidence interval for the mean weight of a bag of flour. Give your limits to 3 decimal places. Assam decides to increase the amount of flour put into the bags.
Explain why the confidence interval has led Assam to take this action. After the increase a random sample of $n$ bags of flour is taken. The sample mean weight of these $n$ bags is 2.995 kg . A $95 \%$ confidence interval for $\mu$ gave a lower limit of less than 2.991 kg .
Find the maximum value of $n$.
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Edexcel S3 2021 October Q6

6. Amala believes that the resting heart rate is lower in men who exercise regularly compared to men who do not exercise regularly. She measures the resting heart rate, $h$, of a random sample of 50 men who exercise regularly and a random sample of 40 men who do not exercise regularly. Her results are summarised in the table below.

\cline { 2 - 6 } \multicolumn{1}{c|}{}

Sample

size

$\sum \boldsymbol { h }$

$\sum \boldsymbol { h } ^ { 2 }$

Calculate the value of $\alpha$ and the value of $\beta$
Test, at the $5 \%$ level of significance, whether there is evidence to support Amala's belief. State your hypotheses clearly.
Explain the significance of the central limit theorem to the test in part (b).
State two assumptions you have made in carrying out the test in part (b).

Edexcel S3 2021 October Q7

A company produces bricks.

The weight of a brick, $B \mathrm {~kg}$, is such that $B \sim \mathrm {~N} \left( 1.96 , \sqrt { 0.003 } ^ { 2 } \right)$
Two bricks are chosen at random.

Find the probability that the difference in weight of the 2 bricks is greater than 0.1 kg A random sample of $n$ bricks is to be taken.
Find the minimum sample size such that the probability of the sample mean being greater than 2 is less than 1\% The bricks are randomly selected and stacked on pallets.
The weight of an empty pallet, $E \mathrm {~kg}$, is such that $E \sim \mathrm {~N} \left( 21.8 , \sqrt { 0.6 } ^ { 2 } \right)$
The random variable $M$ represents the total weight of a pallet stacked with 500 bricks. The random variable $T$ represents the total weight of a container of cement.
Given that $T$ is independent of $M$ and that $T \sim \mathrm {~N} \left( 774 , \sqrt { 1.8 } ^ { 2 } \right)$
calculate $\mathrm { P } ( 4 T > 100 + 3 M )$

Edexcel S3 2018 Specimen 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.

Edexcel S3 2018 Specimen Q2

2. 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.

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.
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.

Edexcel S3 2018 Specimen Q3

3. 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 2018 Specimen Q4

4. 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 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

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Edexcel S3 2018 Specimen 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 2018 Specimen Q6

6. 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 2018 Specimen Q7

A fair six-sided die is labelled with the numbers $1,2,3,4,5$ and 6
(b) Find an approximation for the probability that the mean of the 40 scores is less than 3
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Edexcel S3 2018 Specimen Q8

8. 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 $\mu$. \section*{8. A factory produces steel sheets whose weights $X \mathrm { gg }$, are such $X \sim N ( \mu , \sigma ) ^ { 2 }$} A. A. 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. (3)
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Edexcel S3 Specimen Q1

A report states that employees spend, on average, 80 minutes every working day on personal use of the Internet. A company takes a random sample of 100 employees and finds their mean personal Internet use is 83 minutes with a standard deviation of 15 minutes. The company's managing director claims that his employees spend more time on average on personal use of the Internet than the report states.

Test, at the $5 \%$ level of significance, the managing director's claim. State your hypotheses clearly.

Edexcel S3 Specimen Q2

2. Philip and James are racing car drivers. Philip's lap times, in seconds, are normally distributed with mean 90 and variance 9. James' lap times, in seconds, are normally distributed with mean 91 and variance 12. The lap times of Philip and James are independent. Before a race, they each take a qualifying lap.

Find the probability that James' time for the qualifying lap is less than Philip's. The race is made up of 60 laps. Assuming that they both start from the same starting line and lap times are independent,
find the probability that Philip beats James in the race by more than 2 minutes.

Edexcel S3 Specimen Q3

3. A woodwork teacher measures the width, $w \mathrm {~mm}$, of a board. The measured width, $X \mathrm {~mm}$, is normally distributed with mean $w \mathrm {~mm}$ and standard deviation 0.5 mm .

Find the probability that $X$ is within 0.6 mm of $w$. The same board is measured 16 times and the results are recorded.
Find the probability that the mean of these results is within 0.3 mm of $w$. Given that the mean of these 16 measurements is 35.6 mm ,
find a 98\% confidence interval for $w$.

Edexcel S3 Specimen Q4

A researcher claims that, at a river bend, the water gradually gets deeper as the distance from the inner bank increases. He measures the distance from the inner bank, $b \mathrm {~cm}$, and the depth of a river, $s \mathrm {~cm}$, at seven positions. The results are shown in the table below.

Position	A	B	$C$	D	$E$	$F$	G
Distance from inner bank $b \mathrm {~cm}$	100	200	300	400	500	600	700
Depth $s \mathrm {~cm}$	60	75	85	76	110	120	104

Calculate Spearman's rank correlation coefficient between $b$ and $s$.
Stating your hypotheses clearly, test whether or not the data provides support for the researcher's claim. Use a $1 \%$ level of significance.

Edexcel S3 Specimen Q5

5. A random sample of 100 people were asked if their finances were worse, the same or better than this time last year. The sample was split according to their annual income and the results are shown in the table below.

\backslashbox{Annual income}{Finances}	Worse	Same	Better
Under £15 000	14	11	9
£15000 and above	17	20	29

Test, at the $5 \%$ level of significance, whether or not the relative state of their finances is independent of their income range. State your hypotheses and show your working clearly.
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Edexcel S3 Specimen Q6

6. A total of 228 items are collected from an archaeological site. The distance from the centre of the site is recorded for each item. The results are summarised in the table below.

Distance from the

centre of the site (m)

Test, at the $5 \%$ level of significance, whether or not the data can be modelled by a continuous uniform distribution. State your hypotheses clearly.

Edexcel S3 Specimen Q7

A large company surveyed its staff to investigate the awareness of company policy. The company employs 6000 full-time staff and 4000 part-time staff.
1. Describe how a stratified sample of 200 staff could be taken.
2. Explain an advantage of using a stratified sample rather than a simple random sample.
A random sample of 80 full-time staff and an independent random sample of 80 part-time staff were given a test of policy awareness. The results are summarised in the table below.
Mean score $( \bar { x } )$
Variance of
scores $\left( s ^ { 2 } \right)$
Full-time staff 52 21
Part-time staff 50 19
Stating your hypotheses clearly, test, at the $1 \%$ level of significance, whether or not the mean policy awareness scores for full-time and part-time staff are different.
Explain the significance of the Central Limit Theorem to the test in part (c).
State an assumption you have made in carrying out the test in part (c). After all the staff had completed a training course the 80 full-time staff and the 80 part-time staff were given another test of policy awareness. The value of the test statistic $z$ was 2.53
Comment on the awareness of company policy for the full-time and part-time staff in light of this result. Use a $1 \%$ level of significance.
Interpret your answers to part (c) and part (f).

Grade	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)
Proportion	\(4 \%\)	\(28 \%\)	\(52 \%\)	\(10 \%\)	\(6 \%\)

Rank	1	2	3	4	5	6	7	8	9
Price	\(A\)	\(B\)	\(E\)	\(C\)	\(D\)	\(F\)	\(G\)	\(H\)	\(I\)
Taste	\(A\)	\(B\)	\(F\)	\(E\)	\(H\)	\(G\)	\(I\)	\(C\)	\(D\)

Rank	1	2	3	4	5	6	7	8	9
Judge 1	\(S\)	\(N\)	\(B\)	\(C\)	\(T\)	\(A\)	\(Y\)	\(R\)	\(L\)
Judge 2	\(S\)	\(T\)	\(N\)	\(B\)	\(C\)	\(Y\)	\(L\)	\(A\)	\(R\)

	\(n\)	\(\bar { x }\)	\(\sum x ^ { 2 }\)
Type \(A\) oranges	40	140.4	790258
Type \(B\) oranges	32	134.7	581430

\multirow{2}{*}{}	Age group
	18-49	50-69	Older than 69
Listen to LSB	130	162	65
Do not listen to LSB	78	98	62

Number of accidents	0	1	2	3	4	5 or more
Frequency	40.38	64.61	\(r\)	27.57	11.03	\(s\)

Questions — Edexcel S3 (313 questions)

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