Questions - Edexcel S3 - A-Level Maths

Edexcel S3 Q4

12 marks Standard +0.3

4. For a project a student collects data on engine size and sales over a period of time for the models of cars made by one particular manufacturer. Her results are shown in the table below.

Engine Capacity

(litres)

1.1

1.3

1.6

2.1

2.4

2.6

2.8

3.0

Sales

527

632

840

619

350

425

487

401

Calculate Spearman's rank correlation coefficient for these data.
Stating your hypotheses clearly, test at the $5 \%$ level of significance whether or not there is any evidence of correlation.
Explain why it is more appropriate to use Spearman's rank correlation coefficient for this test than the product moment correlation coefficient.
(2 marks)

Edexcel S3 Q5

12 marks Standard +0.3

5. A child is playing with a set of red and blue wooden cubes. The side length of the red cubes is normally distributed with a mean of 14.5 cm and a variance of $16.0 \mathrm {~cm} ^ { 2 }$. The side length of the blue cubes is normally distributed with a mean of 12.2 cm and a variance of $9.0 \mathrm {~cm} ^ { 2 }$.

Find the probability that a randomly chosen red cube will have a side length of more than 3 cm greater than a randomly chosen blue cube. The child makes two towers, one from 4 red cubes and one from 5 blue cubes. Assuming that the cubes for each colour of tower were chosen at random,
find the probability that the red tower is taller than the blue tower.
Explain why the assumption that the cubes for each tower were chosen at random is unlikely to be realistic.

Edexcel S3 Q6

14 marks Standard +0.3

6. A market researcher recorded the number of adverts for vehicles in each of three categories on ITV, Channel 4 and Channel 5 over a period of time. The results are shown in the table below.

	ITV	Channel 4	Channel 5
Family Saloon	69	35	28
Sports Car	20	28	18
Off-road Vehicle	12	22	8

Stating your hypotheses clearly, test at the $5 \%$ level of significance whether or not there is evidence of the proportion of adverts for each type of vehicle being dependent on the channel.
Suggest a reason for your result in part (a).

Edexcel S3 Q7

14 marks Standard +0.3

7. (a) Briefly state the central limit theorem. A student throws ten dice and records the number of sixes showing. The dice are fair, numbered 1 to 6 on the faces.
(b) Write down the distribution of the number of sixes obtained when the ten dice are thrown.
(c) Find the mean and variance of this distribution. The student throws the ten dice 100 times, recording the number of sixes showing each time.
(d) Find the probability that the mean number of sixes obtained is more than 1.8

Edexcel S3 Q1

5 marks Easy -1.8

A personnel manager has details on all company employees and wishes to consult a sample of them on a possible change to the company's hours of business. She decides to take a stratified sample based on different age groups.
1. Give one advantage of using stratified sampling in this situation.
The manager needs to select a sample of size 10 , without replacement, from a list of 65 employees aged 16 to 25 . She numbers these employees from 01 to 65 in alphabetical order and uses the table of random numbers given in the formula book. She starts with the top of the sixth two-digit column and works down. The first two numbers she writes down are 30 and 47.
Find the other eight numbers in the sample.
Suggest another factor that might be useful to consider in deciding on the strata.
(1 mark)

Edexcel S3 Q2

6 marks Standard +0.3

2. A Geography teacher is interested in the link between mathematical ability and the ability to visualise three-dimensional situations. He gives a group of 15 students a test and records each student's score, $m$, on the mathematics questions and each student's score, $v$, on the visiospatial questions. He calculates the following summary statistics: $$S _ { m m } = 3747.73 , \quad S _ { v v } = 2791.33 , \quad S _ { m v } = 2564.33$$

Calculate the product moment correlation coefficient for these data.
Stating your hypotheses clearly and using a $5 \%$ level of significance test the theory that students who are good at Mathematics tend to have better visio-spatial awareness.
(4 marks)

Edexcel S3 Q3

9 marks Standard +0.3

3. A random variable $X$ is distributed normally with a standard deviation of 6.8 Sixty observations of $X$ are made and found to have a mean of 31.4

Find a 90\% confidence interval for the mean of $X$.
How many observations of $X$ would be needed in order to obtain a $90 \%$ confidence interval for the mean of $X$ with a width of less than 1.5
(5 marks)

Edexcel S3 Q4

12 marks Standard +0.3

4. A paranormal investigator invites couples who believe they have a telepathic connection to participate in a trial. With each couple one person looks at a card with one of five shapes on it and the other person says which of the shapes they think it is. This is repeated six times and the number of correct answers recorded. The results from 120 couples are given below.

Number Correct	0	1	2	3	4	5	6
Number of Couples	26	56	28	8	2	0	0

The investigator wishes to see if this data fits a binomial distribution with parameters $n = 6$ and $p = \frac { 1 } { 5 }$ and calculates to 2 decimal places the expected frequencies given below.

Number Correct	0	1	2	3	4	5	6
Expected Frequency				9.83	1.84	0.18	0.01

Find the other expected frequencies.
Stating your hypotheses clearly, test at the $5 \%$ level of significance whether or not the distribution is an appropriate model.
Comment on your findings.

Edexcel S3 Q5

13 marks Standard +0.3

5. A Policy Unit wished to find out whether attitudes to the European Union varied with age. It conducted a survey asking 200 individuals to which of three age groups they belonged and whether they regarded themselves as generally pro-Europe or Eurosceptic. The results are shown in the table below.

\cline { 2 - 3 } \multicolumn{1}{c\|}{}	Pro-Europe	Eurosceptic
$18 - 34$ years	43	21
$35 - 54$ years	30	36
55 years or over	27	43

Stating your hypotheses clearly, test at the $5 \%$ level of significance whether attitudes to Europe are associated with age.
(11 marks)
The survey also asked people if they voted at the last election. When the above test was repeated using only the results from those who had voted a value of 4.872 was calculated for $\sum \frac { ( O - E ) ^ { 2 } } { E }$. No classes were combined.
Find if this value leads to a different result.

Edexcel S3 Q6

14 marks Challenging +1.2

6. Four swimmers, $A , B , C$ and $D$, are to be used in a $4 \times 100$ metres freestyle relay. The time for each swimmer to complete a leg follows a normal distribution. The mean and standard deviation, in seconds, of the time for each swimmer to complete a leg and the order in which they are to swim are shown in the table below.

	mean	standard deviation
$1 ^ { \text {st } }$ leg $- A$	63.1	1.2
$2 ^ { \text {nd } }$ leg $- B$	65.7	1.5
$3 ^ { \text {rd } } \operatorname { leg } - C$	65.4	1.8
$4 ^ { \text {th } }$ leg - $D$	62.5	0.9

Find the probability that the total time for first two legs is less than the total time for the last two.
(6 marks)
The total time for another team to complete this relay is normally distributed with a mean of 259.0 seconds and a standard deviation of 3.4 seconds. The two teams are to compete over four races.
Find the probability that the first team wins all four races, assuming that the team's performances are not affected by previous results.
(8 marks)

Edexcel S3 Q7

16 marks Standard +0.3

7. A telephone company believes that, for young people, the average length of a telephone call on a land line is longer than on a mobile, due to the difference in price. The company collected data on the time, $t$ minutes, of 500 calls made by young people on mobiles and the data is summarised by $$\Sigma t = 7335 , \quad \Sigma t ^ { 2 } = 172040 .$$

Calculate unbiased estimates of the mean and variance of $t$. For 200 calls made on land lines by the same young people, unbiased estimates of the mean and variance of the call length were 15.9 minutes and 108.5 minutes ${ } ^ { 2 }$ respectively.
Stating your hypotheses clearly, test at the $5 \%$ level whether or not there is evidence that longer calls are made on land lines than on mobiles.
(9 marks)
Explain the importance of the central limit theorem in carrying out the test in part (b).

Edexcel S3 Q7

Standard +0.8

7. A set of scaffolding poles come in two sizes, long and short. The length $L$ of a long pole has the normal distribution $\mathrm { N } \left( 19.7,0.5 ^ { 2 } \right)$. The length $S$ of a short pole has the normal distribution $\mathrm { N } \left( 4.9,0.2 ^ { 2 } \right)$. The random variables $L$ and $S$ are independent. A long pole and a short pole are selected at random.

Find the probability that the length of the long pole is more than 4 times the length of the short pole. Four short poles are selected at random and placed end to end in a row. The random variable $T$ represents the length of the row.
Find the distribution of $T$.
Find $\mathrm { P } ( | L - T | < 0.1 )$.
\end{table}
1. Some biologists were studying a large group of wading birds. A random sample of 36 were measured and the wing length, $x \mathrm {~mm}$ of each wading bird was recorded. The results are summarised as follows
$$\sum x = 6046 \quad \sum x ^ { 2 } = 1016338$$
Calculate unbiased estimates of the mean and the variance of the wing lengths of these birds. Given that the standard deviation of the wing lengths of this particular type of bird is actually 5.1 mm ,
find a $99 \%$ confidence interval for the mean wing length of the birds from this group.
2. Students in a mixed sixth form college are classified as taking courses in either Arts, Science or Humanities. A random sample of students from the college gave the following results \end{table}
1. A telephone directory contains 50000 names. A researcher wishes to select a systematic sample of 100 names from the directory.
2. Explain in detail how the researcher should obtain such a sample.
3. Give one advantage and one disadvantage of
  1. quota sampling,
  2. systematic sampling.
4. The heights of a random sample of 10 imported orchids are measured. The mean height of the sample is found to be 20.1 cm . The heights of the orchids are normally distributed.
Given that the population standard deviation is 0.5 cm ,
estimate limits between which $95 \%$ of the heights of the orchids lie,
find a 98\% confidence interval for the mean height of the orchids. A grower claims that the mean height of this type of orchid is 19.5 cm .
Comment on the grower's claim. Give a reason for your answer.
3. A doctor is interested in the relationship between a person's Body Mass Index (BMI) and their level of fitness. She believes that a lower BMI leads to a greater level of fitness. She randomly selects 10 female 18 year-olds and calculates each individual's BMI. The females then run a race and the doctor records their finishing positions. The results are shown in the table.
Individual $A$ $B$ $C$ $D$ $E$ $F$ $G$ $H$ $I$ $J$
BMI 17.4 21.4 18.9 24.4 19.4 20.1 22.6 18.4 25.8 28.1
Finishing position 3 5 1 9 6 4 10 2 7 8
Calculate Spearman's rank correlation coefficient for these data.
Stating your hypotheses clearly and using a one tailed test with a $5 \%$ level of significance, interpret your rank correlation coefficient.
Give a reason to support the use of the rank correlation coefficient rather than the product moment correlation coefficient with these data.
4. A sample of size 8 is to be taken from a population that is normally distributed with mean 55 and standard deviation 3. Find the probability that the sample mean will be greater than 57.
5. The number of goals scored by a football team is recorded for 100 games. The results are summarised in Table 1 below. \begin{table}[h]
Number of goals Frequency
0 40
1 33
2 14
3 8
4 5
\captionsetup{labelformat=empty} \caption{Table 1}
\end{table}
Calculate the mean number of goals scored per game. The manager claimed that the number of goals scored per match follows a Poisson distribution. He used the answer in part (a) to calculate the expected frequencies given in Table 2. \begin{table}[h]
Number of goals Expected Frequency
0 34.994
1 $r$
2 $s$
3 6.752
$\geqslant 4$ 2.221
\captionsetup{labelformat=empty} \caption{Table 2}
\end{table}
Find the value of $r$ and the value of $s$ giving your answers to 3 decimal places.
Stating your hypotheses clearly, use a $5 \%$ level of significance to test the manager's claim.
1. The lengths of a random sample of 120 limpets taken from the upper shore of a beach had a mean of 4.97 cm and a standard deviation of 0.42 cm . The lengths of a second random sample of 150 limpets taken from the lower shore of the same beach had a mean of 5.05 cm and a standard deviation of 0.67 cm .
2. Test, using a $5 \%$ level of significance, whether or not the mean length of limpets from the upper shore is less than the mean length of limpets from the lower shore. State your hypotheses clearly.
3. State two assumptions you made in carrying out the test in part (a).
4. A company produces climbing ropes. The lengths of the climbing ropes are normally distributed. A random sample of 5 ropes is taken and the length, in metres, of each rope is measured. The results are given below.
  119.9
  120.3
  120.1
  120.4
  120.2
5. Calculate unbiased estimates for the mean and the variance of the lengths of the climbing ropes produced by the company.
The lengths of climbing rope are known to have a standard deviation of 0.2 m . The company wants to make sure that there is a probability of at least 0.90 that the estimate of the population mean, based on a random sample size of $n$, lies within 0.05 m of its true value.
Find the minimum sample size required.

Edexcel S3 Q8

Moderate -0.3

The random variable $A$ is defined as

$$A = 4 X - 3 Y$$ where $X \sim \mathrm {~N} \left( 30,3 ^ { 2 } \right) , Y \sim \mathrm {~N} \left( 20,2 ^ { 2 } \right)$ and $X$ and $Y$ are independent. Find

$\mathrm { E } ( A )$,
$\operatorname { Var } ( A )$. The random variables $Y _ { 1 } , Y _ { 2 } , Y _ { 3 }$ and $Y _ { 4 }$ are independent and each has the same distribution as $Y$. The random variable $B$ is defined as $$B = \sum _ { i = 1 } ^ { 4 } Y _ { i }$$
Find $\mathrm { P } ( B > A )$.
advancing learning, changing lives
1. 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.
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.
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 $w$.
1. 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.
advancing learning, changing lives \includegraphics[max width=\textwidth, alt={}, center]{fb233c8c-e1b7-4ba5-aa4d-c23d5382dc84-055_2632_1828_123_121}
2. A county councillor is investigating the level of hardship, h , of a town and the number of calls per 100 people to the emergency services, c. He collects data for 7 randomly selected towns in the county. The results are shown in the table below.
1. Interviews for a job are carried out by two managers. Candidates are given a score by each manager and the results for a random sample of 8 candidates are shown in the table below.
\includegraphics[max width=\textwidth, alt={}, center]{fb233c8c-e1b7-4ba5-aa4d-c23d5382dc84-081_2642_1833_118_118}
2. A random sample of size n is to be taken from a population that is normally distributed with mean 40 and standard deviation 3 . Find the minimum sample size such that the probability of the sample mean being greater than 42 is less than $5 \%$.
(5)
3. The table below shows the population and the number of council employees for different towns and villages. \end{table} A nswers without working may not gain full credit. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{ $0 - 3$ & 8
\hline $3 - 5$ & 12
\hline $5 - 6$ & 13
\hline $6 - 8$ & 9
\hline $8 - 12$ & 8
\hline \captionsetup{labelformat=empty} \caption{Table 1}
\end{table}
Show that an estimate of $\bar { X } = 5.49$ and an estimate of $S _ { X } ^ { 2 } = 6.88$ The post office manager believes that the customers' waiting times can be modelled by a normal distribution.
Assuming the data is normally distributed, she calculates the expected frequencies for these data and some of these frequencies are shown in Table 2. \begin{table}[h]
Waiting Time $\mathrm { x } < 3$ $3 - 5$ $5 - 6$ $6 - 8$ $\mathrm { x } > 8$
Expected Frequency 8.56 12.73 7.56 a b
\captionsetup{labelformat=empty} \caption{Table 2}
\end{table}
Find the value of a and the value of b .
Test, at the $5 \%$ level of significance, the manager's belief. State your hypotheses clearly.
\section*{Q uestion 4 continued}
1. Blumen is a perfume sold in bottles. The amount of perfume in each bottle is normally distributed. The amount of perfume in a large bottle has mean 50 ml and standard deviation 5 ml . The amount of perfume in a small bottle has mean 15 ml and standard deviation 3 ml .
One large and 3 small bottles of Blumen are chosen at random.
Find the probability that the amount in the large bottle is less than the total amount in the 3 small bottles. A large bottle and a small bottle of Blumen are chosen at random.
Find the probability that the large bottle contains more than 3 times the amount in the small bottle.
\section*{Q uestion 5 continued} 6. Fruit-n-Veg4U M arket Gardens grow tomatoes. They want to improve their yield of tomatoes by at least 1 kg per plant by buying a new variety. The variance of the yield of the old variety of plant is $0.5 \mathrm {~kg} ^ { 2 }$ and the variance of the yield for the new variety of plant is $0.75 \mathrm {~kg} ^ { 2 }$. A random sample of 60 plants of the old variety has a mean yield of 5.5 kg . A random sample of 70 of the new variety has a mean yield of 7 kg .
Stating your hypotheses clearly test, at the $5 \%$ level of significance, whether or not there is evidence that the mean yield of the new variety is more than 1 kg greater than the mean yield of the old variety.
Explain the relevance of the Central Limit Theorem to the test in part (a). \section*{Q uestion 6 continued} \includegraphics[max width=\textwidth, alt={}, center]{fb233c8c-e1b7-4ba5-aa4d-c23d5382dc84-102_46_79_2620_1818}
7. Lambs are born in a shed on M ill Farm. The birth weights, $x \mathrm {~kg}$, of a random sample of 8 newborn lambs are given below. $$\begin{array} { l l l l l l l l } 4.12 & 5.12 & 4.84 & 4.65 & 3.55 & 3.65 & 3.96 & 3.40 \end{array}$$
Calculate unbiased estimates of the mean and variance of the birth weight of lambs born on Mill Farm. A further random sample of 32 lambs is chosen and the unbiased estimates of the mean and variance of the birth weight of lambs from this sample are 4.55 and 0.25 respectively.
Treating the combined sample of 40 lambs as a single sample, estimate the standard error of the mean. The owner of M ill Farm researches the breed of lamb and discovers that the population of birth weights is normally distributed with standard deviation 0.67 kg .
Calculate a $95 \%$ confidence interval for the mean birth weight of this breed of lamb using your combined sample mean.
\section*{Q uestion 7 continued} \end{figure}

Edexcel S3 2015 June Q1

5 marks Easy -1.8

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.

Find the value of $w$. [1]
Find the value of $x$. [1]
Write down the probability that the sample includes both the first name and the second name in the club's membership book. [1]
State one advantage and one disadvantage of systematic sampling in this case. [2]

Edexcel S3 2015 June Q2

9 marks Standard +0.3

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. [5]
Stating your hypotheses clearly, test at the 1\% level of significance, whether or not the two judges are generally in agreement. [4]

Edexcel S3 2015 June Q3

11 marks Standard +0.3

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 [1]

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. [3]
Stating your hypotheses clearly, use a 10\% level of significance to test the motorway supervisor's belief. Show your working clearly. [7]

Edexcel S3 2015 June Q4

11 marks Standard +0.3

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 [5]

Edexcel S3 2015 June Q5

12 marks Standard +0.3

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.

	Male	Female
Distinction	18.5\%	27.5\%
Merit	63.5\%	60.0\%
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. [12]

Edexcel S3 2015 June Q6

13 marks Standard +0.3

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. [4]

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. [7]
Explain the relevance of the Central Limit Theorem to the test in part (b). [1]
State an assumption you have made in carrying out the test in part (b). [1]

Edexcel S3 2015 June Q7

5 marks Moderate -0.3

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.

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]

Edexcel S3 2015 June Q8

9 marks Standard +0.3

A factory produces steel sheets whose weights $X$ kg, are such that $X \sim \text{N}(\mu, \sigma^2)$ 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. [3]
Hence, or otherwise, find a 90\% confidence interval for $\mu$ based on the same sample of sheets. [3]

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 $\mu$. [3]

Edexcel S3 Q1

5 marks Easy -2.0

A hotel has 160 rooms of which 20 are classified as De-luxe, 40 Premier and 100 as Standard. The manager wants to obtain information about room usage in the hotel by taking a 10\% sample of the rooms.

Suggest a suitable sampling method. [1]
Explain in detail how the manager should obtain the sample. [4]

Edexcel S3 Q2

9 marks Standard +0.3

A random sample of 100 classical CDs produced by a record company had a mean playing time of 70.6 minutes and a standard deviation of 9.1 minutes. An independent random sample of 80 CDs produced by a different company had a mean playing time of 67.2 minutes with a standard deviation of 8.4 minutes.

Using a 1\% level of significance, test whether or not there is a difference in the mean playing times of the CDs produced by these two companies. State your hypotheses clearly. [8]
State an assumption you made in carrying out the test in part (a). [1]

Edexcel S3 Q3

10 marks Standard +0.3

The weights of a group of males are normally distributed with mean 80 kg and standard deviation 2.6 kg. A random sample of 10 of these males is selected.

Write down the distribution of $M$, the mean weight, in kg, of this sample. [2]
Find P($M < 78.5$). [3]

The weights of a group of females are normally distributed with mean 59 kg and standard deviation 1.9 kg. A random sample of 6 of the males and 4 of the females enters a lift that can carry a maximum load of 730 kg.

Find the probability that the maximum load will be exceeded when these 10 people enter the lift. [5]

Edexcel S3 Q4

11 marks Standard +0.3

At the end of a season an athletics coach graded a random sample of ten athletes according to their performances throughout the season and their dedication to training. The results, expressed as percentages, are shown in the table below.

Athlete	Performance	Dedication
A	86	72
B	60	69
C	78	59
D	56	68
E	80	80
F	66	84
G	51	65
H	59	55
I	73	79
J	49	53

Calculate the Spearman rank correlation coefficient between performance and dedication. [5]
Stating clearly your hypotheses and using a 10\% level of significance, interpret your rank correlation coefficient. [5]
Give a reason to support the use of the rank correlation coefficient rather than the product moment correlation coefficient with these data. [1]

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

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Edexcel S3 2015 June Q3

Edexcel S3 2015 June Q4

Edexcel S3 2015 June Q5

Edexcel S3 2015 June Q6

Edexcel S3 2015 June Q7

Edexcel S3 2015 June Q8

Edexcel S3 Q1

Edexcel S3 Q2

Edexcel S3 Q3

Edexcel S3 Q4

	mean	standard deviation
\(1 ^ { \text {st } }\) leg \(- A\)	63.1	1.2
\(2 ^ { \text {nd } }\) leg \(- B\)	65.7	1.5
\(3 ^ { \text {rd } } \operatorname { leg } - C\)	65.4	1.8
\(4 ^ { \text {th } }\) leg - \(D\)	62.5	0.9

Individual	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)	\(I\)	\(J\)
BMI	17.4	21.4	18.9	24.4	19.4	20.1	22.6	18.4	25.8	28.1
Finishing position	3	5	1	9	6	4	10	2	7	8

Number of goals	Expected Frequency
0	34.994
1	\(r\)
2	\(s\)
3	6.752
\(\geqslant 4\)	2.221

Waiting Time	\(\mathrm { x } < 3\)	\(3 - 5\)	\(5 - 6\)	\(6 - 8\)	\(\mathrm { x } > 8\)
Expected Frequency	8.56	12.73	7.56	a	b