Questions - Edexcel S3 - A-Level Maths

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 evidence of a correlation between the number of students per member of staff and the student satisfaction score.

Edexcel S3 2013 June Q3

7 marks Easy -1.2

3. A college manager wants to survey students' opinions of enrichment activities. She decides to survey the students on the courses summarised in the table below.

Course	Number of students enrolled
Leisure and Sport	420
Information Technology	337
Health and Social Care	200
Media Studies	43

Each student takes only one course.
The manager has access to the college's information system that holds full details of each of the enrolled students including name, address, telephone number and their course of study. She wants to compare the opinions of students on each course and has a generous budget to pay for the cost of the survey.

Give one advantage and one disadvantage of carrying out this survey using
1. quota sampling,
2. stratified sampling. The manager decides to take a stratified sample of 100 students.
Calculate the number of students to be sampled from each course.
Describe how to choose students for the stratified sample.

Edexcel S3 2013 June Q4

14 marks Standard +0.3

4. Customers at a post office are timed to see how long they wait until being served at the counter. A random sample of 50 customers is chosen and their waiting times, $x$ minutes, are summarised in Table 1. \begin{table}[h]

Waiting time in minutes $( x )$	Frequency
$0 - 3$	8
$3 - 5$	12
$5 - 6$	13
$6 - 8$	9
$8 - 12$	8

\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 $x < 3$ $3 - 5$ $5 - 6$ $6 - 8$ $x > 8$
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.

Edexcel S3 2013 June Q5

12 marks Standard +0.8

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.

Edexcel S3 2013 June Q6

11 marks Standard +0.3

6. Fruit-n-Veg4U Market 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).

Edexcel S3 2013 June Q7

13 marks Moderate -0.5

Lambs are born in a shed on Mill 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 Mill 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.

Edexcel S3 2014 June Q1

11 marks Standard +0.3

A journalist is investigating factors which influence people when they buy a new car. One possible factor is fuel efficiency. The journalist randomly selects 8 car models. Each model's annual sales and fuel efficiency, in km/litre, are shown in the table below.

Car model	$A$	$B$	$C$	$D$	$E$	$F$	$G$	$H$
Annual sales	1800	5400	18100	7100	9300	4800	12200	10700
Fuel efficiency	5.2	18.6	14.8	13.2	18.3	11.9	16.5	17.7

Calculate Spearman's rank correlation coefficient for these data. The journalist believes that car models with higher fuel efficiency will achieve higher sales.
Stating your hypotheses clearly, test whether or not the data support the journalist's belief. Use a $5 \%$ level of significance.
State the assumption necessary for a product moment correlation coefficient to be valid in this case.
The mean and median fuel efficiencies of the car models in the random sample are 14.5 km /litre and 15.65 km /litre respectively. Considering these statistics, as well as the distribution of the fuel efficiency data, state whether or not the data suggest that the assumption in part (c) might be true in this case. Give a reason for your answer. (No further calculations are required.)

Edexcel S3 2014 June Q2

7 marks Standard +0.3

A survey asked a random sample of 200 people their age and the main use of their mobile phone.

The results are shown in Table 1 below. \begin{table}[h]

\multirow{2}{*}{}		Main use of their mobile phone
		Internet	Texts	Phone calls
\multirow{3}{*}{Age}	Under 20	27	14	9
	From 20 to 40	32	34	29
	Over 40	15	19	21

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

\end{table} The data are to be used to test whether or not age and main use of their mobile phone are independent. Table 2 shows the expected frequencies for each group, assuming people's age and main use of their mobile phone are independent. \begin{table}[h]

\multirow{2}{*}{}		Main use of their mobile phone
		Internet	Texts	Phone calls
\multirow{3}{*}{Age}	Under 20	18.5	16.75	14.75
	From 20 to 40	35.15	31.825	28.025
	Over 40	20.35	18.425	16.225

\captionsetup{labelformat=empty} \caption{Table 2}

\end{table}

For users under 20 choosing the Internet as the main use of their mobile phone,
1. verify that the expected frequency is 18.5
2. show that the contribution to the $\chi ^ { 2 }$ test statistic is 3.91 to 3 significant figures.
Given that the $\chi ^ { 2 }$ test statistic for the data is 9.893 to 3 decimal places, test at the $5 \%$ level of significance whether or not age and main use of their mobile phone are independent. State your hypotheses clearly.

Edexcel S3 2014 June Q3

11 marks Standard +0.8

A company produces two types of milk powder, 'Semi-Skimmed' and 'Full Cream'. In tests, each type of milk powder is used to make a large number of cups of coffee. The mass, $S$ grams, of 'Semi-Skimmed' milk powder used in one cup of coffee is modelled by $S \sim \mathrm {~N} \left( 4.9,0.8 ^ { 2 } \right)$. The mass, $C$ grams, of 'Full Cream' milk powder used in one cup of coffee is modelled by $C \sim \mathrm {~N} \left( 2.5,0.4 ^ { 2 } \right)$
1. Two cups of coffee, one with each type of milk powder, are to be selected at random. Find the probability that the mass of 'Semi-Skimmed' milk powder used will be at least double that of the 'Full Cream' milk powder used.
2. 'Semi-Skimmed' milk powder is sold in 500 g packs. Find the probability that one pack will be sufficient for 100 cups of coffee.

Edexcel S3 2014 June Q4

6 marks Challenging +1.2

4. A manufacturing company produces solar panels. The output of each solar panel is normally distributed with standard deviation 6 watts. It is thought that the mean output, $\mu$, is 160 watts. A researcher believes that the mean output of the solar panels is greater than 160 watts. He writes down the output values of 5 randomly selected solar panels. He uses the data to carry out a hypothesis test at the $5 \%$ level of significance. He tests $\mathrm { H } _ { 0 } : \mu = 160$ against $\mathrm { H } _ { 1 } : \mu > 160$ On reporting to his manager, the researcher can only find 4 of the output values. These are shown below $$\begin{array} { l l l l } 168.2 & 157.4 & 173.3 & 161.1 \end{array}$$ Given that the result of the hypothesis test is that there is significant evidence to reject $\mathrm { H } _ { 0 }$ at the $5 \%$ level of significance, calculate the minimum possible missing output value, $\alpha$. Give your answer correct to 1 decimal place.

Edexcel S3 2014 June Q5

13 marks Standard +0.3

5. A student believes that there is a difference in the mean lengths of English and French films. He goes to the university video library and randomly selects a sample of 120 English films and a sample of 70 French films. He notes the length, $x$ minutes, of each of the films in his samples. His data are summarised in the table below.

	$\Sigma x$	$\Sigma x ^ { 2 }$	$s ^ { 2 }$	$n$
English films	10650	956909	98.5	120
French films	6510	615849	151	70

Verify that the unbiased estimate of the variance, $s ^ { 2 }$, of the lengths of English films is 98.5 minutes ${ } ^ { 2 }$
Stating your hypotheses clearly, test, at the 1\% level of significance, whether or not the mean lengths of English and French films are different.
Explain the significance of the Central Limit Theorem to the test in part (b).
The university video library contained 724 English films and 473 French films. Explain how the student could have taken a stratified sample of 190 of these films.

Edexcel S3 2014 June Q6

17 marks Standard +0.3

6. Bags of $\pounds 1$ coins are paid into a bank. Each bag contains 20 coins. The bank manager believes that $5 \%$ of the $\pounds 1$ coins paid into the bank are fakes. He decides to use the distribution $X \sim \mathrm {~B} ( 20,0.05 )$ to model the random variable $X$, the number of fake $\pounds 1$ coins in each bag.

State the assumptions necessary for the binomial distribution to be an appropriate model in this case. The bank manager checks a random sample of 150 bags of $\pounds 1$ coins and records the number of fake coins found in each bag. His results are summarised in Table 1. \begin{table}[h]
Number of fake coins in each bag 0 1 2 3 4 or more
Observed frequency 43 62 26 13 6
Expected frequency 53.8 56.6 $r$ 8.9 $s$
\captionsetup{labelformat=empty} \caption{Table 1}
\end{table}
Calculate the values of $r$ and $s$, giving your answers to 1 decimal place.
Carry out a hypothesis test, at the $5 \%$ significance level, to see if the data supports the bank manager's statistical model. State your hypotheses clearly. Question 6 parts (d) and (e) are continued on page 24 The assistant manager thinks that a binomial distribution is a good model but suggests that the proportion of fake coins is higher than $5 \%$. She calculates the actual proportion of fake coins in the sample and uses this value to carry out a new hypothesis test on the data. Her expected frequencies are shown in Table 2. \begin{table}[h]
Number of fake coins in each bag 0 1 2 3 4 or more
Observed frequency 43 62 26 13 6
Expected frequency 44.5 55.7 33.2 12.5 4.1
\captionsetup{labelformat=empty} \caption{Table 2}
\end{table}
Explain why there are 2 degrees of freedom in this case.
Given that she obtains a $\chi ^ { 2 }$ test statistic of 2.67 , test the assistant manager's hypothesis that the binomial distribution is a good model for the number of fake coins in each bag. Use a $5 \%$ level of significance and state your hypotheses clearly.

Edexcel S3 2014 June Q7

10 marks Standard +0.3

7. A petrol pump is tested regularly to check that the reading on its gauge is accurate. The random variable $X$, in litres, is the quantity of petrol actually dispensed when the gauge reads 10.00 litres. $X$ is known to have distribution $X \sim \mathrm {~N} \left( \mu , 0.08 ^ { 2 } \right)$

Eight random tests gave the following values of $x$ $$\begin{array} { l l l l l l l l } 10.01 & 9.97 & 9.93 & 9.99 & 9.90 & 9.95 & 10.13 & 9.94 \end{array}$$
1. Find a 95\% confidence interval for $\mu$ to 2 decimal places.
2. Use your result to comment on the accuracy of the petrol gauge.
A sample mean of 9.96 litres was obtained from a random sample of $n$ tests. A $90 \%$ confidence interval for $\mu$ gave an upper limit of less than 10.00 litres. Find the minimum value of $n$.

Edexcel S3 2014 June Q1

5 marks Easy -1.8

(a) Explain what you understand by a random sample from a finite population.
(b) Give an example of a situation when it is not possible to take a random sample.

A college lecturer specialising in shoe design wants to change the way in which she organises practical work. She decides to gather ideas from her 75 students. She plans to give a questionnaire to a random sample of 8 of these students.
(c) (i) Describe the sampling frame that she should use.
(ii) Explain in detail how she should use a table of random numbers to obtain her sample.

Edexcel S3 2014 June Q2

7 marks Moderate -0.8

2. The weights of pears in an orchard are assumed to have unknown mean $\mu$ and unknown standard deviation $\sigma$. A random sample of 20 pears is taken and their weights recorded.
The sample is represented by $X _ { 1 } , X _ { 2 } , \ldots , X _ { 20 }$. State whether or not the following are statistics. Give reasons for your answers.

1. $\frac { X _ { 1 } + 3 X _ { 20 } } { 2 }$
2. $\sum _ { i = 1 } ^ { 20 } \left( X _ { i } - \mu \right)$
3. $\sum _ { i = 1 } ^ { 20 } \left( \frac { X _ { i } - \mu } { \sigma } \right)$
Find the mean and variance of $\frac { 3 X _ { 1 } - X _ { 20 } } { 2 }$

Edexcel S3 2014 June Q3

10 marks Standard +0.3

3. A number of males and females were asked to rate their happiness under the headings "not happy", "fairly happy" and "very happy". The results are shown in the table below

		Happiness			\multirow{2}{*}{Total}
\cline { 3 - 5 } \multicolumn{2}{\|c\|}{}	Not happy	Fairly happy	Very happy
\multirow{2}{*}{Gender}	Female	9	43	34	86
\cline { 2 - 6 }	Male	13	25	16	54
Total		22	68	50	140

Stating your hypotheses, test at the $5 \%$ level of significance, whether or not there is evidence of an association between happiness and gender. Show your working clearly.

Edexcel S3 2014 June Q4

6 marks Standard +0.3

The random variable $A$ is defined as

$$A = B + 4 C - 3 D$$ where $B$, $C$ and $D$ are independent random variables with $$B \sim \mathrm {~N} \left( 6,2 ^ { 2 } \right) \quad C \sim \mathrm {~N} \left( 7,3 ^ { 2 } \right) \quad D \sim \mathrm {~N} \left( 4,1.5 ^ { 2 } \right)$$ Find $\mathrm { P } ( A < 45 )$

Edexcel S3 2014 June Q5

13 marks Standard +0.3

5. A research station is doing some work on the germination of a new variety of genetically modified wheat. They planted 120 rows containing 7 seeds in each row.
The number of seeds germinating in each row was recorded. The results are as follows

Number of seeds germinating in each row	0	1	2	3	4	5	6	7
Observed number of rows	2	6	11	19	25	32	16	9

Write down two reasons why a binomial distribution may be a suitable model.
Show that the probability of a randomly selected seed from this sample germinating is 0.6 The research station used a binomial distribution with probability 0.6 of a seed germinating. The expected frequencies were calculated to 2 decimal places. The results are as follows
Number of seeds germinating in each row 0 1 2 3 4 5 6 7
Expected number of rows 0.20 2.06 $s$ 23.22 $t$ 31.35 15.68 3.36
Find the value of $s$ and the value of $t$.
Stating your hypotheses clearly, test, at the $1 \%$ level of significance, whether or not the data can be modelled by a binomial distribution.

Edexcel S3 2014 June Q6

8 marks Standard +0.3

6. A random sample $X _ { 1 } , X _ { 2 } , \ldots , X _ { n }$ is taken from a population with mean $\mu$.

Show that $\bar { X } = \frac { 1 } { n } \left( X _ { 1 } + X _ { 2 } + \ldots + X _ { n } \right)$ is an unbiased estimator of the population mean $\mu$. A company produces small jars of coffee. Five jars of coffee were taken at random and weighed. The weights, in grams, were as follows $$\begin{array} { l l l l l } 197 & 203 & 205 & 201 & 195 \end{array}$$
Calculate unbiased estimates of the population mean and variance of the weights of the jars produced by the company. It is known from previous results that the weights are normally distributed with standard deviation 4.8 g . The manager is going to take a second random sample. He wishes to ensure that there is at least a $95 \%$ probability that the estimate of the population mean is within 1.25 g of its true value.
Find the minimum sample size required.

Edexcel S3 2014 June Q7

10 marks Standard +0.3

7. A machine fills packets with $X$ grams of powder where $X$ is normally distributed with mean $\mu$. Each packet is supposed to contain 1 kg of powder. To comply with regulations, the weight of powder in a randomly selected packet should be such that $\mathrm { P } ( X < \mu - 30 ) = 0.0005$

Show that this requires the standard deviation to be 9.117 g to 3 decimal places. A random sample of 10 packets is selected from the machine. The weight, in grams, of powder in each packet is as follows 999.8991 .61000 .31006 .11008 .2997 .0993 .21000 .0997 .11002 .1
Assuming that the standard deviation of the population is 9.117 g , test, at the $1 \%$ significance level, whether or not the machine is delivering packets with mean weight of less than 1 kg . State your hypotheses clearly.

Edexcel S3 2014 June Q8

16 marks Standard +0.3

8. The heights, in metres, and weights, in kilograms, of a random sample of 9 men are shown in the table below

Man	$A$	$B$	$C$	$D$	$E$	$F$	$G$	$H$	$I$
Height $( x )$	1.68	1.74	1.75	1.76	1.78	1.82	1.84	1.88	1.98
Weight $( y )$	75	76	100	77	90	95	110	96	120

Given that $\mathrm { S } _ { x x } = 0.0632 , \mathrm {~S} _ { y y } = 1957.5556$ and $\mathrm { S } _ { x y } = 9.3433$ calculate, to 3 decimal places, the product moment correlation coefficient between height and weight for these men.
Use your value of the product moment correlation coefficient to test whether or not there is evidence of a positive correlation between the height and weight of men. Use a $5 \%$ significance level. State your hypotheses clearly. Peter does not know the heights or weights of the 9 men. He is given photographs of them and asked to put them in order of increasing weight. He puts them in the order $$A C E B G D I F H$$
Find, to 3 decimal places, Spearman's rank correlation coefficient between Peter's order and the actual order.
Use your value of Spearman's rank correlation coefficient to test for evidence of Peter's ability to correctly order men, by their weight, from their photographs. Use a 5\% significance level and state your hypotheses clearly.

Edexcel S3 2015 June Q1

9 marks Standard +0.3

A mobile library has 160 books for children on its records. The librarian believes that books with fewer pages are borrowed more often. He takes a random sample of 10 books for children.
1. Explain how the librarian should select this random sample.
  (2)
The librarian ranked the 10 books according to how often they had been borrowed, with 1 for the book borrowed the most and 10 for the book borrowed the least. He also recorded the number of pages in each book. The results are in the table below.
Book $A$ $B$ $C$ $D$ $E$ $F$ $G$ $H$ $I$ $J$
Borrowing rank 1 2 3 4 5 6 7 8 9 10
Number of pages 50 212 115 80 301 90 356 283 152 317
Calculate Spearman's rank correlation coefficient for these data.
Test the librarian's belief using a $5 \%$ level of significance. State your hypotheses clearly.

Edexcel S3 2015 June Q2

10 marks Standard +0.3

2. A researcher believes that the mean weight loss of those people using a slimming plan as part of a group is more than 1.5 kg a year greater than the mean weight loss of those using the plan on their own. The mean weight loss of a random sample of 80 people using the plan as part of a group is 8.7 kg with a standard deviation of 2.1 kg . The mean weight loss of a random sample of 65 people using the plan on their own is 6.6 kg with a standard deviation of 1.4 kg .

Stating your hypotheses clearly, test the researcher's claim. Use a $1 \%$ level of significance.
For the test in part (a), state whether or not it is necessary to assume that the weight loss of a person using this plan has a normal distribution. Give a reason for your answer.

Edexcel S3 2015 June Q3

11 marks Moderate -0.8

3. A nursery has 16 staff and 40 children on its records. In preparation for an outing the manager needs an estimate of the mean weight of the people on its records and decides to take a stratified sample of size 14 .

Describe how this stratified sample should be taken. The weights, $x \mathrm {~kg}$, of each of the 14 people selected are summarised as $$\sum x = 437 \text { and } \sum x ^ { 2 } = 26983$$
Find unbiased estimates of the mean and the variance of the weights of all the people on the nursery's records.
Estimate the standard error of the mean. The estimates of the standard error of the mean for the staff and for the children are 5.11 and 1.10 respectively.
Comment on these values with reference to your answer to part (c) and give a reason for any differences.

Edexcel S3 2015 June Q4

9 marks Standard +0.3

The weights of bags of rice, $X \mathrm {~kg}$, have a normal distribution with unknown mean $\mu \mathrm { kg }$ and known standard deviation $\sigma \mathrm { kg }$. A random sample of 100 bags of rice gave a $90 \%$ confidence interval for $\mu$ of $( 0.4633,0.5127 )$.
1. Without carrying out any further calculations, use this confidence interval to test whether or not $\mu = 0.5$
State your hypotheses clearly and write down the significance level you have used. A second random sample, of 150 of these bags of rice, had a mean weight of 0.479 kg .
Calculate a $95 \%$ confidence interval for $\mu$ based on this second sample.

Waiting time in minutes \(( x )\)	Frequency
\(0 - 3\)	8
\(3 - 5\)	12
\(5 - 6\)	13
\(6 - 8\)	9
\(8 - 12\)	8

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

	\(\Sigma x\)	\(\Sigma x ^ { 2 }\)	\(s ^ { 2 }\)	\(n\)
English films	10650	956909	98.5	120
French films	6510	615849	151	70

Number of fake coins in each bag	0	1	2	3	4 or more
Observed frequency	43	62	26	13	6
Expected frequency	53.8	56.6	\(r\)	8.9	\(s\)

Number of seeds germinating in each row	0	1	2	3	4	5	6	7
Expected number of rows	0.20	2.06	\(s\)	23.22	\(t\)	31.35	15.68	3.36

Man	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)	\(I\)
Height \(( x )\)	1.68	1.74	1.75	1.76	1.78	1.82	1.84	1.88	1.98
Weight \(( y )\)	75	76	100	77	90	95	110	96	120

Book	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)	\(I\)	\(J\)
Borrowing rank	1	2	3	4	5	6	7	8	9	10
Number of pages	50	212	115	80	301	90	356	283	152	317

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