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Edexcel S3 2009 June Q3

11 marks Standard +0.3

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. [5]
Stating your hypotheses clearly and using a one tailed test with a 5\% 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]

Edexcel S3 2009 June Q4

5 marks Moderate -0.5

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]

Edexcel S3 2009 June Q5

12 marks Standard +0.3

The number of goals scored by a football team is recorded for 100 games. The results are summarised in Table 1 below.

Number of goals	Frequency
0	40
1	33
2	14
3	8
4	5

Table 1

Calculate the mean number of goals scored per game. [2]

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.

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

Table 2

Find the value of $r$ and the value of $s$ giving your answers to 3 decimal places. [3]
Stating your hypotheses clearly, use a 5\% level of significance to test the manager's claim. [7]

Edexcel S3 2009 June Q6

10 marks Standard +0.3

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.

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. [8]
State two assumptions you made in carrying out the test in part (a). [2]

Edexcel S3 2009 June Q7

11 marks Standard +0.3

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. 120.3 \quad 120.1 \quad 120.4 \quad 120.2 \quad 119.9

Calculate unbiased estimates for the mean and the variance of the lengths of the climbing ropes produced by the company. [5]

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. [6]

Edexcel S3 2009 June Q8

11 marks Standard +0.3

The random variable $A$ is defined as $$A = 4X - 3Y$$ where $X \sim \text{N}(30, 3^2)$, $Y \sim \text{N}(20, 2^2)$ and $X$ and $Y$ are independent. Find

E($A$), [2]
Var($A$). [3]

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 P($B > A$). [6]

Edexcel S3 2011 June Q1

3 marks Moderate -0.5

Explain what you understand by the Central Limit Theorem. [3]

Edexcel S3 2011 June Q2

10 marks Standard +0.3

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.

Town	$A$	$B$	$C$	$D$	$E$	$F$	$G$
$h$	14	20	16	18	37	19	24
$c$	52	45	43	42	61	82	55

Calculate the Spearman's rank correlation coefficient between $h$ and $c$. [6]
Test, at the 5\% level of significance, the councillor's claim. State your hypotheses clearly. [4]

After collecting the data, the councillor thinks there is no correlation between hardship and the number of calls to the emergency services.

Edexcel S3 2011 June Q3

10 marks Standard +0.3

A factory manufactures batches of an electronic component. Each component is manufactured in one of three shifts. A component may have one of two types of defect, $D_1$ or $D_2$, at the end of the manufacturing process. A production manager believes that the type of defect is dependent upon the shift that manufactured the component. He examines 200 randomly selected defective components and classifies them by defect type and shift. The results are shown in the table below.

	$D_1$	$D_2$
First shift	45	18
Second shift	55	20
Third shift	50	12

Stating your hypotheses, test, at the 10\% level of significance, whether or not there is evidence to support the manager's belief. Show your working clearly. [10]

Edexcel S3 2011 June Q4

13 marks Standard +0.3

A shop manager wants to find out if customers spend more money when music is playing in the shop. The amount of money spent by a customer in the shop is £$x$. A random sample of 80 customers, who were shopping without music playing, and an independent random sample of 60 customers, who were shopping with music playing, were surveyed. The results of both samples are summarised in the table below.

	$\sum x$	$\sum x^2$	Unbiased estimate of mean	Unbiased estimate of variance
Customers shopping without music	5320	392000	$\bar{x}$	$s^2$
Customers shopping with music	4140	312000	69.0	446.44

Find the values of $\bar{x}$ and $s^2$. [5]
Test, at the 5\% level of significance, whether or not the mean money spent is greater when music is playing in the shop. State your hypotheses clearly. [8]

Edexcel S3 2011 June Q5

13 marks Standard +0.3

The number of hurricanes per year in a particular region was recorded over 80 years. The results are summarised in Table 1 below.

No of hurricanes, $h$	0	1	2	3	4	5	6	7
Frequency	0	2	5	17	20	12	12	12

Table 1

Write down two assumptions that will support modelling the number of hurricanes per year by a Poisson distribution. [2]
Show that the mean number of hurricanes per year from Table 1 is 4.4875 [2]
Use the answer in part (b) to calculate the expected frequencies $r$ and $s$ given in Table 2 below to 2 decimal places. [3]

$h$	0	1	2	3	4	5	6	7 or more
Expected frequency	0.90	4.04	$r$	13.55	$s$	13.65	10.21	13.39

Table 2

Test, at the 5\% level of significance, whether or not the data can be modelled by a Poisson distribution. State your hypotheses clearly. [6]

Edexcel S3 2011 June Q6

10 marks Standard +0.3

The lifetimes of batteries from manufacturer $A$ are normally distributed with mean 20 hours and standard deviation 5 hours when used in a camera.

Find the mean and standard deviation of the total lifetime of a pack of 6 batteries from manufacturer $A$. [2]

Judy uses a camera that takes one battery at a time. She takes a pack of 6 batteries from manufacturer $A$ to use in her camera on holiday.

Find the probability that the batteries will last for more than 110 hours on her holiday. [2]

The lifetimes of batteries from manufacturer $B$ are normally distributed with mean 35 hours and standard deviation 8 hours when used in a camera.

Find the probability that the total lifetime of a pack of 6 batteries from manufacturer $A$ is more than 4 times the lifetime of a single battery from manufacturer $B$ when used in a camera. [6]

Edexcel S3 2011 June Q7

16 marks Standard +0.3

Roastie's Coffee is sold in packets with a stated weight of 250 g. A supermarket manager claims that the mean weight of the packets is less than the stated weight. She weighs a random sample of 90 packets from their stock and finds that their weights have a mean of 248 g and a standard deviation of 5.4 g.

Using a 5\% level of significance, test whether or not the manager's claim is justified. State your hypotheses clearly. [5]
Find the 98\% confidence interval for the mean weight of a packet of coffee in the supermarket's stock. [4]
State, with a reason, the action you would recommend the manager to take over the weight of a packet of Roastie's Coffee. [2]

Roastie's Coffee company increase the mean weight of their packets to $\mu$ g and reduce the standard deviation to 3 g. The manager takes a sample of size $n$ from these new packets. She uses the sample mean $\bar{X}$ as an estimator of $\mu$.

Find the minimum value of $n$ such that P$(|\bar{X} - \mu| < 1) \geq 0.98$ [5]

Edexcel S3 2016 June Q1

Easy -1.8

State two reasons why stratified sampling might be a more suitable sampling method than simple random sampling. (2)
State two reasons why stratified sampling might be a more suitable sampling method than quota sampling. (2)

Edexcel S3 2016 June Q2

Standard +0.3

A new drug to vaccinate against influenza was given to 110 randomly chosen volunteers. The volunteers were given the drug in one of 3 different concentrations, $A$, $B$ and $C$, and then were monitored to see if they caught influenza. The results are shown in the table below.

	$A$	$B$	$C$
Influenza	12	29	9
No influenza	15	23	22

Test, at the 10\% level of significance, whether or not there is an association between catching influenza and the concentration of the new drug. State your hypotheses and show your working clearly. You should state your expected frequencies to 2 decimal places. (10)

Edexcel S3 2016 June Q3

Moderate -0.3

Describe when you would use Spearman's rank correlation coefficient rather than the product moment correlation coefficient to measure the strength of the relationship between two variables. (1) A shop sells sunglasses and ice cream. For one week in the summer the shopkeeper ranked the daily sales of ice cream and sunglasses. The ranks are shown in the table below.
Sun Mon Tues Weds Thurs Fri Sat
Ice cream 6 4 7 5 3 2 1
Sunglasses 6 5 7 2 3 4 1
Calculate Spearman's rank correlation coefficient for these data. (3)
Test, at the 5\% level of significance, whether or not there is a positive correlation between sales of ice cream and sales of sunglasses. State your hypotheses clearly. (4) The shopkeeper calculates the product moment correlation coefficient from his raw data and finds $r = 0.65$
Using this new coefficient, test, at the 5\% level of significance, whether or not there is a positive correlation between sales of ice cream and sales of sunglasses. (2)
Using your answers to part (c) and part (d), comment on the nature of the relationship between sales of sunglasses and sales of ice cream. (1)

Edexcel S3 2016 June Q4

Standard +0.3

The weights of eggs are normally distributed with mean 60g and standard deviation 5g Sairah chooses 2 eggs at random.

Find the probability that the difference in weight of these 2 eggs is more than 2g (5) Sairah is packing eggs into cartons. The weight of an empty egg carton is normally distributed with mean 40g and standard deviation 1.5g
Find the distribution of the total weight of a carton filled with 12 randomly chosen eggs. (3)
Find the probability that a randomly chosen carton, filled with 12 randomly chosen eggs, weighs more than 800g (2)

Edexcel S3 2016 June Q5

Standard +0.3

A doctor claims there is a higher mean lung capacity in people who exercise regularly compared to people who do not exercise regularly. He measures the lung capacity, $x$, of 35 people who exercise regularly and 42 people who do not exercise regularly. His results are summarised in the table below.

	$n$	$\bar{x}$	$s^2$
Exercise regularly	35	26.3	12.2
Do not exercise regularly	42	24.8	10.1

Test, at the 5\% level of significance, the doctor's claim. State your hypotheses clearly. (6)
State any assumptions you have made in testing the doctor's claim. (2) The doctor decides to add another person who exercises regularly to his data. He measures the person's lung capacity and finds $x = 31.7$
Find the unbiased estimate of the variance for the sample of 36 people who exercise regularly. Give your answer to 3 significant figures. (4)

Edexcel S3 2016 June Q6

Standard +0.3

An airport manager carries out a survey of families and their luggage. Each family is allowed to check in a maximum of 4 suitcases. She observes 50 families at the check-in desk and counts the total number of suitcases each family checks in. The data are summarised in the table below.

Number of suitcases	0	1	2	3	4
Frequency	6	25	12	6	1

The manager claims that the data can be modelled by a binomial distribution with $p = 0.3$

Test the manager's claim at the 5\% level of significance. State your hypotheses clearly. Show your working clearly and give your expected frequencies to 2 decimal places. (8) The manager also carries out a survey of the time taken by passengers to check in. She records the number of passengers that check in during each of 100 five-minute intervals. The manager makes a new claim that these data can be modelled by a Poisson distribution. She calculates the expected frequencies given in the table below.
Number of passengers 0 1 2 3 4 5 or more
Observed frequency 5 40 31 18 6 0
Expected frequency 16.53 29.75 $r$ $s$ 7.23 3.64
Find the value of $r$ and the value of $s$ giving your answers to 2 decimal places. (3)
Stating your hypotheses clearly, use a 1\% level of significance to test the manager's new claim. (6)

Edexcel S3 2016 June Q7

Standard +0.3

A restaurant states that its hamburgers contain 20\% fat. Paul claims that the mean fat content of their hamburgers is less than 20\%. Paul takes a random sample of 50 hamburgers from the restaurant and finds that they contain a mean fat content of 19.5\% with a standard deviation of 1.5\% You may assume that the fat content of hamburgers is normally distributed.

Find the 90\% confidence interval for the mean fat content of hamburgers from the restaurant. (4)
State, with a reason, what action Paul should recommend the restaurant takes over the stated fat content of their hamburgers. (2) The restaurant changes the mean fat content of their hamburgers to $\mu$\% and adjusts the standard deviation to 2\%. Paul takes a sample of size $n$ from this new batch of hamburgers. He uses the sample mean $\bar{X}$ as an estimator of $\mu$.
Find the minimum value of $n$ such that $\mathrm{P}(|\bar{X} - \mu| < 0.5) \geq 0.9$ (5)

Edexcel S3 Q1

5 marks Moderate -0.8

A random sample $X_1, X_2, \ldots, X_{10}$ is taken from a normal population with mean 100 and standard deviation 14.

Write down the distribution of $\overline{X}$, the mean of this sample. [2]
Find $\text{Pr}(|\overline{X} - 100| > 5)$. [3]

Edexcel S3 Q2

6 marks Standard +0.3

A random sample of the invoices, for books purchased by the customers of a large bookshop, was classified by book cover (hardback, paperback) and type of book (novel, textbook, general interest). As part of the analysis of these invoices, an approximate $\chi^2$ statistic was calculated and found to be 11.09. Assuming that there was no need to amalgamate any of the classifications, carry out an appropriate test to determine whether or not there was any association between book cover and type of book. State your hypotheses clearly and use a 5% level of significance. [6]

Edexcel S3 Q3

11 marks Standard +0.3

As part of a research project into the role played by cholesterol in the development of heart disease a random sample of 100 patients was put on a special fish-based diet. A different random sample of 80 patients was kept on a standard high-protein low-fat diet. After several weeks their blood cholesterol was measured and the results summarised in the table below.

Group	Sample size	Mean drop in cholesterol (mg/dl)	Standard deviation
Special diet	100	75	22
Standard diet	80	64	31

Stating your hypotheses clearly and using a 5% level of significance, test whether or not the special diet is more effective in reducing blood cholesterol levels than the standard diet. [9]
Explain briefly any assumptions you made in order to carry out this test. [2]

Edexcel S3 Q4

13 marks Standard +0.8

Breakdowns on a certain stretch of motorway were recorded each day for 80 consecutive days. The results are summarised in the table below.

Number of breakdowns	0	1	2	$>2$
Frequency	38	32	10	0

Edexcel S3 Q5

12 marks Moderate -0.3

The random variable $R$ is defined as $R = X + 4Y$ where $X \sim \text{N}(8, 2^2)$, $Y \sim \text{N}(14, 3^2)$ and $X$ and $Y$ are independent. Find

E$(R)$, [2]
Var$(R)$, [3]
P$(R < 41)$ [3]

The random variables $Y_1$, $Y_2$ and $Y_3$ are independent and each has the same distribution as $Y$. The random variable $S$ is defined as $$S = \sum_{i=1}^{3} Y_i - \frac{1}{2}X.$$

Find Var$(S)$. [4]

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

	\(D_1\)	\(D_2\)
First shift	45	18
Second shift	55	20
Third shift	50	12

	\(\sum x\)	\(\sum x^2\)	Unbiased estimate of mean	Unbiased estimate of variance
Customers shopping without music	5320	392000	\(\bar{x}\)	\(s^2\)
Customers shopping with music	4140	312000	69.0	446.44

	\(n\)	\(\bar{x}\)	\(s^2\)
Exercise regularly	35	26.3	12.2
Do not exercise regularly	42	24.8	10.1

Number of passengers	0	1	2	3	4	5 or more
Observed frequency	5	40	31	18	6	0
Expected frequency	16.53	29.75	\(r\)	\(s\)	7.23	3.64

	Sun	Mon	Tues	Weds	Thurs	Fri	Sat
Ice cream	6	4	7	5	3	2	1
Sunglasses	6	5	7	2	3	4	1

Questions — Edexcel S3 (332 questions)

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Edexcel S3 2009 June Q3

Edexcel S3 2009 June Q4

Edexcel S3 2009 June Q5

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