Questions - Edexcel S3 - A-Level Maths

Assuming that the time for the credits to roll follows a Normal distribution with a standard deviation of 23 seconds, use her data to calculate a $90 \%$ confidence interval for the mean time taken for the credits to roll.
(5 marks)
Find the minimum number of movies she would need to have included in her sample for her confidence interval to have a width of less than 10 seconds.
(5 marks)
Explain why her sample might not be representative of all movies.

Edexcel S3 Q4

4. A hospital administrator is assessing staffing needs for its Accident and Emergency Department at different times of day. The administrator already has data on the number of admissions at different times of day but needs to know if the proportion of the cases that are serious remains constant. Staff are asked to assess whether each person arriving at Accident and Emergency has a "minor" or "serious" problem and the results for three different time periods are shown below.

\cline { 2 - 3 } \multicolumn{1}{c\|}{}	Minor	Serious
8 a.m. - 6 p.m.	45	11
6 p.m. - 2 a.m.	49	22
2 a.m. - 8 a.m.	14	7

Stating your hypotheses clearly, test at the $5 \%$ level of significance whether or not there is evidence of the proportion of serious injuries being different at different times of day.
(11 marks)

Edexcel S3 Q5

5. In a competition, a wine-enthusiast has to rank ten bottles of wine, $A$ to $J$, in order starting with the one he thinks is the most expensive. The table below shows his rankings and the actual order according to price.

Rank	1	2	3	4	5	6	7	8	9	10
Enthusiast	D	$C$	J	$A$	$G$	$F$	$B$	E	I	H
Price	A	$C$	D	$H$	$J$	B	$F$	I	$G$	$E$

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 positive correlation.
Explain briefly how you would have been able to carry out the test if bottles $B$ and $F$ had the same price.

Edexcel S3 Q6

6. A researcher collects data on the height of boys aged between nine and nine and-a-half years and their diet. The data on the height, $V$ cm, of the 80 boys who had always eaten a vegetarian diet is summarised by $$\Sigma V = 10367 , \quad \Sigma V ^ { 2 } = 1350314 .$$

Calculate unbiased estimates of the mean and variance of $V$. The researcher calculates unbiased estimates of the mean and variance of the height of boys whose diet has included meat from a sample of size 280, giving values of 130.5 cm and $96.24 \mathrm {~cm} ^ { 2 }$ respectively.
Stating your hypotheses clearly, test at the $1 \%$ level whether or not there is a significant difference in the heights of boys of this age according to whether or not they have a vegetarian diet.
(8 marks)

Edexcel S3 Q7

7. An examiner believes that once she has marked the first 20 papers the time it takes her to mark one paper for a particular exam follows a Normal distribution. Having already marked more than 20 papers for each of the $P 1$, M1 and S1 modules set one summer, the mean and standard deviation, in seconds, of the time it takes her to mark a paper for each module are as shown in the table below.

\cline { 2 - 3 } \multicolumn{1}{c\|}{}	Mean	Standard Deviation
P1	252	17
M1	314	42
S1	284	29

Find the probability that the difference in the time it takes her to mark two randomly chosen $P 1$ papers is less than 5 seconds.
(6 marks)
Find the probability that it takes her less than 10 hours to mark $45 M 1$ and $80 S 1$ papers. \section*{END}

Edexcel S3 Q1

A researcher wishes to take a sample of size 9 , without replacement, from a list of 72 people involved in the trial of a new computer keyboard. She numbers the people from 01 to 72 and uses the table of random numbers given in the formula book. She starts with the left-hand side of the sixth row of the table and works across the row. The first two numbers she writes down are 56 and 32 .
1. Find the other six numbers in the sample.
2. Give one advantage and one disadvantage of using random numbers when taking a sample.
  (2 marks)
3. The length of time that registered customers spend on each visit to a supermarket's website is normally distributed with a mean of 28.5 minutes and a standard deviation of 7.2 minutes.
Eight visitors to the site are selected at random and the length of time, $T$ minutes, that each stays is recorded.
Write down the distribution of $\bar { T }$, the mean time spent at the site by these eight visitors.
(2 marks)
Find $\mathrm { P } ( 25 < \bar { T } < 30 )$.
(4 marks)

Edexcel S3 Q3

3. The discrete random variable $X$ has the probability distribution given below.

$x$	2	4	7	$k$
$\mathrm { P } ( X = x )$	0.05	0.15	0.3	0.5

Find the mean of $X$ in terms of $k$.
Find the bias in using ( $2 \bar { X } - 5$ ) as an estimator of $k$. Fifty observations of $X$ were made giving a sample mean of 8.34 correct to 3 significant figures.
Calculate an unbiased estimate of $k$.
(2 marks)

Edexcel S3 Q4

4. The mass of waste in filled large dustbin bags is normally distributed with a mean of 6.8 kg and a standard deviation of 1.5 kg . The mass of waste in filled small dustbin bags is normally distributed with a mean of 3.2 kg and a standard deviation of 0.6 kg . One week there are 8 large and 3 small dustbin bags left for collection outside a block of flats. Find the probability that this waste has a total mass of more than 70 kg .
(7 marks)

Edexcel S3 Q5

5. For a project, a student is investigating whether more athletic individuals have better hand-eye coordination. He records the time it takes a number of students to complete a task testing coordination skills and notes whether or not they play for a school sports team. His results are as follows:

\cline { 2 - 4 } \multicolumn{1}{c|}{}

Stating your hypotheses clearly, test at the $5 \%$ level of significance whether or not there is evidence that those who play in a school team complete the task more quickly on average.
(8 marks)

Edexcel S3 Q6

6. Two schools in the same town advertise at the same time for new heads of English and History departments. The number of applicants for each post are shown in the table below.

\cline { 2 - 3 } \multicolumn{1}{c\|}{}	English	History
Highfield School	32	14
Rowntree School	48	26

Stating your hypotheses clearly, test at the $10 \%$ level of significance whether or not there is evidence of the proportion of applicants for each job being different in the two schools.
(11 marks) Turn over

Edexcel S3 Q7

7. A sports scientist wishes to examine the link between resting pulse and fitness. He records the resting pulse, $p$, of 20 volunteers and the length of time, $t$ minutes, that each one can run comfortably at 4 metres per second on a treadmill. The results are summarised by $$\Sigma p = 1176 , \quad \Sigma t = 511 , \quad \Sigma p ^ { 2 } = 70932 , \quad \Sigma t ^ { 2 } = 19213 , \quad \Sigma p t = 27188 .$$

Calculate the product moment correlation coefficient for these data.
Stating your hypotheses clearly, test at the $1 \%$ level of significance whether there is evidence of people with a lower resting pulse having a higher level of fitness as measured by the test.
State an assumption necessary to carry out the test in part (b) and comment on its validity in this case.
(2 marks)

Edexcel S3 Q8

8. A physicist believes that the number of particles emitted by a radioactive source with a long half-life can be modelled by a Poisson distribution. She records the number of particles emitted in 80 successive 5-minute periods and her results are shown in the table below.

No. of Particles	0	1	2	3	4	5 or more
No. of Intervals	23	32	14	8	3	0

Comment on the suitability of a Poisson distribution for this situation.
Show that an unbiased estimate of the mean number of particles emitted in a 5 -minute period is 1.2 and find an unbiased estimate of the variance.
Explain how your answers to part (b) support the fitting of a Poisson distribution.
Stating your hypotheses clearly and using a $5 \%$ level of significance, test whether or not these data can be modelled by a Poisson distribution. END

Edexcel S3 Q1

A charity has 240 volunteers and wishes to consult a sample of them of size 20 .
1. Explain briefly how a systematic sample can be taken using random numbers.
2. Give one advantage and one disadvantage of using systematic sampling compared with simple random sampling.
  (2 marks)
3. A teacher gives each student in his class a list of 30 numbers. All the numbers have been generated at random by a computer from a normal distribution with a fixed mean and variance. The teacher tells the class that the variance of the distribution is 25 and asks each of them to calculate a $95 \%$ confidence interval based on their list of numbers.
The sum of the numbers given to one student is 1419 .
Find the confidence interval that should be obtained by this student. Assuming that all the students calculate their confidence intervals correctly,
state the proportion of the students you would expect to have a confidence interval that includes the true mean of the distribution,
(1 mark)
explain why the probability of any one student's confidence interval including the true mean is not 0.95
(1 mark)

Edexcel S3 Q3

3. A newly promoted manager is present when an experienced manager interviews six candidates, $A , B , C , D , E$ and $F$ for a job. Both managers rank the candidates in order of preference, starting with the best candidate, giving the following lists: $$\begin{array} { l l } \text { Experienced Manager: } & B F A C E D
\text { New Manager: } & F C B D E A \end{array}$$

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 positive correlation.
Comment on whether the new manager needs training in the assessment of candidates at interview.
(1 mark)

Edexcel S3 Q4

4. A student collected data on the number of text messages, $t$, sent by 30 students in her year group in the previous week. Her results are summarised as follows: $$\Sigma t = 1039 , \quad \Sigma t ^ { 2 } = 65393 .$$

Calculate unbiased estimates of the mean and variance of the number of text messages sent by these students per week.
(4 marks)
Another student collected similar data for 20 different students and calculated unbiased estimates of the mean and variance of 32.0 and 963.4 respectively.
Calculate unbiased estimates of the mean and variance for the combined sample of 50 students.
(6 marks)

Edexcel S3 Q5

5. An organic farm produces eggs which it sells through a local shop. The weight of the eggs produced on the farm are normally distributed with a mean of 55 grams and a standard deviation of 3.9 grams.

Find the probability that two of the farm's eggs chosen at random differ in weight by more than 4 grams. The farm sells boxes of six eggs selected at random. The weight of the boxes used are normally distributed with a mean of 28 grams and a standard deviation of 1.2 grams.
Find the probability that a randomly chosen box with six eggs in weighs less than 350 grams.

Edexcel S3 Q6

6. A survey found that of the 320 people questioned who had passed their driving test aged under twenty-five, 104 had been involved in an accident in the two years following their test. Of the 80 people in the survey who were aged twenty-five or over when they passed their test, 16 had been involved in an accident in the following two years.

Draw up a contingency table showing this information. It is desired to test whether the proportion of drivers having accidents within two years of passing their test is different for those who were aged under twenty-five at the time of passing their test than for those aged twenty-five or over.
1. Stating your hypotheses clearly, carry out the test at the $5 \%$ level of significance.
2. Explain clearly why there is only one degree of freedom. It is found that 12 people who were aged under twenty-five when they took their test and had been involved in an accident in the following two years had been omitted from the information given.
Explain why you do not need to repeat the calculation to know the correct result of the test.
(2 marks)

Edexcel S3 Q7

7. A shoe manufacturer sees a report from another country stating that the length of adult male feet is normally distributed with a mean of 22.4 cm and a standard deviation of 2.8 cm . The manufacturer wishes to see if this model is appropriate for his customers and collects data on the length, correct to the nearest cm, of the right foot of a random sample of 200 males giving the following results:

Length (cm)	$\leq 18$	$19 - 21$	$22 - 24$	$25 - 27$	$\geq 28$
No. of Men	24	48	69	41	18

The expected frequencies for the $\leq 18$ and $19 - 21$ groups are calculated as 16.46 and 58.44 respectively, correct to 2 decimal places.

Calculate expected frequencies for the other three classes.
Stating your hypotheses clearly, test at the $10 \%$ level of significance whether or not this data can be modelled by the distribution $\mathrm { N } \left( 22.4,2.8 ^ { 2 } \right)$.
(7 marks)
The manufacturer wishes to refine the model by not assuming a mean and standard deviation.
Explain briefly how the manufacturer should proceed. \section*{END}

Edexcel S3 Q1

A Veterinary Surgeon wishes to survey a stratified sample of size 100 from those people who have pets registered at her surgery. The list below shows the strata to be used and the number in each group.

people who own just dogs - 165 ,
people who own just cats - 140 ,
people who own just small mammals - 105,
others, including those who own more than one type of pet - 90 .
1. Find how many members of each group should be included in the sample.
2. Give two advantages of using stratified sampling.

Edexcel S3 Q2

A psychologist is investigating the numbers people choose when asked to pick a number at random in a given interval. He finds that when asked to pick a number between 0 and 100 people are less likely to pick certain numbers, such as multiples of ten. He believes, however that if people are asked to pick an odd number between 0 and 100 they are equally likely to pick a number ending in any of the digits $1,3,5,7$ or 9 .

To test this theory he asks 80 people to pick an odd number between 0 and 100 and records the last digit of the numbers chosen. His results are shown in the table below.

Last Digit	1	3	5	7	9
Frequency	16	20	14	17	13

Stating your hypotheses clearly and using a 10\% level of significance test the psychologist’s theory.
(9 marks)

Edexcel S3 Q3

3. A clothes manufacturer wishes to find out if adult females have become taller on average since twenty years ago when their mean height was 5 ft 6 inches. Studies over time have shown that the standard deviation of the height of adult females has been fairly constant at 2.3 inches. The manager wishes to test if the mean height is now more than 5 ft 6 inches and takes a sample of 150 adult females.

Stating your hypotheses clearly, find the critical region for the mean height of the sample for a test at the $5 \%$ level of significance. The total height of the females in the sample is 832 ft .
Carry out the test making your conclusion clear.

Edexcel S3 Q4

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.

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

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

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

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

Rank	1	2	3	4	5	6	7	8	9	10
Enthusiast	D	\(C\)	J	\(A\)	\(G\)	\(F\)	\(B\)	E	I	H
Price	A	\(C\)	D	\(H\)	\(J\)	B	\(F\)	I	\(G\)	\(E\)

Length (cm)	\(\leq 18\)	\(19 - 21\)	\(22 - 24\)	\(25 - 27\)	\(\geq 28\)
No. of Men	24	48	69	41	18

\(x\)	2	4	7	\(k\)
\(\mathrm { P } ( X = x )\)	0.05	0.15	0.3	0.5

Questions — Edexcel S3 (313 questions)

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