Questions S3 - A-Level Maths

\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

Edexcel S3 Q1

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

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

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

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

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

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

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).

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

	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

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