Practice Paper 6 (Probability & Statistics 2) (CIE A Level Maths: Probability & Statistics 1)

Practice Paper Questions

1a
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2 marks

In England, it is known that 5% of the population have ginger hair. Kenneth, the owner of a hairdressing salon, has 40 appointments available each day. He models the number of clients with ginger hair that attend his salon in a day using a binomial distribution.

State the assumptions that Kenneth has made by using a binomial distribution.

1b
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2 marks

Assuming all 40 appointments are filled, find the probability that at least one person has ginger hair.

1c
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2 marks

Assuming all 40 appointments are filled each day, the number of clients with ginger hair attending an appointment over a two-day period is denoted G. It can be assumed that clients on each day are independent of each other. Explain why a Poisson distribution can be used to approximate G. State the appropriate Poisson distribution.

1d
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3 marks

Using the Poisson distribution, find the probability at most 5 clients with ginger hair will have an appointment at the salon over the two-day period.

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2a
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3 marks

The masses, M grams, of a random sample of 80 potatoes from a farm are summarised as follows.

n equals 80 space space space space space space space space space space space space sum m equals 17764 space space space space space space space space space space space sum m squared equals 4103225

Calculate unbiased estimates of the population mean and variance.

2b
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3 marks

Calculate a 90% confidence interval for the population mean.

2c
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1 mark

Explain why it was necessary to use the Central Limit theorem in your answer to part (b).

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3a
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4 marks

Frank has a variable tariff for his electricity and gas bills. His monthly electricity bill is $E and his monthly gas bill is $G. E space and space G are independent random variables with distributions N left parenthesis 85 comma 9.4 squared right parenthesis space and space N left parenthesis 53 comma 12.45 right parenthesis respectively.

Find the probability that the total electricity and gas bill in a month exceeds $150.

3b
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5 marks

Frank has a part-time job tutoring college students. His monthly income from this job can be modelled as a Normal distribution with mean $504 and standard deviation $41. Frank uses this income to pay for his gas and electricity bills, he puts the remaining money into his partner’s bank account each month.

(i)
Find the probability that Frank puts between $350 and $450 into his partner’s bank account in a month.
(ii)
What assumption did you make in part (b)(i) regarding his income and his bills.

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4a
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3 marks

The continuous random variable X follows a continuous uniform distribution over the interval left parenthesis 3 comma 28 right parenthesis so that its probability density function is

f left parenthesis x right parenthesis equals open curly brackets table row cell 1 over 25 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space 3 less or equal than x less or equal than 28 end cell row cell 0 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space o t h e r w i s e end cell end table close

Find

(i)
P left parenthesis X less than 15 right parenthesis
(ii)
P left parenthesis 20 less than X less than 30 right parenthesis
(iii)
P left parenthesis X equals 25 right parenthesis
4b
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3 marks
(i)
Write down E open parentheses X close parentheses.
(ii)
Find V a r space open parentheses X close parentheses.

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5a
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3 marks

The number of mistakes made by a student, Priya, in a 20-minute revision period is modelled as a Poisson distribution with mean of 1.2. The number of mistakes made by a different student, Qays, in a 30-minute revision period is modelled as a Poisson distribution with a mean of 2.2.

Find the probability that Priya makes exactly 2 mistakes and Qays makes exactly 1 mistake within a one-hour revision period. Write your answer in the form a over b e to the power of c where a comma space b comma space and space c are integers to be found. State any assumptions that are needed.

5b
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4 marks

The number of mistakes made by Priya and Qays in a one-hour revision period are added together. Given that they make exactly 9 mistakes in total in a one-hour revision period, find the probability that Priya made exactly 5 mistakes in that same revision period.

5c
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2 marks

Given that Priya makes exactly 5 mistakes in a one-hour revision period, find the probability that Priya and Qays made exactly 9 mistakes in total in that same revision period.

5d
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3 marks

In a week before a test, Priya has ten 30-minute revision periods. Estimate the number of these revision periods during which Priya will make a mistake.

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6a
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1 mark

The IQ of a student at Calculus High can be modelled as a random variable with the distribution straight N left parenthesis 126 comma 50 right parenthesis . The headteacher decides to play classical music during lunchtimes and suspects that this has caused a change in the average IQ of the students.

Write suitable null and alternative hypotheses to test the headteacher’s suspicion.

6b
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5 marks

The headteacher selects 10 students and asks them to complete an IQ test.  Their scores are:

127, 127, 129, 130, 130, 132, 132, 132, 133, 138

Test, at the 5% level of significance, whether there is evidence to support the headteacher’s suspicion.

6c
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1 mark

It was later discovered that the 10 students used in the sample were all in the same advanced classes.

Comment on the validity of the conclusion of the test based on this information.

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