Practice Paper Statistics 2 (Edexcel International A Level Maths: Pure 2)

Practice Paper Questions

1a
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2 marks

For a jellyfish population in a certain area of the ocean, there is a 95% chance that any given jellyfish contains microplastic particles in its body.

State any assumptions that are required to model the number of jellyfish containing microplastic particles in their bodies in a sample of size n as a binomial distribution.

1b
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3 marks

Using this model, for a sample size of 40, find the probability of

(i)
exactly 38 jellyfish
(ii)
all the jellyfish

having microplastic particles in their bodies.

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2a
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4 marks

A “double yolker” is an egg which contains two yolks.  It is known that the probability of a chicken laying a double yolker is 0.1%.  A chicken farmer, Paolo, claims that double yolkers are rarer than the stated 0.1%.  To test his claim, Paolo records that his chickens lay 1217 eggs in a month and he uses these as his sample.  He discovers that none of these eggs are double yolkers.

Test, at the 5% level of significance, whether there is evidence to support Paolo’s claim that double yolkers are rarer than 0.1%. State your hypotheses clearly.

2b
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4 marks

Paolo decides to take a larger sample so extends his test to six months.  During this time, a sample of 7300 eggs is formed and two of them are double yolkers.

Use a suitable approximation to show that there is evidence, at the 5% level of significance, to support Paolo’s claim that double yolkers are rarer than 0.1%.

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3a
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2 marks

A high school principal claims that lost phones are handed in to the school office at an average rate of 1.3 phones per day.  The school secretary believes that the rate is higher than this, so she conducts a hypothesis test at the 10% significance level over a 5-day period and finds that 10 phones are handed in to the school office during that time.

State suitable null and alternative hypotheses for the secretary’s test.

3b
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4 marks

Test, at the 10% level of significance, whether there is evidence to support the secretary’s belief.

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4a
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4 marks

The random variable X space tilde P o left parenthesis 10 right parenthesis.

Find:

(i)
P left parenthesis X less than 8 right parenthesis
(ii)
P left parenthesis X greater or equal than 11 right parenthesis
(iii)
P left parenthesis 5 less or equal than X less or equal than 14 right parenthesis.
4b
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2 marks

Find the largest integer k such P left parenthesis X greater than k right parenthesis greater than 0.25.

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5a
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3 marks

The continuous random variable X has probability density function

straight f open parentheses x close parentheses equals open curly brackets table row cell k over x cubed end cell row 0 end table close space space space space space space space space space space space space space space space space space table row cell 2 less or equal than x less or equal than a end cell row cell otherwise. end cell end table

Show that k equals fraction numerator 8 a squared over denominator a squared minus 4 end fraction.

5b
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5 marks

Given that straight E left parenthesis X right parenthesis equals 20 over 7 comma show that a equals 5 and hence find the value of k.

5c
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3 marks

Find the exact value of Var left parenthesis X right parenthesis .

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6a
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4 marks

Frank is the owner of a factory which has recently opened near a school where Hilda is the headteacher.  Before the factory opened, the attendance rate at the school was good 90% of the time.  Hilda claims that the proportion of days when the attendance rate is good has decreased and she suspects this is due to the fumes from the factory making the children sick.  Frank disagrees and claims that the factory has made no difference to the attendance rate.  To test their claims a sample of 40 days is taken and on 32 days the attendance rate is good.

If space X tilde B left parenthesis 40 comma 0.9 right parenthesis then:

P left parenthesis X less than 32 right parenthesis equals 0.015495 space
P left parenthesis X equals 32 right parenthesis equals 0.026407 space
P left parenthesis X greater than 32 right parenthesis equals 0.958098

Stating your hypotheses clearly, test Hilda’s claim using a 5% level of significance. Give your answer in context.

6b
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1 mark

Explain whether the outcome of the test supports Hilda’s suspicion.

6c
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3 marks

Stating your hypotheses clearly, test Frank’s claim using a 5% level of significance.

6d
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1 mark

Suggest a reason why Frank might have chosen to use a two-tailed test.

6e
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2 marks

State, with a reason, whether a one-tailed test or a two-tailed test would have been more appropriate for this scenario.

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7a
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4 marks

The continuous random variable, X, has a cumulative distribution function straight F open parentheses x close parentheses given by

straight F open parentheses x close parentheses equals open curly brackets table row 0 cell x less than 3 end cell row cell a x squared plus b x end cell cell 3 less or equal than x less or equal than 5 end cell row 1 cell x greater than 5 end cell end table close

Show that a equals 1 over 10 and find the value of b.

7b
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3 marks

Fully define the probability density function straight f open parentheses x close parentheses.

7c
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1 mark

Write down the mode of X.

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8a
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2 marks

Due to a manufacturing irregularity, 41% of Adventure Dude action figures were produced with two left hands.  Although not especially rare, and therefore not especially collectible, these so-called ‘double left’ figures are nonetheless considered to be collector’s items by hard-core Adventure Dude fanatics.

A vintage toy shop has obtained 100 Adventure Dude action figures.  These may be assumed to represent a random sample.

Find the probability that exactly 45 of the 100 figures are ‘double left’ figures. Give your answer to 6 decimal places.

8b
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4 marks

Use an appropriate normal approximation to approximate the probability that exactly 45 of the 100 figures are ‘double left’ figures.

8c
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2 marks

Find the percentage error when using your normal approximation from part (b) to estimate the probability that exactly 45 of the 100 figures are ‘double left’ figures. Give your answer correct to two decimal places.

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9a
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1 mark

The random variable W tilde B left parenthesis 980 comma 0.4 right parenthesis.

Explain why a normal distribution can be used to approximate W.

9b
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7 marks

Find, using the appropriate normal approximation:

(i)
P left parenthesis 386 less than W less than 398 right parenthesis
(ii)
P left parenthesis W greater or equal than 400 right parenthesis
9c
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4 marks

Using the appropriate normal approximation, find the smallest value of k (where k element of straight integer numbers)  such that  P left parenthesis k less than W less than left parenthesis 784 minus k right parenthesis right parenthesis less than 0.5.

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