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

Practice Paper Questions

1a
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4 marks

The continuous random variable X has probability density function straight f open parentheses x close parentheses given by

straight f open parentheses x close parentheses equals open curly brackets table row cell 1 fifth open parentheses x minus 1 close parentheses end cell cell 1 less or equal than x less or equal than k end cell row cell 2 minus 2 over 5 x end cell cell k less than x less or equal than 4 end cell row 0 cell otherwise. end cell end table close

Find the value of k.

1b
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2 marks

Find P left parenthesis X less than 3.5 right parenthesis.

1c
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3 marks

Show that straight E left parenthesis X right parenthesis equals 3.

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2a
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2 marks

The diagram below shows the probability density function, f open parentheses t close parentheses, of a random variable T.

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For a less or equal than t less or equal than b comma space space f left parenthesis t right parenthesis equals 3 over 32 left parenthesis 10 t minus t squared minus 21 right parenthesis, elsewhere f left parenthesis t right parenthesis equals 0.

Find the values of a space and space b.

2b
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4 marks

State the value of E left parenthesis T right parenthesis and find V a r left parenthesis T right parenthesis.

2c
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1 mark

Find P left parenthesis negative 2 less or equal than T less or equal than 5 right parenthesis.

2d
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2 marks

Given that P left parenthesis 4 less or equal than T less or equal than 6 right parenthesis equals 11 over 16, find P left parenthesis T greater or equal than 6 right parenthesis.

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3a
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2 marks

A snowboarder is trying to perform the Poptart trick.
The snowboarder has a success rate of 25% of completing the trick.

The snowboarder will model the number of times they can expect to successfully complete the Poptart trick, out of their next 12 attempts, using the random variable X tilde B left parenthesis 12 comma 0.25 right parenthesis.

(i)
Give a reason why the model is suitable in this case.

(ii)
Suggest a reason why the model may not be suitable in this case.
3b
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2 marks

Using the model, find the probability that the snowboarder

(i)
successfully completes the Poptart trick more than 3 times in their next 12 attempts

(ii)
fails to successfully complete the trick on any of their next 12 attempts.

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4a
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1 mark

The random variable W tilde B left parenthesis 1200 comma 0.6 right parenthesis.

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

4b
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7 marks

Find, using the appropriate normal approximation:

(i)
P left parenthesis 700 less than W less or equal than 730 right parenthesis
(ii)
P left parenthesis W greater or equal than 719 right parenthesis
4c
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4 marks

Using the normal approximation, find the largest value of k (where k is an integer) such that  P left parenthesis left parenthesis 720 minus k right parenthesis less than W less than left parenthesis 720 plus k right parenthesis right parenthesis less than 0.5.

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5a
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4 marks

Whilst writing an essay, Gamu notices that she makes spelling mistakes at a rate of 7 for every 150 words. Gamu models the number of spelling mistakes she makes using a Poisson distribution.

Find the maximum number of words Gamu can write before the probability of her making a spelling mistake exceeds 0.75.

5b
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7 marks

Gamu is asked to write three short essays by her lecturer. She writes one containing 100 words, one containing 200 words and one containing 250 words. An essay is returned by Gamu’s lecturer if more than 1% of its words contain spelling mistakes.

Find the probability that all three short essays are returned.

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6a
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5 marks

It is thought that 2% of all giant huntsman spiders have a leg span of over 30 cm when fully grown.  Residents in a village in Laos believe that in their village the proportion of these spiders with a leg span of over 30 cm is different to 2%.  To test their theory, they gather a sample of 500 fully grown giant huntsman spiders, measure their leg spans and then release them back into the wild.

Using an approximating distribution, find the critical region for the residents’ hypothesis test using a 10% significance level. Clearly state your sampling distribution and hypotheses

6b
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1 mark

Using the approximating distribution, estimate the probability that the residents will incorrectly conclude that the proportion of spiders in their village, with a leg span of over 30 cm, is not 2%.

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7a
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3 marks

The continuous random variable, X, has probability density function straight f open parentheses x close parentheses given by

straight f left parenthesis x right parenthesis equals open curly brackets table row cell x over 5 end cell cell 0 less than x less or equal than 2 end cell row cell fraction numerator 10 minus 2 x over denominator 15 end fraction end cell cell 2 less than x less or equal than 5 end cell row 0 cell otherwise. end cell end table close

Sketch the probability density function of X.

7b
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1 mark

Write down the mode of X.

7c
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7 marks

Fully define the cumulative distribution function straight F open parentheses x close parentheses of X.

7d
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3 marks

Find the median of X.

7e
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2 marks

Comment on the skewness of the distribution of X. Justify your answer.

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8a
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1 mark

A chocolatier produces his trademark homemade chocolate truffle, the Deliciously Decadent, in batches of 50.  He always tastes two randomly-selected chocolates from each batch to check the quality of the batch.

Explain why the chocolatier takes a sample rather than a census to check for quality.

8b
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2 marks

Suggest a suitable sampling frame and identify the sampling units.

8c
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2 marks

If a chocolate is up to standard, the chocolatier assigns it the number 1.  If not then the chocolate is assigned the number 0.

Using this numbering system, list all of the possible samples that could result when the chocolatier quality tests a batch of chocolates.

8d
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3 marks

After conducting many quality tests the chocolatier finds that 5% of chocolates are not up to standard, regardless of which batch they have come from.

Using your answer from part (c), and clearly defining any random variables you use, find the sampling distribution of the mean value of the sample when quality testing a single batch of chocolates.

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