Practice Paper Statistics 2 (Edexcel International A Level Maths: Pure 3)

The diagram below shows the probability density function, , of a random variable .

For , elsewhere .

Find the values of .

State the value of  and find .

Find .

Given that , find .

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

Using the model, find the probability that the snowboarder

The random variable .

Explain why a normal distribution can be used to approximate .

Find, using the appropriate normal approximation:

Using the normal approximation, find the largest value of  (where  is an integer) such that  .

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.

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.

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

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%.

The continuous random variable, , has probability density function  given by

Sketch the probability density function of .

Write down the mode of .

Fully define the cumulative distribution function  of .

Find the median of .

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

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.

Suggest a suitable sampling frame and identify the sampling units.

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.

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.