Practice Paper 6 (Probability & Statistics 2) (CIE A Level Maths: Pure 3)

Remy has a biased coin with the probability of it landing on tails being 0.05.

Using a Poisson approximation, find the probability that the coin lands on tails more than 4 times when Remy flips it 60 times.

Remy flips the coin 2000 times.

Using a normal approximation, find the probability that the coin lands on tails fewer than 120 times when Remy flips it 2000 times.

Roger has a keen fascination with rabbits that live on Easter Island, in particular the lengths of their ears. The mean length of a rabbit’s ear on Easter Island is cm. He takes a random sample of 40 rabbits and finds that a 90% confidence interval for is 8.3 to 9.9 cm.

Using the same sample, find a 99% confidence interval for . Give the end-points correct to 1 decimal place.

Roger wants to find a confidence interval for  with a width of 1.28 cm. Find the confidence level that Roger should use.

Roger takes 100 random samples and calculates a 95% confidence interval for each one. How many of the confidence intervals would be expected to contain the true population mean, .

The continuous random variable, , has probability density function

Show that .

The random variable is approximated by .

Explain why this approximation is valid and state the value of .

Hence find the percentage error when a Poisson distribution is used to approximate .

The distributions of the independent random variables  are  respectively.

Find the probability that

Explain why your answers to (a) part (i) and (ii) are different.

The standard normal distribution is denoted by .

(i)

Write down a formula that links the standard normal distribution, , to the distribution  .

The population mean of the random variable is being tested using a null hypothesis  against the alternative hypothesis  .  A random sample of  observations is taken from the population and the sample mean is calculated as 28.

Using a 5% level of significance, there is not enough evidence to reject the null hypothesis.

Show that

Hence find the greatest possible value for the sample size,  .