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

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

Calculate unbiased estimates of the population mean and variance.

Calculate a 90% confidence interval for the population mean.

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

Frank has a variable tariff for his electricity and gas bills. His monthly electricity bill is $E and his monthly gas bill is $G.  are independent random variables with distributions  respectively.

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

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.

The continuous random variable  follows a continuous uniform distribution over the interval  so that its probability density function is

Find

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  where  are integers to be found. State any assumptions that are needed.

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.

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.

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.

The IQ of a student at Calculus High can be modelled as a random variable with the distribution . 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.

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.

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.