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

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.

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%.

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.

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

The random variable .

Find:

.

Find the largest integer  such .

The continuous random variable  has probability density function

Show that 

Given that  show that  and hence find the value of .

Find the exact value of .

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  then:

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

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

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

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

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

The continuous random variable, , has a cumulative distribution function  given by

Show that  and find the value of .

Fully define the probability density function .

Write down the mode of .

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.

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

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.

The random variable .

Explain why a normal distribution can be used to approximate .

Find, using the appropriate normal approximation:

Using the appropriate normal approximation, find the smallest value of (where )  such that  .