Practice Paper Statistics 2 (Edexcel International A Level Maths: Pure 1)

Chloe’s sister likes to play a game in which she hides a coin in one of her hands and asks Chloe to guess which hand it is in.  Chloe claims that she can predict which hand the coin is hidden in more often than not.  Her sister thinks Chloe is just guessing so she decides to conduct a hypothesis test to test Chloe’s claim.

State suitable null and alternative hypotheses for Chloe’s sister’s test.

They play the game 100 times, and Chloe guesses correctly 60 times.

Using a suitable approximation, test at the 5% significance level whether or not Chloe’s claim is justified.

Justify the use of your approximation in part (b).

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

Write down an expression for  in terms of .

Write down an expression for  in terms of .

Find a simplified expression for  in terms of .

Given that  find the value of .

Harry is trying to draw specific lengths without measuring equipment. He draws a straight line and stops when he thinks it is 10 cm long. The actual length of Harry’s line, cm, can be modelled by the probability density function

Estimate the lower quartile of the lengths of Harry’s lines.

Harry tries to draw a 10 cm line 80 times.

Write down how many of Harry’s line you would expect to be less than the lower quartile.

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

Explain why .

Show that .

Show that  and hence find the values of  and .

Write down the median of .

Find the lower quartile and the upper quartile of .

Describe the skewness of . Justify your answer.

Grace, a grumpy toddler, attends nursery five days a week. The number of tantrums that Grace has in a day follows a Poisson distribution with variance 3.14.

Find the probability that Grace has exactly 17 tantrums during a week at nursery.

Find the probability that Grace has fewer than four tantrums in a two-day period at nursery.

Given that Grace has fewer than four tantrums at nursery one day, find the probability that she had no tantrums at nursery that day.

A walk-in vaccination centre can model the number of people arriving every 5 minutes by the random variable  with distribution Po(0.6).

State two assumptions required for the Poisson model to be valid.

The centre launches a new advertising campaign and conducts a hypothesis test to see if it has increased the number of people arriving for a vaccine. They choose a half hour period at random and find that 7 people arrive for a vaccine during this time.

Clearly stating your hypotheses, test at the 10% significance level whether the number of arrivals has increased.

A teacher keeps bronze, silver and gold star stickers as prizes for his students.  The teacher takes a sample of five stickers at random from a full pack.

The school office contains a large number of the bronze, silver and gold star stickers in the ratio 5:3:2. At the end of each term the students are awarded 50 points for a gold star, 20 points for a silver star and 10 points for a bronze star. Their final score is the product of the points their stars are worth. The teacher takes a sample of 3 stickers at random to show his students how to find the product of the points their stickers are worth.

Given that the teacher chose his sample only from bronze and silver stars, write down the sampling distribution of the product of the values the stickers are worth. Clearly define your random variable and statistic.

An amusement park has a challenge where guests attempt to remain on a mechanical bull for as long as possible. The times, in seconds, that the guests stay on the mechanical bull can be modelled using a continuous random variable, , with probability density function .

Specify fully the cumulative distribution function .

Find the median length of time that a guest is able to stay on the mechanical bull.

Given that a guest has remained on the mechanical bull for 5 seconds, find the probability that the guest will still be on the mechanical bull after another 5 seconds.