Practice Paper Statistics 1 (Edexcel International A Level Maths: Pure 4)

An A level music teacher is collecting data on the number of hours his students spend rehearsing their  final piece, , and the number of mistakes made in their exam, .  He calculates the following summary data of ten of his students.

Show that the product moment correlation coefficient for these data is , correct to 3 decimal places. 

The music teacher calculates the equation of the regression line of  on  to be  .

Show that  correct to 3 significant figures and find the value of .

Tim has just moved to a new town and is trying to choose a doctor’s surgery to join, HealthHut or FitFirst. He wants to register with the one where patients get seen faster. He takes of sample of 150 patients from HealthHut and calculates the range of waiting times as 45 minutes and the variance as 121 minutes².

An outlier is defined as a value which is more than 2 standard deviations away from the mean.

Prove that the sample contains an outlier.

Tim finds out that the outlier is a valid piece of data and decides to keep the value in his sample.

Which pair of statistical measures would be more appropriate to use when using the sample to compare the doctor’s surgeries: the mean and standard deviation or the median and interquartile range? Give a reason for your answer.

The box plots below show the waiting times for the two surgeries.

Given that there is only one outlier for HealthHut, label it on the box plot with a cross (×).

Compare the two distributions of waiting times in context.

An advertisement for a charity is shown on TV at the same time every weekday for four weeks.  To assess the impact of the advert, the charity’s manager decides to record the number of donations the charity receives each day in the hour after the advert is broadcast.  The results are listed below: 

                                    21       27       24       31       17

                                    22       25       26       27       9

                                    32       29       25       24       40

                                    23       22       19       12       14 

Represent these data in a sorted stem-and-leaf diagram.

The manager decides that the advert is not cost effective unless the median number of donations per day in the hour after broadcast is at least 25. Determine whether the manager should continue to run the TV advert.

Give one advantage of using a stem-and-leaf diagram as opposed to grouping the data into a frequency table.

A pharmacy sells face masks in a variety of sizes.  Their sales over a week are recorded in the table below:

Rosco is a somewhat inept rural county sheriff who frequently finds himself involved in car chases with well-meaning local entrepreneurs.  During any given car chase, Rosco inevitably runs into one of three obstacles – a damaged bridge (with probability 0.47), an oil slick (with probability 0.32), or a pigpen at the end of a dead-end road.

If he encounters a damaged bridge there is a 25% chance that he will make it across safely; otherwise he lands in the river and ends up covered in mud.  If he encounters an oil slick there is a 40% chance that his car will spin around and he will end up continuing his hot pursuit in the wrong direction; otherwise he goes off the road into a farm pond and ends up covered in mud.  If he encounters a pigpen at the end of a dead-end road there is a 15% chance he will stop his car in time; otherwise he drives into the muddy end of the pigpen while the pigs sit at the other end laughing.  If he drives into the muddy end of a pigpen there is a 20% chance he will only end up covered in mud; otherwise he ends up covered in mud and other things that are found in pigpens.

Draw a tree diagram to represent this information.

Find the probability that in the course of a randomly chosen car chase

Given that Rosco ends up covered in mud in the course of a randomly chosen car chase, find the probability that he didn’t encounter an oil slick. Give your answer as an exact value.

In the course of a particular day Rosco finds himself engaged in three separate car chases with well-meaning local entrepreneurs.  The car chases may be considered to be independent events.

Determine the probability that on that day Rosco will not end up covered in other things that are found in pigpens.

A spinner has three sectors labelled 0, 1 and 2. Let  be the random variable denoting the number the spinner lands on when spun. The probability distribution table for  is shown below: 

It is given that  and . 

Write down the value of .

Find the values of .

Susie spins the spinner twice and adds together the two numbers to calculate her score, . Tommy spins the spinner once and doubles the number to calculate his score, . Each spin of the spinner is independent of all other spins. 

Draw up the probability distribution table for:

,

.

Which player is most likely to get a score that is bigger than 2?

A biased four-sided dice is rolled and the number that it lands on is denoted   which has probability distribution shown below.

The cumulative distribution function of  is given by:

Find the values of and .

Saskia and Tamara place a game by rolling the dice. Saskia’s score is the number the dice lands on and Tamara’s score is the reciprocal of that number.

Find the probability that Saskia’s score is bigger than Tamara’s score.

Paul enjoys solving sudoku puzzles. The lengths of time he spends on sudokus in a week are normally distributed with a mean of 2048 minutes and a standard deviation of 64 minutes.

Find the probability that in a given month Paul spends less than 1945 minutes solving sudoku puzzles.

Estimate the number of weeks in a year that Paul spends between 2019 and 2091 minutes solving sudoku puzzles.

Assuming it takes Paul exactly 10 minutes to solve any sudoku puzzle, find the greatest number, , of sudoku puzzle such that the probability of Paul solving less than  puzzles in a week is less than 0.01.