Statistical Measures (Edexcel A Level Maths: Statistics): Exam Questions

Using the data from the large data set, Simon produced the following summary statistics for the daily mean air temperature, °C, for Beijing in 2015

Show that, to 3 significant figures, the standard deviation is 5.19 °C.

Each member of a group of 27 people was timed when completing a puzzle.

The time taken, minutes, for each member of the group was recorded.

For these 27 people and

Calculate the mean time taken to complete the puzzle.

Calculate the standard deviation of the times taken to complete the puzzle.

Dian uses the large data set to investigate the Daily Total Rainfall, mm, for Camborne.

Write down how a value of is recorded in the large data set.

Dian uses the data for the 31 days of August 2015 for Camborne and calculates the following statistics

Use these statistics to calculate

(i) the mean of the Daily Total Rainfall in Camborne for August 2015,

(ii) the standard deviation of the Daily Total Rainfall in Camborne for August 2015.

Fran sits three Maths papers and six Science papers during her final A Level exams.  She achieves a mean score of 62 across the three Maths exam papers, and needs an overall mean score of 78 across all nine papers to get into her chosen University.  After getting the results of four out of her six Science papers, her mean score in Science is 84.5.

Given that each of the nine papers is weighted equally when working out the mean scores, calculate the mean score she must achieve on her final two science papers in order to gain a place at University.

Coffee4Life manufactures reusable coffee cups out of coffee plant waste.  Coffee cups are tested to see how many times they can be used before they begin to disintegrate.  A sample of 15 cups are tested, giving the following results for numbers of uses:

31    36    41    43    47

49    51    56    58    62

62    63    68    69    72

(i) Write down the modal number of times a cup can be used.

(ii) Find the values of the lower quartile, median and upper quartile.

The advertising department at Coffee4Life designs an advert which says;

“If used once a day,   of our cups last longer than 9 weeks.”

Explain the mistake that the advertising department has made, and state how the advert could be reworded to make it correct.

A machine is set to fill sacks of potatoes to a weight of 50 kg.  In a random sample, the masses, in kg, of seven sacks of potatoes were recorded.

The values are coded by subtracting 50 kg from the masses and then halving the new masses.

The mean of the coded data is 3.96 kg.

Calculate the mean mass of the seven sacks of potatoes in the sample.

The standard deviation of the coded data is 2.57 kg.

Calculate the standard deviation of the masses of the seven sacks of potatoes in the sample.

A random sample of 50 students were asked how long they spent revising for their Maths exam in the 24 hours before the exam.  The results are shown in the table below:

For this data, use linear interpolation to show that an estimate for the median is 169 minutes to the nearest minute.

Using  to represent the mid-point of each class, and .

Estimate the mean and the standard deviation of the amount of time students spent revising.

The speeds (s), to the nearest mile per hour, of 80 vehicles passing a speed camera were recorded and are grouped in the table below. 

(i)

Write down the modal class for this data.

(ii)

Write down the class group that contains the median.

(i) Assuming that ≥35 means ‘at least 35 mph but less than 40 mph’, calculate an estimate for the mean speed of the 80 vehicles.

(ii) It is now discovered that ≥35 means ‘at least 35 mph but less than 60 mph’. Without further calculation, state with a reason whether this would cause an increase, a decrease or no change to the value of the estimated mean.

Stav is studying the large data set for September 2015.

He codes the variable Daily Mean Pressure, , using the formula .

The data for all 30 days from Hurn are summarised by

State the units of the variable .

Find the mean Daily Mean Pressure for these 30 days.

Find the standard deviation of Daily Mean Pressure for these 30 days.

Ben is studying the Daily Total Rainfall, x mm, in Leeming for 1987.

He used all the data from the large data set and summarised the information in the following table.

Explain how the data will need to be cleaned before Ben can start to calculate statistics such as the mean and standard deviation.

Using all 184 of these values, Ben estimates and

Calculate estimates for

(i) the mean Daily Total Rainfall,

(ii) the standard deviation of the Daily Total Rainfall.

Ben suggests using the statistic calculated in part (b)(i) to estimate the annual mean Daily Total Rainfall in Leeming for 1987.

Using your knowledge of the large data set,

(i) give a reason why these data would not be suitable,

(ii) state, giving a reason, how you would expect the estimate in part (b)(i) to differ from the actual annual mean Daily Total Rainfall in Leeming for 1987.

The table below gives information about the ages of passengers on an airline.

There were no passengers aged 90 or over.

Use linear interpolation to estimate the median age.

A veterinary nurse records the masses of puppies (in kg) at birth and again at their eight-week check-up.  The table below summarises the gains in mass of 50 small breed puppies over their first eight weeks.

Use linear interpolation to estimate the median of the weight gain of the 50 puppies.

Give a reason why it is not possible to determine the exact median for this data.

The veterinary nurse decides to monitor any puppies whose gain in mass during their first 8 weeks was less than 0.8 kg. 

(i) Estimate the number of puppies whose gain in mass is below 0.8 kg.

(ii) Explain the assumption you have made in part (b)(i) and why the vet would need more information before determining for certain how many puppies would need to be monitored.

A machine is set to fill sacks of potatoes to a target weight of 50 kg, although the actual weight of the sacks () can vary from that target.  

To test the accuracy of the machine, a random sample of 20 sacks is taken and the values of are recorded.  

The mean and standard deviation of are found to be -1.8 and 3.1 respectively. 

Write down the mean and standard deviation of .

Calculate the value of

(i)

(ii) ²

Another 10 sacks of potatoes are sampled and the mean weight of these is found to be 51.2 kg. 

Calculate the mean of all 30 sacks of potatoes.

Comment on the accuracy of the machine.

The ages, years, of 200 people attending a vaccination clinic in one day are summarised by the following:    and  .

Calculate the mean and standard deviation of the ages of the people attending the clinic that day.

One person choose not to get the vaccine.

The mean of the 199 people who got the vaccine is exactly 36. 

(i) Calculate the age of the person who did not get the vaccine.

(ii) Calculate the standard deviation of the ages of the 199 people who got the vaccine.

, ,  and  are 4 integers written in order of size, starting with the smallest. 

The sum of ,  and is 70
The mean of , ,  and  is 25
The range of the 4 integers is 14.

Work out the median of , ,  and 

Whilst in lockdown, 100 people were asked to record the length of time, rounded to the nearest minute, that they spent exercising on a particular day. 

The results are summarised in the table below:

Using the midpoints, an estimate of mean time spent exercising based on this table is 35.4 minutes.

Find the values of and .

Two friends, Anna and Connor, are playing a gaming app on their phones.  As they play, they can choose from three different booster options.  They are unaware that each of the three options are charging them automatically from their mobile accounts.  The number of in-app purchases they each make are shown in the table below.

(i) The mean and standard deviation of the cost of Anna’s in-app purchases are £0.50 and £0 respectively.  Write down the cost of a single in-app purchase to ‘Level-up’.

(ii) Given that the mean cost of Connor’s in-app purchases is £0.38, find the standard deviation of the costs of Connor’s purchases.

During initial training for the Royal Air Force new recruits must sit an aptitude test.  Test scores for the latest round of recruits are shown in the table below:

Recruits who score below the 25th percentile are disqualified.

Calculate an estimate for the score recruits must have achieved to avoid disqualification. 

Those who score in the top 30% move on to the next stage of training and the rest must re-sit the test.

One of the recruits, Amelia, achieves a score of 231.  Estimate whether Amelia will need to re-sit the test or will be moved on to the next stage of training. 

Whilst in lockdown, a group of people were asked to record the length of time, t hours, they spent browsing the internet on a particular day. 

The results are summarised in the table below.

From this data, an A Level Statistics student used the midpoints and calculated that the estimated mean time spent browsing the internet is 5 hours and 15 minutes.

Show that the estimated standard deviation is 2 hours and 24 minutes to the nearest minute.

Zisien measures the speeds,  miles per hour, of a number of cars passing her house one day.  She knows that the speed limit is 30 miles per hour so she decides to use the coding  when she records the data. 

Zisien finds that  .

Zisien claims that more than half of the cars in the sample were going over the speed limit because .

Explain why Zisien's reasoning is incorrect.

Zisien finds that and  .

Calculate the standard deviation of the speeds of the cars in the sample.

Zisien’s sister, Ying, used the code =  – 20 to record the data for the same cars.

Ying discovers that the median of her coded data is 9.4. 

Does this information support Zisien's claim in part (a)? Give a reason for your answer.

Wildlife researchers are studying the swimming speeds, kmph, of two species of penguin, the emperor penguin and the gentoo penguin.  The mean swimming speed of 40 gentoo penguins was found to be 31.4 kmph and the standard deviation was found to be 3.8 kmph.

Allowing to represent the swimming speeds of the gentoo penguins,

(i) show that ,

(ii) calculate the value of  .

The swimming speeds of 20 emperor penguins () were also recorded and the mean swimming speed of all 60 penguins surveyed was found to be 24.1 kmph. Given that  ,  calculate the mean and standard deviation of the 20 emperor penguins.

Some entomologists were studying the amount of time two different species of butterflies spent cocooned.  The table shows the means and standard deviations of the time spent cocooned, measured in days, by 15 Monarch butterflies and 25 Common Blue butterflies.

Given that the overall mean time for all 40 butterflies was 11.93 days, calculate the mean number of days the Monarch butterflies spent cocooned and complete the table.

Calculate the overall standard deviation of the time spent cocooned by all 40 butterflies.

Lab technicians were studying the effect of caffeine on mice.  The resting heart rates,  beats per minute (bpm), of some mice were recorded and the results were summarised by   and   , where is a constant.

Given that the variance of the resting heart rates was found to be 10 bpm², calculate the two possible options for the number of mice in the study.

The mean resting heart rate is found to be 605 bpm.  Using this information, find the two possible options for the value of .

Roger has been looking at some data on the daily mean air temperature, t, in two different locations, Perth and Jacksonville, taken from the large data set.  All the data is taken from the month of July in 2015.

Unfortunately, some of the information has been lost and Roger does not know which data is for which location.

Complete the table.

Using your knowledge of the large data set, state which of the locations is most likely to be Jacksonville, giving a reason for your answer.