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

A random sample of ten CO2 emissions was selected from the Large Data Set.

The emissions in grams per kilogram were:

13, 45, 45, 0, 49, 77, 49, 49, 49, 78

Find the standard deviation of the sample.

An environmentalist calculated the average CO2 emissions for cars in the Large Data Set registered in 2002 and in 2016.

The averages are listed below.

The environmentalist claims that the average CO2 emissions for 2002 and 2016 combined is 145.8

Determine whether this claim is correct. Fully justify your answer.

Denzel wants to buy a car with a propulsion type other than petrol or diesel.

He takes a sample, from the Large Data Set, of the ${\text{CO}}_2$ emissions, in g/km, of cars with one particular propulsion type.

The sample is as follows.

82, 13, 96, 49, 96, 92, 70, 81

Using your knowledge of the Large Data Set, state which propulsion type this sample is for, giving a reason for your answer.

Calculate the mean of the sample.

Calculate the standard deviation of the sample.

Denzel claims that the value 13 is an outlier.

(i) Any value more than 2 standard deviations from the mean can be regarded as an outlier. Verify that Denzel's claim is correct.

(ii) State what effect, if any, removing the value 13 from the sample would have on the standard deviation.

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,  $\frac{3}{4}$ 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.

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.

$a$, $b$, $c$ and $d$ are 4 integers written in order of size, starting with the smallest. 

The sum of $a$, $b$ and $c$ is 70
The mean of $a$, $b$, $c$ and $d$ is 25
The range of the 4 integers is 14.

Work out the median of $a$, $b$, $c$ and $d$

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 $a$ and $b$.

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.

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.

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.

The ages, $x$ years, of 200 people attending a vaccination clinic in one day are summarised by the following:  $\mathrm{\Sigma }x=7211$  and  $\mathrm{\Sigma }{x}^2=275360$.

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.

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. 

Zisien measures the speeds, $x$ 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  $y=x-30$ when she records the data. 

Zisien finds that  $\overline{y}=0.67$.

Zisien claims that more than half of the cars in the sample were going over the speed limit because $\overline{y}>0$.

Explain why Zisien's reasoning is incorrect.

Zisien finds that $\mathrm{\Sigma }y=13.4$ and  $\mathrm{\Sigma }{y}^2=1470$.

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

Zisien’s sister, Ying, used the code $z$ = $x$ – 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, $x$ 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 $x_G$ to represent the swimming speeds of the gentoo penguins,

(i) show that $\sum x_G=1256$,

(ii) calculate the value of  $\sum {x_G}^2$.

The swimming speeds of 20 emperor penguins ($x_E$) were also recorded and the mean swimming speed of all 60 penguins surveyed was found to be 24.1 kmph. Given that  $\sum x^2=41891$,  calculate the mean and standard deviation of the 20 emperor penguins.

Lab technicians were studying the effect of caffeine on mice.  The resting heart rates, $x$ beats per minute (bpm), of some mice were recorded and the results were summarised by $\mathrm{\Sigma }\left(x-a\right)=150$  and   $\mathrm{\Sigma }{\left(x-a\right)}^2=1050$, where $a$ 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 $a$.