Working with Data (AQA A Level Maths: Statistics): Exam Questions

A random sample of eight cars was selected from the Large Data Set.

The masses of these cars, in kilograms, were as follows.

950, 989, 1247, 1415, 1506, 1680, 1833, 2040

It is given that, for the population of cars in the Large Data Set, the lower quartile is 1167 kg, the median is 1393 kg and the upper quartile is 1570 kg.

(i) It was decided to remove any of the masses which fall outside the following interval.

$\text{median}-1.5\times \text{IQR}\le \text{mass}\le \text{median}+1.5\times \text{IQR}$

Show that only one of the eight masses in the sample should be removed.

(ii) Write down the statistical name for the mass that should be removed in part (i).

The table below is an extract from the Large Data Set.

(i) Calculate the mean and standard deviation of CO2 emissions in the table.

(ii) Any value more than 2 standard deviations from the mean can be identified as an outlier. Determine, using this definition of an outlier, if there are any outliers in this sample of CO2 emissions. Fully justify your answer.

Maria claims that the last line in the table must contain two errors.

Use your knowledge of the Large Data Set to comment on Maria's claim.

A hotel manager recorded the number of towels that went missing at the end of each day for 12 days. The results are below.

$2,4,1,0,3,4,3.2,9,3,2,4,5$

Explain how the data will need to be cleaned before the manager can calculate summary statistics.

The manager cleans the data as required. For the remaining $11$ days,

$n=11,\Sigma x=37,\Sigma x^2=181$

Calculate the mean and the standard deviation for the number of towels missing per day.

An outlier is defined as any data value lying more than $2$ standard deviations away from the mean.

Determine, giving a reason, whether there are any outliers in the cleaned data.

State, giving a reason, whether the manager should remove this outlier from the data set.

The cumulative frequency diagram below shows completion times for 100 competitors at the 2019 Rubik’s cube championships.  The quickest completion time was 9.8 seconds and the slowest time was 52.4 seconds.

The grid below shows a box plot of the 2020 championship data.  Draw a box plot on the grid to represent the 2019 championship data.

(i) Compare the distributions of the completion times for the 2019 and 2020 championships.

(ii) Given that the 2020 championships happened after the global pandemic, during which many competitors spent months at home, interpret your findings from part (b)(i).

Students at two karate schools, Miyagi Dojo and Cobra Kicks, measured the force, in newtons, with which they could perform a particular style of hit.

The mean and standard deviation for the students at Cobra Kicks are $\overline{x}=1740\text{ N}$ and $\sigma =251\text{ N}$ (both to 4 significant figures).

For the students at Miyagi Dojo, the summary statistics are

$n=12,\Sigma x=21873,\Sigma x^2=41532545$

Calculate the mean and standard deviation for the force with which the students at Miyagi Dojo can hit.

Compare the distributions of hitting force for the two karate schools.

The heights, in metres, of a flock of $20$ flamingos are recorded and shown below:

An outlier is an observation that falls either more than $1.5\times \text{IQR}$ above the upper quartile or less than $1.5\times \text{IQR}$ below the lower quartile.

(i) Find the median.

(ii) Find the interquartile range.

(iii) Determine, giving a reason, whether there are any outliers in the data.

Using your answers to part (a), draw a box plot for the data.

The number of packages processed daily at a sorting office over a $14$-day period are given below:

$237,264,308,313,319,352,378$

$378,405,421,428,450,465,583$

Given that $\Sigma x=5301$ and $\Sigma x^2=2113195$, calculate the mean and standard deviation for the number of daily packages processed.

An outlier is defined as any data value lying more than $2$ standard deviations away from the mean.

Determine, giving a reason, whether there are any outliers in the data.

By removing the outlier identified in part (b), clean the data and recalculate the mean and standard deviation.

The cumulative frequency diagram below shows the distribution of income of $120$ managers across a supermarket chain.

The income of a sample of $120$ other employees across the supermarket chain are recorded in the table below.

On the grid above, draw a cumulative frequency graph to show the data for the other employees.

Compare the income of the managers and the other employees.

Ms Chew is an accountant who is examining the length of time it takes her to complete jobs for her clients.  Ms Chew looks at her spreadsheet and lists the number of hours it took her to complete her last 12 jobs:

‘-’ represents a job for which the length of time taken was not recorded.

An outlier is an observation which lies more than  ±2  standard deviations away from the mean.

By first cleaning the data, show that 21 is the only outlier.

Ms Chew looks at her handwritten records and finds that the value 21 was typed into the spreadsheet incorrectly.  It should have been 12.

Without further calculations, explain the effect this would have on the:

(i) mean

(ii) standard deviation

(iii) median.

As part of an experiment, $15$ maths teachers are asked to solve a puzzle and their times, in minutes, are recorded:

An outlier is an observation which lies more than $\pm 2$ standard deviations away from the mean.

Show that there is exactly one outlier.

State, with a reason, whether the mean or the median would be the most suitable measure of central tendency for these data.

15 history teachers also completed the riddle; their times are shown below in the box plot:

Explain what the cross (×) represents on the box plot above. Interpret this in context.

Compare the distributions of the times taken to complete the puzzle by the two sets of teachers.

Hugo, a newly appointed HR administrator for a company, has been asked to investigate the number of absences within the IT department.  The department contains 23 employees, and the box plot below summarises the data for the number of days that individual employees were absent during the previous quarter.

An outlier is an observation that falls either more than 1.5 (interquartile range) above the upper quartile or less than 1.5 (interquartile range) below the lower quartile.

Show that these data have an outlier, and state its value.

For the 23 employees within the department, Hugo has the summary statistics:

 $\mathrm{\Sigma }x=286$ and  $\mathrm{\Sigma }{x}^2=4328$$x$²

Hugo investigates the employee corresponding to the outlier value found in part (a) and discovers that this employee had a long-term illness.  Hugo decides not to include that value in the data for the department.

Assuming that there are no other outliers, calculate the mean and standard deviation of the number of days absent for the remaining employees.

Sam, a zoologist, is a member of a group researching the masses of gentoo penguins.  The research group takes a sample of 100 male and 100 female penguins and records their masses.

An outlier is an observation that falls either more than 1.5 $\times$(interquartile range) above the upper quartile or less than 1.5 $\times$  (interquartile range) below the lower quartile.

Given that values are outliers if they are less than 4.2kg or more than 8.5kg, calculate the upper and lower quartiles for the mass of the 200 gentoo penguins.

Casey is another member of Sam's research group. She believes that the masses of male and female gentoo penguins follow different distributions. The cumulative frequency graphs below show the masses of the male and female gentoo penguins in the sample.

Use the graphs to compare the distributions of the masses of male and female gentoo penguins.

Marya is consistently late for work. David, Marya’s boss, records the number of minutes that she is late during the next six days. David calculates the mean is 18 minutes and the variance is 210 minutes². On one of the six days, Marya was 50 minutes late.

Show that 50 is an outlier, using the definition that outliers are more than 2 standard deviations away from the mean.

(i) Give a reason why the value of $50$ should be excluded from the data set.

(ii) Give a reason why the value of $50$ should be included in the data set.

Marya tells David that she was $50$ minutes late that day because her car broke down on the way to work, and she shows him the breakdown receipt as evidence.

David agrees to remove the $50$ from the data set. Calculate the new mean and standard deviation for the remaining values.

The cumulative frequency graph below shows the information about the lengths of time taken for 80 students to run a lap of the sports hall.

Complete the table below:

Hence estimate the mean and the standard deviation of the times.

Given that the fastest time was 21 seconds and the slowest time was 100 seconds, show that these values are outliers using the definition that an outlier is more than 2 standard deviations away from the mean.

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.