Statistical Measures (OCR A Level Maths: Statistics)

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

Fran sits three Maths papers and six Science papers during her final A Level exams.  She achieves a mean score of 62.7% across the three Maths exam papers, and needs an overall mean score of 78.5% 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 84.2%.

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

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.

The lengths ( cm) of a sample of nine otters, measured to the nearest centimetre by a wildlife research team, are:

                        76     77      91      65       63      83      92      61      88

Calculate the mean and standard deviation of the nine recorded lengths.

During initial small aircraft pilot training candidates must sit an aptitude test. Grades for the latest 28 candidates are shown in the table below (0 is the lowest grade, 8 is the highest):

Grade	Frequency
0	2
1	3
2	2
3	3
4	4
5	2
6	5
7	4
8	3

Candidates in the bottom 25% are disqualified.

Calculate the grade candidates must achieve to avoid disqualification.

Those who score in the top 25% move on to the next stage of training while the rest (other than those who have been disqualified) must re-sit the test.

One of the candidates, Amelia, achieves grade 6. Determine whether Amelia will need to re-sit the test or will be moved on to the next stage of training.

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:

Time  (minutes)	Number of students
	5
	6
	17
	14
	8

Write down the modal group for these data.

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.

From the large data set, data relating to the number of people cycling to work in each Local Authority in the North East for both 2001 and 2011 is given above.

After major investments in cycle lanes across the North East, a government assistant uses the data above to suggest that these investments have encouraged almost 20% more people in each Local Authority in the North East to cycle to work.

For the data given above, compare the mean number of people per Local Authority who cycle to work across the North East in 2001 and 2011 and state whether or not this data supports the government's claim that they have encouraged almost 20% more people to cycle to work across the North East between 2001 and 2011.

Newcastle Upon Tyne was the first Local Authority where cycle lane improvements were introduced, a rolling programme of improvements was then carried out across the North East.

The government assistant says by removing some of the data the average percentage increase could reach nearly 30%. Suggest one way the government could justify removing some of the Local Authority data before recalculating the mean percentage increase.

, ,  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 

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. 

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

Estimate the mean weight gain of the 50 puppies.

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

The head veterinarian directs the nurse to monitor all puppies whose weight gain lies within the bottom 20% of the data set.

Explain why, without further information, the veterinary nurse would have to monitor over 50% of the puppies to be sure of satisfying this instruction.

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:

Given that the estimate of the mean time spent exercising based on this table is 35.4 minutes, find the values of and .

The ages, years, of 200 people attending a vaccination clinic in one day are summarised by the following:  = 7211  and  = 275 360.

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

One person chooses not to get the vaccine, so their data is discounted. The new mean is exactly 36.  Calculate the age of the person who left and the standard deviation of the remaining people.

In January of 2021, the UK government announced a nationwide lockdown to control the spread of the coronavirus.  The table below shows the means and standard deviations of the average amounts of time spent indoors per day by some people in London, UK and in Wellington, New Zealand, in January of 2021.

Suggest a reason, in the context of the question, for why

Based on the data in the table, do you think the government in New Zealand had imposed the same restrictions as those in the UK?  Give a reason for your answer.

Calculate the overall mean for the average amounts of time spent indoors by all 40 people.

The large data set contains data on methods of travel to work from 2001 and 2011. A selection of data relating to the number of people cycling to work in each Local Authority in the North East for both 2001 and 2011 is given above.

Improvements to cycle lanes, to encourage more people to cycle to work, have been rolled out across the North East in phases since 2001. Three local authorities have yet to make such improvements.

Identify which Local Authorities are most likely to be the final three to make improvements to their cycle lanes. Give a reason for your answer.

After the investments in cycle lanes across the North East, the government claims that its investments have encouraged almost 30% more people in the North East to cycle to work.

Excluding the authorities identified in part (a), compare the mean number of people per Local Authority who cycle to work across the North East in 2001 and 2011 and state whether or not this data supports the government's claim that they have encouraged almost 30% more people to cycle to work across the North East between 2001 and 2011.

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 calculated that the estimated mean time spent browsing the internet is 5 hours and 15 minutes. Calculate the value of  and find the estimated variance of the length of time spent browsing the internet.

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.

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, show that =1256  and 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  =41891,  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.

Hattie’s homeroom teacher decides to summarise the number of minutes, t, she has been late to school during the last year in preparation for a parents’ meeting.  The results are shown in the table below.

Time t (mins)	Frequency f
-10 ≤ t < -5	3
-5 ≤ t < 0	19
0 ≤ t < 5	32
5 ≤ t < 10
10 ≤ t < 20	53
20 ≤ t < 60	24

Write down, in the context of the question, what the time interval -10 ≤ t < -5  represents.

Hattie happens to notice that on three of the days she was recorded as being 40 minutes late for school, she had actually arrived 40 minutes early.  Calculate the corrected estimate for the mean amount of time she was late for school last year.

The large data set contains data on the numbers of people in employment using different methods of travelling to work for the Local Authorities in England and Wales in 2011. The number of people who walked to work across England and Wales in 2011 has been summarised in the table below, with the frequency referring to the number of Local Authorities within each range.

Number of people	Frequency
	137
	143
	40
	6
	5
	0
	3

The estimated mean number of people per local authority that walked to work in 2011 is calculated from the table to be 7313, to the nearest integer. Using that information, and the figures from the table above, find the value of .

Zisien measures the speeds, x miles per hour, of several cars passing her house one day.
She knows that the speed limit is 30 miles per hour.

She finds that  and .

Is it more likely that more of the cars in Zisien's sample were going over or under the speed limit that day? Give a reason for your answer.

Given that the mean speed of the cars in Zisien's sample, , is 30.67 miles per hour, calculate the standard deviation of .

Zisien's sister, Ying, worked out the median of the speeds of the cars and found it was 27.43 miles per hour.

Use this information to comment on your answer to (a).

Botanists at a research centre are carrying out research on a new type of fertiliser. They collect data on the heights of one group of geraniums growing without the fertiliser (the control group) and of another group growing with the fertiliser (the experimental group). They take care to keep all other growing conditions the same for both groups.

The table below shows the heights of the control group of geraniums 15 weeks after planting.

Height (cm)
Cumulative Frequency	2	7	12	19	34	39	40

The data for the group of geraniums growing in the experimental group were summarised as follows:

The shortest plant in the experimental group was 15.2 cm and the tallest was 33.5 cm.

Compare the distribution of the heights of the plants in the two groups.