Statistical Measures (Cambridge (CIE) AS Maths: Probability & Statistics 1): Exam Questions

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

(i) Write down the mode for this data.

(ii) Explain why, in this case, the mode from part (i) would not be particularly helpful to the shop owner when reordering masks.

(iii) Given that the shop is open every day of the week, calculate the mean number of masks sold per day.

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 16 cups are tested, giving the following results for numbers of uses:

                     31    36    40    41    

                     47    49    51    56    

                      58    62    62    62

                     64    68    69    70    

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

(ii) Find the median number of times a cup can be used.

(iii) 25% of the results are greater than or equal to $p$ uses.

Find the value of $p$.

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.

The lengths ($l$ 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.

A machine is set to fill sacks of potatoes to a weight of $50kg$.

In a random sample of 7 sacks the following weights in kg were recorded:$\sum \left(w-50\right)=-14.6$, where $w$$kg$ is the weight of a sack of potatoes.

Show that the mean weight of the sampled sacks of potatoes is less than $50kg$.

Given that $\sum {\left(w-50\right)}^2=170.2$, find the standard deviation of the sampled sacks of potatoes.

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); 

Candidate 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 Math exam in the 24 hours before the exam.  The results are shown in the table below:

Write down the modal group for these data.

Using $x$ to represent the mid-point of each class,  $\sum xf=8340$ and $\sum x^{2}f=1636200$.

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

A college needs to standardise the test scores of all students before submitting them to the Exam Board.  The scores are standardised by using the coding $\sum \left(x-53\right)$, where $x$ represents the raw test score. The college calculates the mean standardised test score to be 42 and the standard deviation to be 3.1.  Find the mean and standard deviation of the raw test scores.

The times, in seconds, taken by 8 skaters to finish a 1000 m race are listed below:

            $7577828479818076$

By using an appropriate assumed mean, calculate the true mean and true standard deviation of the speed skaters' times.

A golfer records her scores for her last 8 rounds of golf, which are given below.

            $7375817276998078$

(i) Find the median $\left(Q_2\right)$, the lower quartile $\left(Q_1\right)$ and the upper quartile $\left(Q_3\right)$.

(ii) Find the interquartile range.

An outlier is defined as any value x such that

         $x<Q_1-1.5\times \left(Q_3-Q_1\right)\mathrm{or}x>Q_3+1.5\times \left(Q_3-Q_1\right)$

(i) Find the boundaries that determine outliers for the golfer's  scores.

(ii) Hence determine if any of the golfer's  last 8 scores were outliers.

$a$, $b$, $c$ 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 $d$

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 $\ge 35$ 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 $\ge 35$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 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 dataset.

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

Workers at an elephant sanctuary measure the health of their elephants by weighing the amount of dung ($dkg$) each one produces. The data for the mass of dung produced in one day by 18 elephants can be summarised as  $\sum d=895\mathrm{and}\sum d^2=45810.$

Calculate the mean and variance of the amount of dung produced that day.

An outlier is identified as any data value, $d$, such that

               $\left|d-\overline{d}\right|>2\times \mathrm{standard}\mathrm{deviation}$

Determine the weights below and above which the amount of dung the elephant produces in a day would classify the elephant as an outlier.

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

To test the accuracy of the machine, a random sample of 20 sacks is taken  which is summarised by $\sum \left(w-50\right)=20\mathrm{and}\sum {\left(w-50\right)}^2=220$.

Calculate the exact values of the mean and standard deviation of $w$.

Calculate the value of

(i) $\sum w$

(ii) $\sum w^2$

Another 10 sacks of potatoes are sampled and the mean of these is found to be 49 kg.  Calculate the mean of all 30 sacks of potatoes and comment on the accuracy of the machine.

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 $a$ and $b$.

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

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

(i) the mean in London is higher than the mean in Wellington

(ii) the standard deviation in London is lower than the standard deviation in Wellington.

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.

In the Tokyo 2020 Paralympics, the Women's Individual Bronze Medial Archery match was contested between the USA's Lia Coryell and Great Britain's Victoria Rumary.

Each competitor fired three arrows from each of five "ends" and they acquired a total score for each end. The competitor with the highest overall score for the five ends won the match and bronze medal.

Using an assumed mean of 25,Lia's scores for her five ends are summarised as $\sum d=-2\mathrm{and}\sum d^2=6.$

Using an assumed mean of 25, Victoria's scores for her five ends are summarised as $\sum d=6and\sum d^2=26$.

Explain how, without any calculations, you can determine that Great Britain won the bronze medal.

Find the mean and standard deviation of the scores for both Lia and Victoria.

Compare Lia's and Victoria's mean scores and standard deviations.

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 $a$ and find the estimated variance of the length of time spent browsing the internet.

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. 

She finds that  $\sum \left(x-30\right)=13.4\mathrm{and}\sum {\left(x-30\right)}^2=1470.$

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, $\overline{x}$ , is 30.67 miles per hour, calculate the standard deviation of $x$.

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).

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.

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. 

(i) Write down the modal class interval for the heights of geraniums in the control group.

(ii) Find the class interval which contains the median.

(iii) Find the smallest and largest possible values for the interquartile range of the heights of the control group.

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

      ${\mathrm{Q}}_1=23.4\mathrm{cm}{\mathrm{Q}}_2=27.1\mathrm{cm}{\mathrm{Q}}_3=28.5\mathrm{cm}$

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.

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 $31.4\mathrm{kmph}$and the standard deviation was found to be $3.8\mathrm{kmph}$.

Allowing $x_G$ to represent the swimming speeds of the gentoo penguins, show that $\sum x_G=1256$  and 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 $24.1\mathrm{kmph}$. Given that  $\sum x^2=41891$,  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.

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, write down the two possible options for the value of $a$.

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.

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

(i) Using $x$ to represent the mid-point of each class, write an expression in terms of a for $\sum fx$, giving your answer in simplified form.  

(ii) Given that $\sum fx=2132.5\mathrm{and}\sum f{x}^2=53568.75$calculate the estimated mean and standard deviation of the amount of time Hattie was late for school last year.

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 data is split into two sets such that the first has $n_1$ data values and  ${\overline{x}}_1=a_1+{\overline{d}}_1$ and the second has $n_2$ data values $\overline{x}{}_2=a_2+{\overline{d}}_2$, where $a_i$ are assumed means for each set and $n_i{\overline{d}}_i=\sum \left(x_i-a_i\right).$

Show that

         $\overline{x}=\frac{n_1\left({\overline{d}}_1+a_1\right)+n_2\left({\overline{d}}_2+a_2\right)}{n_1+n_2}$

Given that $\overline{x}$ ≠ 0,  $\overline{x}=2{\overline{x}}_1$ and  ${\overline{x}}_2=4{\overline{x}}_1$, show that $n_1=2n_2$.