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

A factory aims to produce 500 plastic garden chairs per day. The number of chairs made per day, $c$, over a period of 10 working days is summarised by

$\Sigma \left(c-500\right)=-15$ and $\Sigma {\left(c-500\right)}^2=381$.

Find the mean number of chairs made per day over the 10 days.

Find the standard deviation of the number of chairs made per day over the 10 days.

On one of the 10 days, a mechanical problem meant that only 485 chairs were made.

Suggest a measure of central tendency, other than the mean, that would be less affected by this extreme value.

Fran sits three Mathematics papers and six Science papers in her final examinations. Each of the nine papers is weighted equally.

Her mean mark on the three Mathematics papers is 62.7%. To gain a place at her chosen university, she needs an overall mean mark of 78.5% across all nine papers. Her mean mark on four of her Science papers is 84.2%.

Calculate the mean mark Fran must achieve on her final two Science papers to gain a place at university.

Coffee4Life manufactures reusable coffee cups from coffee plant waste. The cups are tested to see how many times they can be used before they begin to disintegrate. A sample of 16 cups is tested, giving the following 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.

A machine is set to fill sacks of potatoes to a target weight of 50 kg. The weights, $w$ kg, of a random sample of 7 sacks are summarised by

$\Sigma (w-50)$ = −14.6.

Show that the mean weight of the sacks in the sample is 47.9 kg, correct to 3 significant figures.

Given also that $\Sigma {\left(w-50\right)}^2$ = 170.2, find the standard deviation of the weights of these 7 sacks.

During initial training for small aircraft pilots, candidates must sit an aptitude test. The grades of the latest 28 candidates are shown in the table below, where 0 is the lowest grade and 8 is the highest.

Candidates in the bottom 25% are disqualified.

Calculate the grade candidates must achieve to avoid disqualification.

Candidates 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 move on to the next stage of training.

The times, $t$ seconds, taken by 8 speed skaters to finish a 1000 m race are summarised by

$\Sigma (t-75)=34$ and $\Sigma {\left(t-75\right)}^2=212$.

Find the mean and the standard deviation of these times.

$a$, $b$, $c$ 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, $s$ miles per hour, of 80 vehicles passing a speed camera were recorded and are summarised in the table below.

(i) Write down the modal class for these data.

(ii) Write down the class that contains the median.

(i) Assuming that $s\ge 35$ means $35\le s<40$, calculate an estimate of the mean speed of the 80 vehicles.

(ii) It is now discovered that $s\ge 35$ means $35\le s<60$. Without further calculation, state with a reason whether this would cause an increase, a decrease or no change to the estimated mean.

Two archers, $A$ and $B$, compete in an archery match. Each archer shoots three arrows in each of five rounds, and the archer with the higher total score over the five rounds wins.

The scores, $x$, for the five rounds are coded using $d=x-25$, and the coded scores are summarised as follows.

Archer $A$: $\sum d=-2$ and $\sum d^2=6$

Archer $B$: $\sum d=6$ and $\sum d^2=26$

Explain how, without any calculation, you can tell that archer $B$ won the match.

Find the mean and the standard deviation of the scores for each archer.

Compare the scores of the two archers.

The table below summarises the weight gain, $w$ kg, of 50 small-breed puppies over their first eight weeks.

Calculate an estimate of the mean weight gain of the 50 puppies.

Give a reason why it is not possible to find the exact median of these data.

A veterinary nurse is told to monitor all the puppies whose weight gain is in the bottom 20% of the data.

Explain why, without further information, the nurse would have to monitor more than 50% of the puppies to be sure of following this instruction.

A machine is set to fill sacks of potatoes to a target weight of 50 kg, although the actual weight, $w$ kg, of a sack can vary. To test the machine, a random sample of 20 sacks is taken, and the weights are summarised by

$\sum \left(w-50\right)=20$ and $\sum {\left(w-50\right)}^2=220$.

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

Find the value of

(i) $\sum w$,

(ii) $\sum w^2$.

Another random sample of 10 sacks is taken, and the mean weight of these sacks is 49 kg.

Find the mean weight of all 30 sacks, and comment on the accuracy of the machine.

The time, $t$ minutes, that each of 100 people spent exercising on a particular day is summarised in the table below.

Given that an estimate of the mean time spent exercising is 35.4 minutes, find the values of $a$ and $b$.

The ages, $x$ years, of the 200 people attending a vaccination clinic on one day are summarised by

$\sum x=7211$ and $\sum x^2=275360$.

Calculate the mean and the standard deviation of the ages of these people.

One person leaves the clinic without being vaccinated, and their age is removed from the data. The mean age of the remaining 199 people is exactly 36 years.

Find the age of the person who left, and the standard deviation of the ages of the remaining 199 people.

A group of people recorded the length of time, $t$ hours, that they spent browsing the internet on a particular day. The results are summarised in the table below.

An estimate of the mean time spent browsing the internet is 5 hours and 15 minutes.

Find the value of $a$, and calculate an estimate of the variance of the time spent browsing the internet.

The speeds, $x$ miles per hour, of a sample of cars passing a house on a road with a speed limit of 30 miles per hour are recorded. The speeds are summarised by

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

State, with a reason, whether the mean speed of the cars in the sample is greater than or less than the speed limit.

Given that the mean speed of the cars in the sample is 30.67 miles per hour, find the standard deviation of $x$.

The median speed of the cars in the sample is 27.43 miles per hour.

State, with a reason, the type of skewness of the speeds, and state what the median tells you about the speeds of the cars compared with the speed limit.

Wildlife researchers are studying the swimming speeds, $x{\text{ km h}}^{-1}$, of two species of penguin, the gentoo penguin and the emperor penguin. For 40 gentoo penguins, the mean swimming speed is 31.4 ${\text{km h}}^{-1}$ and the standard deviation is 3.8 ${\text{km h}}^{-1}$.

Using $x_G$ to denote the swimming speeds of the gentoo penguins, show that $\sum x_G=1256$ and find the value of $\sum x_G^2$.

The swimming speeds, $x_E{\text{ km h}}^{-1}$, of 20 emperor penguins are also recorded. For all 60 penguins, the mean swimming speed is 24.1 ${\text{km h}}^{-1}$ and $\sum x^2=41891$.

Find the mean and the standard deviation of the swimming speeds of the 20 emperor penguins.

The times, in days, spent cocooned by 15 Monarch butterflies and 25 Common Blue butterflies were recorded. The table shows the means and the standard deviations of the times.

Given that the mean time for all 40 butterflies is 11.93 days, find the mean time for the Monarch butterflies and complete the table.

Find the standard deviation of the times spent cocooned by all 40 butterflies.

A teacher records the number of minutes, $t$, that a student, Hattie, is late for school each day during a school year. The results are shown in the table below.

Write down, in the context of the question, what the class −10 ≤ t < −5 represents.

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

(ii) Given that $\sum fx$ = 2132.5 and $\sum f{x}^2$ = 53568.75, calculate estimates of the mean and the standard deviation of the number of minutes Hattie was late for school.

Hattie notices that on three of the days recorded in the class 20 ≤ t < 60 as being 40 minutes late, she had actually arrived 40 minutes early.

Find a corrected estimate of the mean number of minutes Hattie was late for school.

Two friends, Anna and Connor, are playing a game on their phones. As they play, they can buy three different boosters, each at a fixed cost. The numbers of purchases they each make are shown in the table below.

(i) The mean and the standard deviation of the costs, in dollars, of Anna's purchases are 0.50 and 0 respectively. Write down the cost, in dollars, of a single Level-up purchase.

(ii) Given that the mean cost, in dollars, of Connor's purchases is 0.38, find the standard deviation of the costs of Connor's purchases.

Botanists are testing a new fertiliser. They grow one group of geraniums without the fertiliser (the control group) and another group with it (the experimental group), keeping all other growing conditions the same.

The heights, $h$ cm, of the 40 plants in the control group 15 weeks after planting are summarised in the table below.

For the heights of the control group,

(i) write down the modal class,

(ii) find the class that contains the median,

(iii) find the smallest and largest possible values of the interquartile range.

For the experimental group, the lower quartile, median and upper quartile of the heights are 23.4 cm, 27.1 cm and 28.5 cm respectively. The shortest plant is 15.2 cm and the tallest is 33.5 cm.

Compare the heights of the plants in the two groups.

The resting heart rates, $x$ beats per minute (bpm), of a sample of mice are summarised by

$\sum \left(x-a\right)=150$ and $\sum {\left(x-a\right)}^2=1050$,

where $a$ is a constant.

Given that the variance of the resting heart rates is 10, find the two possible values of the number of mice in the sample.

Given also that the mean resting heart rate is 605 bpm, find the two possible values of $a$.