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Past Papers

## Statistics

NOTE: Clicking an entry in the right column will take you directly to that question!

(BLUE = Paper 1, RED = Paper 2)

Since 2015, the following subtopic... Has explicitly appeared in...

Mean, Mode & Median

Stem and Leaf Diagrams

Bar Charts

Pie Charts

Histograms

Scatter Diagrams

Normal Distributions

Margin of Error

Confidence Intervals

Hypothesis Tests

## 2022 Paper 2 Question 3(a)

(a) A list of $$7$$ numbers is shown below.

\begin{align}17,8,9,8,14,11,28\end{align}

(i) Find the mean of these $$7$$ numbers, correct to $$1$$ decimal place.

(ii) Find the median of these $$7$$ numbers.

(iii) One more number is added to the list. The new median is $$10.5$$.
Find the number that was added to the list.

(i) $$13.6$$

(ii) $$11$$

(iii) $$10$$

Solution

(i)

\begin{align}\mbox{Mean}&=\frac{17+8+9+8+14+11+28}{7}\\&\approx13.6\end{align}

(ii)

\begin{align}8,8,9,11,14,17,28\end{align}

\begin{align}\downarrow\end{align}

\begin{align}\mbox{Median}=11\end{align}

(iii)

\begin{align}\frac{x+11}{2}=10.5\end{align}

\begin{align}\downarrow\end{align}

\begin{align}x+11=21\end{align}

\begin{align}\downarrow\end{align}

\begin{align}x=10\end{align}

Video Walkthrough

## 2022 Paper 2 Question 4

(a) A group of students sat an exam. Each student was given a grade.
The following table shows how many students got each grade.

Grade Distinction High Merit Merit Achieved

Number of students

$$8$$

$$12$$

$$39$$

$$13$$

Complete the pie chart below to show this information.
Label each sector clearly. Show your working out.

The sector for Distinction is already given.
It has an angle of $$40^{\circ}$$.

Solution
Video Walkthrough
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(b) A large group of people took a reading test.
The scores were normally distributed, with a mean of $$100$$ and a standard deviation of $$20$$.

Use the empirical rule to answer parts (b)(i) and (b)(ii).

(i) What percentage of the people had scores between $$80$$ and $$120$$?

(ii) The top $$2.5\%$$ of scores were given a grade of ‘Exceptional’.
What was the least score that was needed to get this grade?

(i) $$68\%$$

(ii) $$140$$

Solution

(i) $$68\%$$

(ii)

\begin{align}100+2\times20=140\end{align}

Video Walkthrough
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(c) The scores of six people on this test were as follows:

\begin{align}104,82,94,113,98,105\end{align}

Find the range and the standard deviation of these numbers.
Give the standard deviation correct to $$1$$ decimal place.

Range: $$31$$

Standard deviation: $$9.8$$

Solution

Range is $$113-82=31$$ and standard deviation is $$\approx 9.8$$.

Video Walkthrough

## 2022 Paper 2 Question 7(a)

An animal shelter takes cats and dogs.

(a) The staff record the weight of each animal in the shelter on two different days: the day they are taken in to the shelter (Day X) and a number of days later (Day Y).

The table below shows the weights of $$10$$ dogs (labelled A to J) on these two days.

Animal A B C D E F G H I J

Weight Day X (kg)

$$4.5$$

$$4.9$$

$$5.3$$

$$5.3$$

$$5.5$$

$$5.7$$

$$6.7$$

$$6.9$$

$$7.3$$

$$7.4$$

Weight Day Y (kg)

$$4.8$$

$$5.4$$

$$5.5$$

$$6.1$$

$$6.3$$

$$6.0$$

$$7.0$$

$$7.6$$

$$8.1$$

$$7.9$$

(i) Draw a back-to-back stem-and-leaf plot to show this information.

(ii) What does the stem-and-leaf plot show about the weights of the dogs on Day X and Day Y?

(iii) $$r$$ is the correlation coefficient between the weight on Day X and the weight on Day Y.
Based on the data in the table, which of the following is most likely value of $$r$$?

\begin{align}-0.9&&-0.2&&0.2&&0.9\end{align}

Give a reason for your answer, based on the data in the table.

(i)

(ii) The weight of the dogs has increased from Day X to Day Y.

(iii) $$0.9$$ as there is a strong, positive linear relationship.

Solution

(i)

(ii) The weight of the dogs has increased from Day X to Day Y.

(iii) $$0.9$$ as there is a strong, positive linear relationship.

Video Walkthrough

## 2021 Paper 2 Question 5

(a) The heights of a given population are normally distributed. $$95%$$ of the population fall within the height range $$[147.9\mbox{ cm}\leftrightarrow 178.7\mbox{ cm}]$$.
Using the empirical rule, find the mean and the standard deviation of the distribution of heights in the population.

$$\mbox{Mean}=163.3\mbox{ cm}$$ and $$\mbox{Standard deviation}=7.7\mbox{ cm}$$

Solution

\begin{align}\mbox{Mean}&=\frac{147.9+178.7}{2}\\&=163.3\mbox{ cm}\end{align}

and

\begin{align}\mbox{Standard deviation}&=\frac{178.7-147.9}{4}\\&=7.7\mbox{ cm}\end{align}

Video Walkthrough
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(b) The following table shows data on the height and arm span of twenty teenagers.

Gender Height (cm) Arm Span (cm)

$$1$$

$$F$$

$$162$$

$$155$$

$$2$$

$$F$$

$$169$$

$$174$$

$$3$$

$$F$$

$$164$$

$$144$$

$$4$$

$$F$$

$$157$$

$$160$$

$$5$$

$$F$$

$$160$$

$$160$$

$$6$$

$$F$$

$$170$$

$$170$$

$$7$$

$$F$$

$$174$$

$$170$$

$$8$$

$$F$$

$$159$$

$$154$$

$$9$$

$$F$$

$$153$$

$$148$$

$$10$$

$$F$$

$$167$$

$$167$$

$$11$$

$$F$$

$$150$$

$$152$$

$$12$$

$$F$$

$$139$$

$$129$$

$$13$$

$$M$$

$$168$$

$$169$$

$$14$$

$$M$$

$$147$$

$$150$$

$$15$$

$$M$$

$$134$$

$$121$$

$$16$$

$$M$$

$$177$$

$$176$$

$$17$$

$$M$$

$$172$$

$$170$$

$$18$$

$$M$$

$$160$$

$$163$$

$$19$$

$$M$$

$$152$$

$$151$$

$$20$$

$$M$$

$$153$$

$$151$$

(i) Complete the scatter plot below by adding in the data for the males (the data in the shaded section of the table).

(ii) Three numbers are shown below.

\begin{align}0.2 && 0.9 && -0.6\end{align}

One of the numbers is the correlation coefficient of the data shown in the table above.
State the one which you think best represents the correlation coefficient between height and arm span for this data set.

(iii) Explain what this correlation coefficient tells us about the relationship between height and arm span for this data set.

(i)

(ii) $$0.9$$

(iii) There is a strong positive linear relationship.

Solution

(i)

(ii) $$0.9$$

(iii) There is a strong positive linear relationship.

Video Walkthrough

## 2021 Paper 2 Question 7

(a) The following table shows the age profiles of all of the people in Ireland, aged between $$15$$ and $$65$$, who subscribed to an online streaming platform.

Age (groups) $$15$$ - $$25$$ $$25$$ - $$35$$ $$35$$ - $$45$$ $$45$$ - $$55$$ $$55$$ - $$65$$

Percentage (%) of subscribers

$$24$$

$$27$$

$$25$$

$$16$$

$$8$$

(i) Draw a Histogram to represent the data.

(ii) In a survey of $$1000$$ subscribers, aged between $$15$$ and $$65$$, how many people would you
expect to be between $$25$$ years and $$55$$ years?

(iii) Use the table or the histogram to write down the age group which contains the median age.

(iv) Use the mid-interval values of the groups in the table to estimate the mean age of the subscribers. Give your answer correct to one decimal place.

(i)

(ii) $$680$$

(iii) $$25$$-$$35$$ age group

(iv) $$35.7\mbox{ years old}$$

Solution

(i)

(ii)

\begin{align}1{,}000\times(0.27+0.25+0.16)=680\end{align}

(iii) $$25$$-$$35$$ age group.

(iv)

\begin{align}\mbox{Mean}&=0.24\times20+0.27\times30+0.25\times40+0.16\times50+0.08\times60\\&=35.7\mbox{ years old}\end{align}

Video Walkthrough
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(b) Prior to the COVID-19 pandemic, Netflix held the largest share of the online streaming
market with $$65\%$$ of all streaming subscribers in the $$15$$ to $$65$$ age group. Aiden thought that this percentage might have changed during the pandemic and surveyed a sample of subscribers in this age group to get data.
He then carried out a Hypothesis Test to test his theory.

(i) If $$540$$ people responded to his survey, calculate the margin of error of his survey.

(ii) His survey revealed that $$372$$ of the responders said they now have a Netflix account.
Use your answer to Part (b)(i) above to create a $$95\%$$ confidence interval for the percentage of the population that have a Netflix account.

(iii) Use your answer to Part (b)(ii) above to conduct the Hypothesis Test, at the $$5\%$$ level of significance to test Aiden’s claim that the percentage of subscribers to Netflix had changed. State clearly your null hypothesis, your alternative hypothesis and give your conclusion in the context of the question.

(i) $$4.3\%$$

(ii) $$64.6\%<\hat{p}<73.2\%$$

(iii)

Null hypothesis: The percentage of subscribers to Netflix had changed.

Alternative hypothesis: The percentage of subscribers to Netflix had not changed.

Calculations: $$65%$$ is is within the confidence interval above.

Conclusion: The percentage of subscribers to Netflix had not changed.

Solution

(i)

\begin{align}\mbox{Margin of error}&=\frac{1}{\sqrt{n}}\\&=\frac{1}{\sqrt{540}}\\&\approx0.043\\&=4.3\%\end{align}

(ii)

\begin{align}\frac{372}{540}-4.3<\hat{p}<\frac{372}{540}+4.3\end{align}

\begin{align}\downarrow\end{align}

\begin{align}64.6\%<\hat{p}<73.2\%\end{align}

(iii)

Null hypothesis: The percentage of subscribers to Netflix had changed.

Alternative hypothesis: The percentage of subscribers to Netflix had not changed.

Calculations: $$65%$$ is is within the confidence interval above.

Conclusion: The percentage of subscribers to Netflix had not changed.

Video Walkthrough

## 2020 Paper 2 Question 3

(a) In a survey, $$1000$$ people are selected at random and asked some questions about online shopping.

(i) Find the margin of error of the survey.
Give your answer as a percentage correct to $$1$$ decimal place.

(ii) Of those asked, $$762$$ said they believe it is safe to give their credit card details when shopping online. Use your answer to Part a(i) above to create a $$95\%$$ confidence interval for the percentage of people who believe it is safe to give their credit card details when shopping online.

(iii) An online media company claims that 80% of people believe it is safe to give their credit card details when shopping online.
Conduct a hypothesis test, at the 5% level of significance, to test the company’s claim.
Give your conclusion in the context of the question and give a reason for your conclusion.

(i) $$3.2\%$$

(ii) $$73\%<\hat{p}<79.4\%$$

(iii)

Null hypothesis: The percentage of people who believe it is safe to give their credit card details when shopping online is $$80\%$$.

Alternative hypothesis: The percentage of people who believe it is safe to give their credit card details when shopping online is not $$80\%$$.

Calculations: $$80%$$ is not within the confidence interval above.

Conclusion: The percentage of people who believe it is safe to give their credit card details when shopping online is not $$80\%$$.

Solution

(i)

\begin{align}\mbox{Margin of error}&=\frac{1}{\sqrt{n}}\\&=\frac{1}{\sqrt{1000}}\\&=0.03162…\\&\approx3.2\%\end{align}

(ii)

\begin{align}\hat{p}-\frac{1}{\sqrt{n}}<\hat{p}<\hat{p}+\frac{1}{\sqrt{n}}\end{align}

\begin{align}\downarrow\end{align}

\begin{align}76.2\%-3.2\%<\hat{p}<76.2\%+3.2\%\end{align}

\begin{align}\downarrow\end{align}

\begin{align}73\%<\hat{p}<79.4\%\end{align}

(iii)

Null hypothesis: The percentage of people who believe it is safe to give their credit card details when shopping online is $$80\%$$.

Alternative hypothesis: The percentage of people who believe it is safe to give their credit card details when shopping online is not $$80\%$$.

Calculations: $$80%$$ is not within the confidence interval above.

Conclusion: The percentage of people who believe it is safe to give their credit card details when shopping online is not $$80\%$$.

Video Walkthrough
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(b)

(i) An aptitude test was taken by $$6500$$ candidates.
The test scores were normally distributed.
The mean score was $$480$$ and the standard deviation was $$90$$.
On the distribution shown below, the shaded area represents all candidates who were within one standard deviation of the mean.
Write the value of A and the value of B into the boxes below.

(ii) Use the Empirical Rule to estimate the number of candidates in the shaded region.

(i) $$A=390$$ and $$B=570$$

(ii) $$4{,}420$$

Solution

(i)

\begin{align}A&=480-90\\&=390\end{align}

and

\begin{align}B&=480+90\\&=570\end{align}

(ii)

\begin{align}0.68\times6{,}500=4{,}420\end{align}

Video Walkthrough

## 2020 Paper 2 Question 8

(a) The table below shows data (measured in kilotonnes) on how municipal waste was dealt with in Ireland from $$2005$$ to $$2011$$.

Year Total Waste Collected (Kilotonnes) Waste Sent to Landfill (Kilotonnes)

$$2005$$

$$2779$$

$$1833$$

$$2006$$

$$3100$$

$$1981$$

$$2007$$

$$3175$$

$$2015$$

$$2008$$

$$3104$$

$$1939$$

$$2009$$

$$2825$$

$$1724$$

$$2010$$

$$2580$$

$$1496$$

$$2011$$

$$2547$$

$$1344$$

###### Source: Environmental Protection Agency

(i) Find the difference between the percentage of the waste collected which was sent to landfill in $$2005$$ and the percentage of waste sent to landfill in 2011.
Give your answer correct to $$1$$ decimal place.

(ii) Find the mean amount of waste collected each year, from $$2005$$ to $$2011$$.
Give your answer correct to $$1$$ decimal place.

(iii) The mean amount of waste sent to landfill in the $$4$$ years from $$2009$$ to $$2012$$ was $$1398$$ kilotonnes. Find the amount of waste sent to landfill in $$2012$$.

(i) $$13.2\%$$

(ii) $$2872.9\mbox{ kt}$$

(iii) $$1028\mbox{ kt}$$

Solution

(i)

\begin{align}\left(\frac{1833}{2779}-\frac{1344}{2547}\right)\times100\approx13.2\%\end{align}

(ii)

\begin{align}\frac{2779+3100+3175+3104+2825+2580+2547}{7}\approx2872.9\mbox{ kt}\end{align}

(iii)

\begin{align}\frac{1724+1496+1344+x}{4}=1398\end{align}

\begin{align}\downarrow\end{align}

\begin{align}\frac{4564+x}{4}=1398\end{align}

\begin{align}\downarrow\end{align}

\begin{align}x&=(4)(1398)-4564\\&=1028\mbox{ kt}\end{align}

Video Walkthrough
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(b) The table below shows the percentages of all the different types of energy sources used in each of $$13$$ European countries during $$2014$$.

Country Renewable Oil Solid Fuel Gas Electricity Other

Austria

$$25$$

$$31$$

$$8$$

$$17$$

$$13$$

$$6$$

Bulgaria

$$8$$

$$19$$

$$28$$

$$11$$

$$11$$

$$23$$

Cyprus

$$5$$

$$82$$

$$0$$

$$0$$

$$13$$

$$0$$

Denmark

$$31$$

$$11$$

$$13$$

$$12$$

$$13$$

France

$$7$$

$$26$$

$$3$$

$$11$$

$$12$$

$$41$$

Germany

$$10$$

$$29$$

$$22$$

$$17$$

$$12$$

$$10$$

Greece

$$9$$

$$43$$

$$24$$

$$9$$

$$15$$

$$0$$

Ireland

$$6$$

$$42$$

$$13$$

$$24$$

$$15$$

$$0$$

Latvia

$$31$$

$$26$$

$$1$$

$$20$$

$$11$$

$$11$$

Luxembourg

$$4$$

$$61$$

$$1$$

$$19$$

$$12$$

$$3$$

Malta

$$2$$

$$82$$

$$0$$

$$0$$

$$16$$

$$0$$

Netherlands

$$4$$

$$38$$

$$10$$

$$33$$

$$10$$

$$5$$

Portugal

$$21$$

$$38$$

$$10$$

$$13$$

$$15$$

$$3$$

(i) What percentage of the energy used in Denmark in $$2014$$ was renewable?

(ii) Name the four countries that used a lower percentage of renewable energy than Ireland.

(iii) Name a country which did not use gas as a fuel source.

(iv) The table below shows the percentage of the different types of energy that were used in Ireland during $$2014$$.
It also shows one of the angles in a pie chart to represent this data.
The angle for renewable energy ($$6\%$$) is $$22^{\circ}$$.
Complete the table to show the angle for each energy type.
Give each angle correct to the nearest degree.

Type Percentage Angle

Renewable

$$6$$

$$22^{\circ}$$

Oil

$$42$$

Solid Fuel

$$13$$

Gas

$$24$$

Electricity

$$15$$

(v) Complete the pie chart below to show the energy types used in Ireland in $$2014$$.
Label each section clearly.

(i) $$20\%$$

(ii) Cyprus, Luxembourg, Malta and Netherlands.

(iii) Cyprus or Malta

(iv)

Type Percentage Angle

Renewable

$$6$$

$$22^{\circ}$$

Oil

$$42$$

$$151^{\circ}$$

Solid Fuel

$$13$$

$$47^{\circ}$$

Gas

$$24$$

$$86^{\circ}$$

Electricity

$$15$$

$$54^{\circ}$$

(v)

Solution

(i)

\begin{align}100-31-11-13-12-13=20\%\end{align}

(ii) Cyprus, Luxembourg, Malta and Netherlands.

(iii) Cyprus or Malta

(iv)

Type Percentage Angle

Renewable

$$6$$

$$22^{\circ}$$

Oil

$$42$$

$$151^{\circ}$$

Solid Fuel

$$13$$

$$47^{\circ}$$

Gas

$$24$$

$$86^{\circ}$$

Electricity

$$15$$

$$54^{\circ}$$

(v)

Video Walkthrough

## 2019 Paper 2 Question 1(a)-(b)

A business has $$28$$ employees.
Their ages, in years, are given below.

\begin{align}32&&41&&57&&64&&19&&21&&35\\ 18&&43&&54&&63&&65&&33&&22\\ 39&&58&&18&&42&&20&&34&&21\\ 49&&33&&55&&34&&57&&43&&63\end{align}

(a) Complete the stem-and-leaf diagram, showing the ages of all $$28$$ employees.

Solution
Video Walkthrough
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(b) Find the percentage of employees who are older than $$40$$ years of age.

$$50\%$$

Solution

\begin{align}\frac{14}{28}=50\%\end{align}

Video Walkthrough

## 2019 Paper 2 Question 7

Table A below shows the price index for thirteen countries in four food categories for the year 2015. (Source: Central Statistics Office.)
It shows, for example, that if you paid €$$132$$ for meat in Austria, then the same quantity of similar meat would cost you €$$117$$ in Germany, €$$106$$ in Ireland, €$$63$$ in Lithuania and so on.
The description is the same for the other food categories.

Country Meat Fish Milk, Cheese and Eggs Fruits, Vegetables and Potatoes

Austria

$$132$$

$$125$$

$$105$$

$$125$$

Germany

$$117$$

$$109$$

$$91$$

$$111$$

Ireland

$$106$$

$$108$$

$$128$$

$$136$$

Lithuania

$$63$$

$$73$$

$$85$$

$$77$$

Macedonia

$$56$$

$$59$$

$$71$$

$$48$$

Netherlands

$$111$$

$$99$$

$$93$$

$$104$$

Norway

$$157$$

$$117$$

$$175$$

$$150$$

Poland

$$54$$

$$64$$

$$65$$

$$62$$

Spain

$$85$$

$$89$$

$$96$$

$$95$$

Sweden

$$131$$

$$115$$

$$116$$

$$137$$

Switzerland

$$254$$

$$177$$

$$148$$

$$172$$

Turkey

$$78$$

$$105$$

$$122$$

$$86$$

United Kingdom

$$112$$

$$105$$

$$118$$

$$116$$

(a) Complete Table B below, to show the maximum value, the minimum value and the range of the data for the four food categories.

Meat Fish Milk, Cheese and Eggs Fruits, Vegetables and Potatoes

Maximum

$$254$$

Minimum

$$54$$

Range

$$200$$

Meat Fish Milk, Cheese and Eggs Fruits, Vegetables and Potatoes

Maximum

$$254$$

$$177$$

$$175$$

$$172$$

Minimum

$$54$$

$$59$$

$$65$$

$$48$$

Range

$$200$$

$$118$$

$$110$$

$$124$$

Solution
Meat Fish Milk, Cheese and Eggs Fruits, Vegetables and Potatoes

Maximum

$$254$$

$$177$$

$$175$$

$$172$$

Minimum

$$54$$

$$59$$

$$65$$

$$48$$

Range

$$200$$

$$118$$

$$110$$

$$124$$

Video Walkthrough
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(b) Write the data for the Fruits, Vegetables and Potatoes category in increasing order and hence find the median for that food category.

\begin{align}48&&62&&77&&86&&95&&104&&111&&116&&125&&136&&137&&150&&172\end{align}

\begin{align}\mbox{Median}=111\mbox{ euro}\end{align}

Solution

\begin{align}48&&62&&77&&86&&95&&104&&111&&116&&125&&136&&137&&150&&172\end{align}

\begin{align}\downarrow\end{align}

\begin{align}\mbox{Median}=111\mbox{ euro}\end{align}

Video Walkthrough
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(c)

(i) Find the mean of the data in the Meat category.

(ii) Use your calculator to find the standard deviation for the data in the Meat category.

(iii) List all the countries in the Meat category that are not within one standard deviation of the mean.

(i) $$112\mbox{ euro}$$

(ii) $$51\mbox{ euro}$$

(iii) Macedonia, Poland and Switzerland.

Solution

(i)

\begin{align}\mbox{Mean}&=\frac{132+117+106+63+56+111+157+54+85+131+254+78+112}{13}\\&=112\mbox{ euro}\end{align}

(ii)

\begin{align}\mbox{Standard Deviation}=51\mbox{ euro}\end{align}

(iii) Macedonia, Poland and Switzerland.

Video Walkthrough
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(d) A customer in Ireland buys some meat, fish, milk, cheese, eggs, fruits, vegetables and potatoes as detailed in Table C.
A customer in Poland buys the same items in the same quantities.
Complete Table C.

Country Meat Fish Milk, Cheese and Eggs Fruits, Vegetables and Potatoes Total Cost

Ireland

$$36.04$$

$$31.32$$

$$24.32$$

$$31.28$$

$$122.96$$

Poland

$$18.36$$

Country Meat Fish Milk, Cheese and Eggs Fruits, Vegetables and Potatoes Total Cost

Ireland

$$36.04$$

$$31.32$$

$$24.32$$

$$31.28$$

$$122.96$$

Poland

$$18.36$$

$$18.56$$

$$12.35$$

$$14.26$$

$$63.53$$

Solution
Country Meat Fish Milk, Cheese and Eggs Fruits, Vegetables and Potatoes Total Cost

Ireland

$$36.04$$

$$31.32$$

$$24.32$$

$$31.28$$

$$122.96$$

Poland

$$18.36$$

$$18.56$$

$$12.35$$