Leaving Certificate AppliedMathematical Applications

General Support MaterialResearch Topic 2011

N.B. This material does not assume to, in any way, cover the research topic or the questions, which might appear on the examination paper. It is intended as a general support to be used along with other sources and each teacher should interpret the research guidelines for themselves.

Statistics and Probability

Contents

Note to teacher

SEC Guidelines

Some useful resources

Organising surveys. Constructing charts

Means, Modes and Medians

Statistics Questions

Counting Techniques

Probability Worksheets

Probability Questions

Note to teacher

The exercises vary in difficulty. Some parts are included for the more able student. Please read through these notes and edit to suit your own group.

Some of the exercises are taken from past exam papers.

Some of the lessons are from Jim Marsden’s C.D. on Leaving Certificate Applied Mathematical Applications (Thank you Jim)

There is a vast amount of digital resources to support this topic. I have reviewed and selected some that you can copy and use as worksheets.

Some useful resourcesMaterials:

Dice, Playing Cards, Coins (Experiments on probability)Measuring Tapes (Student height)Thermometer (daily temperature)Simple Rain Gauge (daily rainfall)Set of newspapers (for analyzing stats)

WEBSITES:

cso.ie/studentcorner – has stats on each county and interactive tables on census results.censusatschool.ie - (You need to register as teacher before your class can use this site) has lots of interactive material. Eg : Phase 9 Results - use for asking questions on different charts. Resources - Strand 1: Statistics Strand 1: Probability projectmaths.ie - digital resources to support this topichttp://www.cimt.plymouth.ac.uk/oldfp.htm - more digital resourceswww.scoilnet.ie

DVDs

J.E Marsden Presentations. (CD of Worksheets, Fieldwork, Lesson plans, Key assignments, Activities, Puzzles) – Email: gereilish@eircom.net

Fun Maths CD, Jim Marsden, – Email: gereilish@eircom.net

Organising a survey.

Survey your class using this worksheet

Here are the results of a national survey. Carry out a similar survey on a group in your school.

This looks much better but there are still a couple of things missing.Can you spot them? That’s right – we need to know what the two colours in each bar mean and also where the data came from.

At last, a barchart that gives us some useful information. Always remember:

TITLE LABELS KEY

Now draw a useful bar chart of either the data given below or some data youhave collected yourself.

Travel Method Walk Bus Car Cycle Train. . OtherNumber of People 16 8 12 4 0 1

LCA results 2010

What % of females achieved a merit ---------------?

What % of males achieved a distinction ---------------?

What % of females achieved a pass ---------------?

What % of males achieved a merit ---------------?

Construct a Pie Chart for the LCA results for males in 2010Construct a Pie Chart for the LCA results for females in 2010Construct a Bar Chart for the LCA results for males in 2010Construct a Bar Chart for the LCA results for males in 2010

When looking at some statistics it is useful to look at a middle value. The Median is the middle number when a set of numbers is arranged in increasing or decreasing order. Numbers on five cards dealt from a pack and arranged in ascending order read 2, 3, 6, 8, and 9. The Median in this case is 6. If 6 cards were taken and read 3, 4, 5, 7, 8, and 9, there is no middle number and so in that case we take the average of the middle two 5+7 divided by 2 = 6.

Find the Median in the following sets of data. (Remember to arrange them in increasing order first)

1 Scores recorded in a darts match for one player for five innings:66, 103, 78, 89 and 58 Rearrange

Median = _________

2 Degree Celsius recordings for a given week at a sports ground were:23, 17, 22, 16, 14, 21 and 18. Rearrange

Median = _________

3 Rainfall recordings at a tennis court over a twelve month period were: 115, 99, 77, 23, 40, 55, 38, 30, 63, 70, 108 and 120 (mm)Rearrange

Median = _________

4 Times recorded in a 1500m race for seven competitors were:5min 15sec, 4min 45sec, 5min 05sec, 4min 30sec, 4min 25sec, 5min 20sec and 4min 35sec.

Rearrange Median = _________

In Statistics the Mean or Arithmetic Mean is the most widely used information value. It is the average value for a set of numbers. Numbers on five cards dealt from a pack read 2, 4, 7, 8, and 9. The Mean in this case is found by adding all the numbers and then dividing this result by the number of cards, i.e. 5. The sum of 2, 4, 7, 8, and 9 is 30. Dividing this by 5 we find the Mean is 6. If 6 cards were taken and read 3, 4, 5, 7, 8, and 9, then their sum is 36 and divided by 6 the Mean is 6.

Find the Mean in the following sets of data. (Remember to add them all together and then divide by the number of values in the set)

1 Scores recorded in a darts match for one player for five innings:66, 103, 78, 89 and 58

Mean = _________

2 Degree Celsius recordings for a given week at a sports ground were:23, 17, 22, 16, 14, 21 and 18.

Mean = _________

3 Rainfall recordings at a tennis court over a twelve month period were:115, 99, 77, 23, 40, 55, 38, 30, 63, 70, 108 and 120 (mm)

Mean = _________

4 Times recorded in a 1500m race for seven competitors were:5min 15sec, 4min 45sec, 5min 05sec, 4min 30sec, 4min 25sec, 5min 20sec and 4min 35sec.

Mean = _________

On a separate sheet (graph paper if you wish) make a trend graph to represent the results as given in question 3 and then draw a horizontal line to represent the Mean and the Median rainfall.

Compare the results for MEAN with those for MEDIAN

Mode

The MODE is the number that occurs most frequently

Write down the Mode for the following sets of Statistics:-

1 2, 3, 2, 5, 6, 2, 8, 2, 5, 9 and 10

Mode = ________

2 Hours overtime for a number of employees was as follows:-Pat 2, Ann 5, Jim 25, Hugh 4, Mary 4 and Liam 4

Mode = _______

3 Pocket money given by parents to first year students each week:-Bob €5, Dave €5, Anne €20, Judy €4, Jill €5 and Peter €4.

Mode = _______

4 Write down why you think that Mode is a more appropriate form of information in questions 2 and 3 than Median or Mean?

5 In a number of games of snooker a player recorded the following scores for each trip trips to the table:23, 8, 46, 112, 0, 46, 65, 34, 42, 38, 51, 36, 46, 34, 9, 15, 46, 17, 32, 34, 52, 44, 38, 45, 39, 52, 47, 46, 22

What is the MODE for these scores? _________

1 On a ring-board game involving 60 students the scores for each ring thrown were tallied and a frequency table set up. Complete the table. Each student throws only one ring in the first round.

Score Tally Frequency0 11 423 4 5678910 Total

(a) calculate the total tally and frequency columns (to check that 60 scores are accounted for)

(b) What is the modal score? The MODE is the number (score) that occurs most frequently

(c) If the second round only included those who scored 6 or higher, how many players would go to the second round?

Make a bar chart to represent your findings

2 Students carried out a sample survey of 100 cars arriving at Fairyhouse racecourse to determine where punters came from, based on the registration of their cars. The results were recorded as shown in the table

MH MH D LK MH MH D D LH LS LD MN CN C KE MH C D WX RNCE WW DL WX W KE WW CN WX LK WH G WD WH MH LM D KE D WHMH G RN CW KY WW DL WX LH CN OY TN CE TN SO C WW KE D RND MO G C KK CW LM MH LK WX OY G LH TS C D KE D MH SOMH CE RN KE KY MH MO CN MN LS G TN C WH D KE MH C MH WH

There are 30 different index marks in the sample. Make out your own table under the headings

Index County or city Tally FrequencyCN Cavan 4

Strategy of Checking: is your total tally correct? Is your total frequency correct?

Did you use any method to ensure that no index in the table was included more than

once?

(a) what is the modal index?

(b) If this information was reflected for a day on which there were 1500 cars present how many cars would there be for each of the five highest index marks?

(c) draw a graph to represent the information from (b)

Measure the height of each student in cms.

All measurements are to the nearest cm.Range 140-

150150-160

160-170

170-180

180-190

190-200

200-210

210-220

220-230

Number

Fraction of totalDecimal equivalent%

Use a ‘strategy of checking’, on the % line, to confirm that your work is correct

Make a bar chart and pie chart to represent your findings

Trend or bar chart

1 Get six regular household bills covering a known period. Telephone, mobile phone, gas or electricity would be suitable

2 You may take the totals or only one element from the bills and record the usage over the period (on a phone bill you might just take all local calls or internet charges –you decide)

3 Make a suitable table to record the amounts used and the cost involved for the period

4 Make a graph to illustrate the usage for the period

5 Calculate the total paid over the period

6 Calculate the average daily usage

7 Calculate the average cost per day

Pie chart

Work in small groups and carry out a survey of students (whole school or year or class as most appropriate) to ascertain the numbers who take part in any sport/s, the name of the sport, is the person male or female and the age of the person.

It will be helpful to design a common survey/tally sheet which will assist in doing the survey and collating the results from each of the groups to obtain the complete results. When all the facts are gathered and collated they will be displayed on a master sheet from which each student can complete the following:

1(a) calculate and tabulate the numbers of males and females who engage in any

sport and the number of each sex who do not engage in any sport and relate to total students

(b) tabulate the numbers (male plus female) for the five most popular sports(c) tabulate the numbers separately for males and females according to age (age

should be the age in years on the date of the survey) who engage in any sport and who do not engage in any sport

2 From your table at (a) make a pie chart to illustrate your results

3 From the results at (b) make a pie chart to illustrate your results

Statistics Questions

The table below shows successive two-monthly electricity bills for a household over the course of a year, in euro.

Period Jan/Feb Mar/Apr May/Jun Jul/Aug Sep/Oct Nov/DecBill (€) 220 165 135 120 140 215

(a) Draw a bar chart, to illustrate this data.(b) Calculate the average two-monthly bill, correct to the nearest euro.

A survey of fuels used in home heating is carried out.The results are shown in the table below.

FUEL OIL GAS COAL OTHERPercentage 40% 30% 25% 5%

(a) Draw a pie chart to illustrate this data.(b) If 200 families used oil, calculate the total number of families that were surveyed.

A survey of car sales is taken in an area.The results are shown in the table below.

Make of Car Ford Renault Toyota Skoda OpelNumber of Car Sales 800 400 750 300 550

(a) Draw, a bar chart to illustrate this data.(b) The total sales of Toyota cars is €13 875 000. Calculate the average price paid for a Toyota car.(c) Name another type of chart or graph on which the information from the survey could be shown.

The temperature is measured at noon each day for a week. The results are recorded in the followingtable:

Day Mon Tues Wed Thurs Fri Sat SunTemp (ºC) 20 15 19 12 16 13 17

(a) Draw a trend graph to represent this information.(b) Calculate the average noon temperature for the week. (c) The above temperatures and the noon temperatures for the following week are as follows:

20, 15, 19, 12, 16, 13, 17, 21, 18, 16, 13, 15, 14 and 13.

Complete the following frequency table:

Temperature 12 13 14 15 16 17 18 19 20 21No. of Days

(d) Write down the modal temperature for the 14 days.

The heights of six students are 153 cm, 162 cm, 161 cm, 155 cm, 161 cm and 159 cm.

What is the average height of the six students?

A school basketball team played 7 matches.Their scores were 71, 57, 26, 48, 68, 42 and 60.What was the median score?

The weights of five students are 45·6 kg, 47·1 kg, 43·7 kg, 44·6 kg and 50·6 kg.Calculate the average weight of the five students.

Counting Techniques

To examine the Fundamental Principle of Counting.

1. John is buying a car and he has a choice of 3 makes in his price range. Eachcar comes in 4 colours. How many cars does he have to choose from, if theseare the only options to be considered? Show calculations.______________________________________________________________________________________________________________________________2. A café offers 5 types of bread and 6 types of sandwich filling. How manydifferent sandwiches can they offer their customers, using only 1 filling at atime? (Note each slice of bread in a sandwich must be of the same type.) Showcalculations.______________________________________________________________________________________________________________________________3. A T shirt factory has 5 designs and each come in 6 colours. How manydifferent types of T shirts can be produced? Show calculations.______________________________________________________________________________________________________________________________4. A café offers starters, main courses and desserts. It has 4 kinds of starters, 6kinds of main courses and 5 desserts.a. How many choices is it offering its customers, if each takes a starter,main course and dessert? Show calculations._________________________________________________________b. How many choices is it offering its customers, if each customer onlytakes a starter and main course? Show calculations._________________________________________________________5. Sheelagh is buying a jumper and she has 12 jumpers to choose from. Eachbrand of jumper comes in 4 colours. How many brands is she choosing from?Show calculations.______________________________________________________________________________________________________________________________6. A café offers its customers 18 choices and each customer has a starter, maincourse and dessert. it offers 3 main courses and 2 starters. How many differentdesserts does it offer?Show calculations.

Probability - Coins

Divide into groups of 2 or 3. Toss a coin 10 times and record your results to show the number of heads out of 10 attempts and the number of tails.

Heads FractionNo out of 10

Decimal Tails FractionNo out of 10

Decimal

Group 1

Group 2

Group 3

Group 4

Group 5

Group 6

Group 7

Average for 7 groups __________ _________ ____________ _________

There were two possibilities in each toss - a Head or a Tail 1 in 2 of any toss being a Head or a Tail. We say that when a coin is tossed once the probability is 1 in 2 that it will be a Head or 1 in 2 that it will be a Tail. This can be expressed as 1/ or 0.5 or 50%

Do you think you could predict the results of your next 10 tosses exactly? _____________

Probability - Dice

Divide into groups of 2 or 3. Throw a dice 6 times and record your results to show the number of times each number comes up. Record your

Number on dice 1 2 3 4 5 6

Group 1

Group 2

Group 3

Group 4

Average for groups 1's ___ 2's ___ 3's ___ 4's ___ 5's ___ 6's ___

There were six possibilities in each throw of the dice. What is the probability of a 2 showing with 1 throw of the dice ________________

What is the probability of a 2 showing with 2 throws of the dice ________________

What is the probability of a 2 showing with 4 throws of the dice ________________

What is the probability of a 2 showing with 6 throws of the dice ________________

Probability - Cards

Questions to familiarise students with the deck/pack of cards

1. How many cards are in the deck/pack?__________________________________________2. How many different types of cards, i.e. suits, are there?__________________________3. Name the ‘suits’.______________________________________________________________4. How many black suits are there? ______________________________________________5. How many red suits are there?_________________________________________________6. How many cards in each suit?__________________________________________________7. List the cards in any one suit.___________________________________________________8. What is the name given to the card with an A written on it? _____________________9. What is a ‘face’ (picture) card?_________________________________________________10. How many ‘face’ cards are there in each suit?___________________________________11. Name the ‘face’ cards. (Aces are not considered to be ‘face’ cards.)________________12. How many numbered cards does each suit have?________________________________13. How many even numbered cards does each suit have?___________________________14. How many odd numbered cards does each suit have?____________________________15. How many non-numbered (lettered) cards are in each suit?_______________________16. Which card is non-numbered and is not a face (picture) card?_____________________17. How many black cards are there in a fair deck?__________________________________18. How many red cards are there in a fair deck?____________________________________19. In the whole deck:a. How many even numbered cards are there?_____________________________________b. How many odd numbered cards are there?______________________________________c. How many lettered cards are there?____________________________________________d. How many picture (face) cards are there?_______________________________________e. How many of each kind of card are there in the full pack? (i.e. how many 1’s, 2’s, …Aces, Kings, etc)?______________________________________________________________

f. What is the lowest numbered card you can pick in any suit?______________________

Probability questions for a fair Deck of Cards

1. What is the probability that the card is red?_____________________________________2. What is the probability that the card is a from the heart suit?_____________________3. What is the probability that the card is a Queen?________________________________4. What is the probability that the card is the Queen of Spades?____________________5. What is the probability that the card is a picture card?_ __________________________6. What is the probability that the card is a two-eyed Jack?_________________________7. What is the probability that the card is a one-eyed King?_________________________8. What is the probability that the card is even numbered?_________________________9. What is the probability that the card has a number on it?________________________10. Not all the cards look the same when you turn them around. What is the probability that the card does not look exactly the same when it is turned around?___________

Work in groups of 3 or four with a pack of 52 cards. Start by answering the following questions:-1 How many diamonds in the pack of cards? ________2 What is the probability of cutting a diamond on the first try? _______

Each group cuts the pack 52 times (shuffle the pack between each try) and records their results which are then entered in the following table -

Number of times each suit was cut

Diamond Spade Heart Club

Group 1

Group 2

Group 3

Group 4

Group 5

Total

How many times out of the total 260 tries did each suit show?Diamond _____ Spade _____ Heart _____ Club ______

How many times out of 52 tries would you expect to cut an ace? _______Can you simplify this? ______

What is the probability of cutting a red king in 52 tries? _________ or ___ _______ What is the probability of cutting a picture card in 52 tries? _________ or _________

What is the probability of cutting a queen in 52 tries? _________ or _________

Probability Questions

There are 6 red balls and 5 black balls in a bag.A ball is chosen at random from the bag.What is the probability that the chosen ball is red?

A die is thrown.(i) Write out all possible outcomes.

(ii) What is the probability of getting a 4? (iii) How often would you expect to get a 4 if the die was thrown 300 times?

A letter is picked at random from the word STATISTICS.What is the probability that the letter chosen is T?

A day of the week is chosen at random.What is the probability that it begins with the letter ‘S’?

A bag contains 5 red balls and 6 white balls.A ball is picked at random.What is the probability that the ball chosen is red? --------------

David has fourteen socks in a box. Eight are black and the rest are white.David picks out a sock at random.Find the probability that the sock is white -----------

.

A coin is tossed. Write down the probability of getting a tail.

The coin is tossed 400 times.How many times would you expect to get a tail?

A die is thrown. What is the probability of getting a 5?

The numbers 1 to 101 are used in a raffle.What is the probability that the first number chosen is an odd number?

There are 6 red balls and 4 white balls in a bag. A ball is picked at random out of the bag. What is the probability that the ball is white?

The numbers 1 to 50 are used in a lottery. What is the probability that the first number chosen contains the digit 5?

A box contains 50 raffle tickets, each having a different whole number from 1 to 50. If a ticket is drawn at random,

what is the probability that the number chosen is less than 10?