A TASK A LITTLE BIT OF HISTORY

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1 Saisics & Probabili Daa Represenaion & Inerpreaion Use scaer plos o invesigae and commen on relaionships beween wo numerical variables. use auhenic daa o consruc scaer plos, make comparisons and draw conclusions. Invesigae and describe bivariae numerical daa where he independen variable is ime. invesigae biodiversi changes in Ausralia since European occupaion. consruc and inerpre daa displas represening bivariae daa over ime. Evaluae saisical repors in he media and oher places b linking claims o displas, saisics and represenaive daa. invesigae he use of saisics in repors regarding he growh of Ausralia s rade wih oher counries of he Asia region. evaluae saisical repors comparing he life epecanc of Aboriginal and Torres Srai Islander people wih ha of he Ausralian populaion as a whole. Use informaion echnologies o invesigae bivariae numerical daa ses. Where appropriae use a sraigh line o describe he relaionship allowing for variaion. invesigae differen echniques for finding a line of bes fi. Saisics means never having o sa ou are cerain. A TASK The longer a person's legs, he faser he can run? Collec daa o answer he research quesion. Use a scaerplo o displa he daa. Analse he daa. Mach he analsis wih he research quesion. Publish our findings. A LITTLE BIT OF HISTORY 1986 Space Shule Challenger disinegraed afer he failure of o-ring seals. I is claimed ha improper analsis of o-ring emperaure failure led o he disaser. I has been suggesed ha proper analsis of a scaerplo and a line of bes fi ma have avoided he disaser. Space Shule Challenger

2 Bivariae Daa Bivariae daa is represened b wo variables. Wha is he relaionship beween he wo variables? Are he values of one variable relaed o he values of anoher variable? Can he values of one variable be used o predic he values of anoher variable? Value of Ausralian epors o China Year $Abillion This is an eample of bivariae daa. The wo variables are Year and $Abillion. Is here a relaionship beween Year and he value of epors o China? Can a predicion be made abou he value of epors o China in 2012? Scaerplos Value of Ausralian epors o China Year $Abillion See Technolog for drawing scaerplos. Value of Ausralian impors from China Year $Abillion Scaerplos give a visual represenaion of how he values of one variable are relaed o he values of anoher variable. $Abillion $Abillion Value of Ausralian epors o China Year The dependen variable, Year, is placed on he -ais. $Abillion depends on he Year, hus $Abillion is he dependen variable. Year is he independen variable. Value of Ausralian impors from China Year 246

3 Eercise For each of he following ables of bivariae daa: a) indicae which variable is dependen and which variable is independen. b) draw a scaerplo. c) aemp o predic he value of epors or impors in d) Wha confidence migh ou have in our predicion? i) Value of Ausralian ii) epors o Indonesia Year $Abillion Value of Ausralian impors from Indonesia Year $Abillion a) Indicae which variable is dependen and which variable is independen. b) Draw a scaerplo. c) Would ou advise ha he number of ads per da be increased? Number of TV advers per da and dail sales $Sales Ads a) Indicae which variable is dependen and which variable is independen. b) Draw a scaerplo. c) Can ou give advice abou he mos economical speed? Speed and diesel use, lires per 100km, of a large ruck L/100km km/h Chaper 18 Saisics 2 247

4 Correlaion A correlaion indicaes a relaionship beween wo variables. Tpes of relaionships linear or non-linear posiive or negaive srong or weak The above scaerplo indicaes no relaionship beween he wo variables. The above scaerplo indicaes a posiive linear relaionship wih a srong correlaion beween he wo variables. The above scaerplo indicaes a negaive possibl linear relaionship wih a weak correlaion beween he wo variables. Eercise 18.2 Describe he relaionship beween he wo variables as suggesed b each of he following scaerplos: Posiive slopes up. Negaive slopes down

5 Eercise Horse feeding, worming, and medicaion requires a reasonable esimae of he horse's weigh. The following able shows bivariae daa for 20 horses. a) Draw a scaerplo of he daa. b) Describe he relaionship as suggesed b he scaerplo. c) If a worm pase dose is 5g per 100 kg weigh, wha dose needs o be given o a horse wih a girh of 161 cm? d) How confiden are ou ha ou are giving he horse a reasonable amoun of worm pase? Measuremen of girh (cm) and weigh (kg) Girh Weigh Girh Weigh The following able shows bivariae daa of shoe size and spelling es scores for primar school sudens. Shoe size and spelling es scores for a) Draw a scaerplo of he daa. primar school sudens b) Describe he relaionship as suggesed b he scaerplo. c) I seems weird o hink ha spelling abili depends on shoe size. Can ou eplain he relaionship? d) Wha relaionship would ou epec beween shoe size and spelling abili for high school sudens? e) Tes our hpohesis. Size Tes Size Tes A srong correlaion beween A and B doesn' mean ha A causes B. I can be quie possible ha anoher facor C causes boh A and B. Is i possible ha increasing age causes an increase in shoe size? Is i also possible ha increasing age can cause increasing spelling es scores? Chaper 18 Saisics 2 249

6 Time Series A ime series is a sequence of daa measured a regular ime inervals. Time series are usuall visuall represened b daa poins joined b a line. Tpes of ime series rends linear or non-linear upward or downward no rend Prices Ausralian All Ordinaries share inde $A Ausralian Gross Domesic Produc Volage Capacior discharge Dae Dae Time The above ime series plo indicaes no rend. The above ime series plo indicaes a linear upward rend. The above ime series plo indicaes a non-linear downward rend. Eercise 18.4 Describe each of he following ime series plos: A rend is an indicaion of a general direcion

7 Eercise Rainfall has a significan impac on ma aspecs of Ausralian life and here is a keen ineres in weaher forecass. a) Draw a ime series plo of he daa. b) Describe he ime series plo. c) Aemp o make a forecas of he mean annual rainfall for he ne five ears. d) How confiden are ou of our forecas? Ausralian five-ear mean annual rainfall Year mm Year mm Temperaure also has a significan impac on ma aspecs of Ausralian life and here is a keen ineres in weaher forecass. a) Draw a ime series plo of he daa. b) Describe he ime series plo. c) Aemp o make a forecas of he mean annual emperaure for he ne five ears. d) How confiden are ou of our forecas? Ausralian five-ear mean annual emperaure Year C Year C I is hough ha sea surface emperaures affec he Ausralian climae. The following able refers o sea surface emperaures in he Ausralian region. a) Draw a ime series plo of he daa. b) Describe he ime series plo. c) Aemp o make a forecas of he mean annual sea surface emperaures for he ne five ears. d) How confiden are ou of our forecas? Ausralian five-ear mean annual sea surface emperaures Year C Year C Chaper 18 Saisics 2 251

8 Line of Bes Fi A 'line of bes fi' is a sraigh line ha bes represens he daa on a scaerplo. The line ma pass hrough some of he poins or none of he poins. Eercise A suden adjuss he power suppl unil a required curren is reached and hen reads he volage drop across he resisor. The following able is produced: a) Produce a scaerplo. b) Use echnolog o find a line of bes fi c) Use he line of bes fi o predic he volage drop across he resisor when he curren is 20 microamps). Commen. d) Use he line of bes fi o predic he volage drop across he resisor when he curren is 120 microamps). Commen. e) Use he poins from c) and d) o plo he line of bes fi. A + - V R The scaerplo and he line of bes fi can be obained from a spreadshee, a graphics calculaor, or online (for eample, 'line of bes fi apple'). See Technolog 18. Ohms Law eperimen Curren (microamps) Volage (millivols) b) Line of bes fi: Volage = 0.61 Curren 0.50 (gradien(m) = 0.61, -inercep = 0.50) c) curren = 20 microamps, Volage = 0.61 Curren 0.50 Volage = Volage = 12 milliamps The scaerplo suggess a srong linear correlaion beween he wo variables. The small number of poins suggess ha some care be aken in reling on using he line of bes fi o make predicions d) curren = 120 microamps, Volage = 0.61 Curren 0.50 Volage = Volage = 73 milliamps The curren of 120 microamps is ouside he range of he small number of poins aken (ma curren = 95 microamps). Care would need o be aken wih his predicion even hough he few poins sugges a srong linear correlaion. e) Plo he wo poins (20,12) and (120,73) and draw he line of bes fi hrough he wo poins. 252

9 Onl male crickes chirp. The chirping sound is made b rubbing a wing agains comb-like eeh on he boom of he oher wing. 2 Is here a linear relaionship beween he emperaure and he frequenc of cricke chirps? Some daa is shown: a) Produce a scaerplo. b) Use echnolog o find a line of bes fi. c) Use he line of bes fi o predic he chirp frequenc a a emperaure of 23 C. Commen. d) Use he line of bes fi o predic he chirp frequenc a a emperaure of 18 C. Commen. e) Use he poins from c) and d) o plo he line of bes fi. Temp ( C) Cricke chirps Frequenc (chirps/sec) The Gross Domesic produc (GDP) of a counr is he value of all goods and services produced b a counr. The GDP is generall acceped as a measure of a counr's econom. Ausralian GDP is shown: a) Produce a scaerplo. b) Use echnolog o find a line of bes fi. c) Use he line of bes fi o predic he Ausralian GDP in Commen. d) Use he line of bes fi o predic he Ausralian GDP in Commen. e) Use he poins from c) and d) o plo he line of bes fi. Populaion of Ausralia = 23 million Populaion of China = 1400 million 4 The Gross Domesic produc (GDP) of a counr is he value of all goods and services produced b a counr. The GDP is generall acceped as a measure of a counr's econom. China's GDP is shown: a) Produce a scaerplo. b) Use echnolog o find a line of bes fi. c) Use he line of bes fi o predic China's GDP in Commen. d) Use he line of bes fi o predic China's GDP in Commen. e) Use he poins from c) and d) o plo he line of bes fi. Ausralian GDP Year $billion GDP of China Year $billion Chaper 18 Saisics 2 253

10 Menal Compuaion Eercise Spell Scaerplo 2 Name wo properies of an isosceles riangle 3 Name wo properies of a square 4 If he angle sum of a polgon = (n 2) 180, wha is he angle sum of a quadrilaeral? 5 Wha is he gradien of he line = + 3? 6 Wha is he gradien of he line perpendicular o = 2 1? 7 Wha angle does he line = + 2 make wih he -ais? 8 In he riangle, wha is cos 60? 9 Facorise: = ( + 3)( + 1) 10 Two sides of a righ-angled riangle are 1 and 2, wha is he hpoenuse? You need o be a good menal ahlee because man everda problems are solved menall. (n 2) 180 = (4 2) 180 = 360 Angle sum of a quadrilaeral is 360 If wo lines are perpendicular hen he produc of heir gradiens is -1. m 1 m 2 = 'All hings are difficul before he are eas' - Thomas Fuller. Eercise Spell Correlaion 2 Name wo properies of an equilaeral riangle 2 3 Name wo properies of a recangle 4 If he angle sum of a polgon = (n 2) 180, 45 1 wha is he angle sum of a heagon? 5 Wha is he gradien of he line = 3 + 2? 6 Wha is he gradien of he line parallel o = 3 1? 7 Wha angle does he line = + 2 make wih he -ais? 8 In he riangle, wha is sin 45? 9 Facorise: Two sides of a righ-angled riangle are 1 and 3, wha is he hpoenuse? 1 'Opporuni is missed b mos people Eercise 18.9 because i is dressed in overalls and 1 Spell Non-linear looks like work' - Thomas Edison. 2 Name wo properies of a rhombus 3 Name wo properies of a parallelogram 4 If he angle sum of a polgon = (n 2) 180, wha is he angle sum of a ocagon? 30 5 Wha is he gradien of he line = 2 + 5? Wha is he gradien of he line perpendicular o = 2 3? 60 7 Wha angle does he line = 2 make wih he -ais? 1 8 In he riangle, wha is cos 30? 9 Facorise: Two sides of a righ-angled riangle are 2 and 3, wha is he hpoenuse? Nuclear Phsiciss work on undersanding nuclear science. Relevan school subjecs are English, Mahemaics, Phsics. Courses generall involve a Universi degree wih a major in nuclear phsics. 254

11 Compeiion Quesions Eercise Wha angle does carbon make a he cenre of he pie char? 15% The human bod 15% 70% Ogen Carbon Oher maerials Carbon angle = Build mahs muscle and prepare for mahemaics compeiions a he same ime. = 54 2 Wha angle does counr 3 Wha angle does China music make a he cenre of make a he cenre of he pie char? he pie char? Music preference Ausralian epors Rock Oher 25% 45% 30% Counr 25% 45% 30% Japan Oher China 4 The graph shows he value of Ausralian epors o China. When was he bigges increase in epors o China? $Abillion Value of Ausralian epors o China Hin: Where is he larges gradien? Use a ruler o find he seepes slope Year 5 A 20 cm conainer, iniiall full of waer, is leaking hrough a hole in he boom. Find he fases change in he deph of waer in cm/s. Deph (cm) Hin: change in gradien = = rae of change change in Time (seconds) Chaper 18 Saisics 2 255

12 Technolog Technolog 18.1 Spreadshees Spreadshees will draw Scaerplos Year $Abillion Highligh daa and inser a scaer char. Righ-Click various pars of he plo o forma pars o our liking Line of bes fi = LINEST(B1:B6,A1:A6,1,1) = or $Abillion = 11.3Year Technolog 18.2 Graphics Calculaors A graphics calculaor will draw Scaerplos. 1 Press STAT hen Edi and ener he independen variable in L1 and he dependen variable in L2 2 Press STATPLOT hen selec Plo 1 and selec he shown opions. 3 Se he Window 4 Press ZOOM hen choose Zoomsa A graphics calculaor will also draw and calculae he 'Line of Bes Fi'. 4 Press STAT CALC LinReg(a+b) L1, L2, VARS YVARS 1 Y1 GRAPH = or $Abillion = 11.3Year

13 A Couple of Puzzles Eercise Each birhda I have had a birhda wih he appropriae number of candles. If I have blown ou 120 candles, how old am I? ? ? ? A Game Twen is plaed b wo people wih he numbers 1 o 20. The winner is he person who crosses ou he number 20. Pla a couple of games and r o deermine a winning sraeg Saring a 1, ake i in urns o cross ou 1, 2, 3 or 4 numbers. The numbers mus be in order. A Swee Trick 1 This rick is a variaion of a rick in a previous chaper in which his grid of numbers was used. 2 Change he grid b adding, or subracing, he same number o each number in an row or column. 3 This grid happened b adding 2 o he 1s row, 3 o he 4h row, 5 o he 4h column, and subracing 5 from he 2nd column. 4 Your audience chooses an uncrossed number and ou cross off he oher numbers in he same row and column. 5 Repea unil all numbers are eiher chosen or crossed. 6 The sum of he four numbers equals equals he sum of each diagonal. Tr a 55 square, a 33 square, ec. Embellish he rick wih creaive ideas Chaper 18 Saisics 2 257

14 Invesigaions Invesigaion 18.1 Underake real-life research. 1 Form a eam and brainsorm an appropriae problem or issue. 2 Plan he research Define he overall research quesion. Define subse research quesions. Decide how o obain daa o answer he research quesions. Consider was of ensuring ha he daa collecion is unbiased. Consider he equipmen needed for he research. 3 Conduc he research Rehearse he daa collecion mehod. Collec he daa. 4 Analse he daa Look for errors/ouliers and decide wha o do wih he errors/ouliers. Calculae he appropriae descripive saisics. Choose an appropriae mehod of presenaion (Scaergrams, Hisograms ec). 5 Repor he conclusions Mach he analsis wih he research quesions. Do he answers o he research quesions indicae furher research quesions? Invesigaion 18.2 Line of bes fi and he spreadshee Invesigae A spreadshee can be used o calculae and draw 'lines of bes fi'. How? A line of bes fi is a sraigh line ha bes represens he daa on a scaer plo. Invesigaion 18.3 Line of bes fi Invesigae Differen 'lines of bes fi'. A line of bes fi is also known as he rend line. 258

15 Chaper Review 1 Eercise Bivariae daa relaing driver's age and road casualies (deah or serious injur) was colleced in Grea Briian in a) Draw a scaerplo of he daa. Driver's age and casualiies Age Casualies b) Describe he relaionship as suggesed b he scaerplo. c) Which driver is more likel o be a road casuali, a 19 ear-old or a 23 ear-old? d) Predic he number of casualiies for he ear-old group. How confiden are ou in making our predicion? 1 (<17) (17) (18) (19) (20-24) (25-29) (30-39) (40-59) Esimaes have been made of Ausralia's meric onnes of carbon dioide emissions (M CO 2 - e). a) Draw a ime series plo of he daa. b) Describe he ime series plo. c) Aemp o make a forecas of he mean annual rainfall for he ne five ears. d) How confiden are ou of our forecas? Emissions Year MT CO 2 - e In an eperimen measuring he growh of micro-organisms in a liquid culure, i was noed ha as he number of micro-organisms increased, he culure became darker. Bivariae daa was colleced on he opical densi (OD) of he culure and he concenraion (C) of he organisms in he culure. a) Produce a scaerplo. b) Use echnolog o find a line of bes fi. Opical Densi OD C c) Use he line of bes fi o predic he concenraion of micro-organisms wih an opical densi of 5. Commen d) Use he line of bes fi o predic he 12 7 concenraion of micro-organisms 15 8 wih an opical densi of 30. Commen e) Use he poins from c) and d) o plo he line of bes fi Chaper 18 Saisics 2 259

16 Chaper Review 2 Eercise Global mean sea level se o base level of 0 mm in 1990 is shown in he able. a) Draw a scaerplo of he daa. b) Describe he relaionship as suggesed b he scaerplo. c) Use he scaerplo o esimae he sea level in d) Wha confidence migh ou have in our esimaion? Global mean sea levels Year Sea level (mm) Esimaes have been made of Global mean sea level over he las 120 housand ears se o a base of 0 m wen housand ears ago. a) Draw a ime series plo of he daa. b) Describe he ime series plo. c) Aemp o make a forecas of he global mean sea level in 10 housand ears ime. d) How confiden are ou of our forecas? Years Global mean sea levels Sea level (m) 1 ( ears ago) ( ears ago) ( ears ago) 83 4 ( ears ago) 80 5 ( ears ago) 75 6 ( ears ago) 75 7 ( ears ago) 55 8 ( ears ago) 30 9 ( ears ago) 0 10 ( ears ago) (now) The relaionship beween he number of bedrooms and he price of a house is being eamined b deailing adverisemens of houses for sale (resriced o one suburb onl). Bedrooms and house prices a) Produce a scaerplo. Bedrooms $Price b) Use echnolog o find a line of bes fi c) Use he line of bes fi o iniiall se he price of a 3 bedroom house. Commen c) Use he line of bes fi o iniiall se he price of a 5 bedroom house. Commen e) Use he poins from c) and d) o plo he line of bes fi. 260