Exploring randomness

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Jeffry Rogers
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Explorig radomess swers Skills hek Write i asedig order:,, 9, 9, 9, 9, 9,,,,, a Media is etwee 9 ad 9 9.

kg d Rage 9 kg e f Lower quartile th oservatio 9 kg Upper quartile 9 th oservatio th oservatio + kg IQR 9.

Total umer of ways umer of ways where all have row eyes 9 Time to get home (mis) Numer of studets t < t < t < t < Exerise a Disrete (as

1 Explorig radomess swers Skills hek Write i asedig order:,, 9, 9, 9, 9, 9,,,,, a Media is etwee 9 ad 9 9. kg Mode kg (most frequet) Mea total umer of oservatios 9. kg d Rage 9 kg e f Lower quartile th oservatio 9 kg Upper quartile 9 th oservatio th oservatio + kg IQR 9.. kg a Cotiuous Numer of teahers y a Cotiuous ge i years Mass (kg) x Numer of hikes w < w < w < w < a Cotiuous a! Total umer of ways umer of ways where all have row eyes 9 Time to get home (mis) Numer of studets t < t < t < t < Exerise a Disrete (as they are asked for a aswer i whole miutes) Numer of studets y 9 Time spet studyig maths (miutes) t t < t < t < t < mis a First diagram D Seod diagram Third diagram C Oxford Uiversity ress : this may e reprodued for lass use solely for the purhaser s istitute Worked solutios: Chapter

2 Exerise a goal (highest frequey of ) a h < (highest frequey of ) a t (miutes) Frequey CF t < t < 9 t < 9 t < t < t < t < 9 t < Cumulative frequey y Waitig time (miutes) % waitig loger tha miutes % % Estimates from tale ad graph: Mea. mis, Media. mis Modal iterval is t < mis Cumulative frequey Height (m) Frequey h < h < h < h < Height (m) CF y x Height (m) t Numer less tha m % m % % d Mea. m Media 9 m Modal lass is h < m a Mode Media Use: + a a or + a+ + oth give a This makes the set imodal at ad. a Series is l a + l a + l a + l a +... This is a G with ommo ratio r Sie r <, this overges with sum l a l a r - r- å la ( r ) r- ( r) æ ö la - ç è ø laæ ö ç çè ø la æ ö ç çè <. l a ø æ ö ç. çè ø Mea Need Exerise C rrage i order:,,,,,,,,,,, a Rage 9 m Media th readig. m LQ th readig +. m d UQ 9 th readig +. m e IQR.. m a From graph, media m.. m y Cumulative frequey % of the oxers have a reah with a maximum differee of. m Legth (mm) x Oxford Uiversity ress : this may e reprodued for lass use solely for the purhaser s istitute Worked solutios: Chapter

3 a y Cumulative frequey i ii t Time (miutes) From graph, media m IQR... mis + + p 9 p + + p + q p + q q a From graph, umer of studets Lower quartile th th oservatio 9 Upper quartile th oservatio Middle % lie etwee 9 ad a 9, Numer gettig more tha 9 % awarded grade %.% a mis IQR UQ LQ mis mis Exerise D a + + a + ( a ) + ( ) Solve to fid that either a, or a,. Give that a a,. a a a a a + a Mea a Variae Mea a+ + a+ Variae. a Mea 9. Stadard deviatio. IQR 9 9 x y x y x y () 9 x y x y x y 9 9 () From, y x,so x x x x x x x x + x x + x + x x is positive, so x ad y (from ()) The last sore sums are ad sore sums are,,,, 9, Rage 9 IQR 9... k k k k k a Mea k Variae x + + ( + ) + ( + ) + x k k k k 9k k + k 9 k + k k Mea d The variae will e uhaged as the spread of the data aout the mea is uaffeted. r a a Mea a r Exerise E,,, a,, a a a,,,,, d,,, e a d e a Oxford Uiversity ress : this may e reprodued for lass use solely for the purhaser s istitute Worked solutios: Chapter

4 d e Exerise F Freh x x x Malay x + x + x + x x (Freh ad Malay) Golf 9 x iao a (golf ut ot piao) (piao ut ot golf ) 9 a... C a X YXY X Y ( ) X Y X Y a ( ) Exerise G (Sophie ad Jerome seleted) a (two lies) (two differet piees) a R,R,R æö ç çè ø æö ç çè ø (ot all same olor) R, R, R Y,Y,Y 9 a (all orage) (all differet olors) 9 (at least oe gree) (o gree) 9 a i (o sores o st shoot) (ill misses) (o hits)... ii (ill sores o rd shoot) (ill misses, o misses, ill misses, o misses, ill hits) iii (ill sores o th shoot) (ill misses, times, o misses, times, o hits)... (ill wis) p p. p. p. +. p Oxford Uiversity ress : this may e reprodued for lass use solely for the purhaser s istitute Worked solutios: Chapter

5 (o wis) (ill wis) p... a w y x z U Exerise H a U ( ).... h h h... a ( )... ( ) \ a ( ) ( ) ( ). U a x x x x ut >,, x so x x C C x y z w x y w x y z C w x z x y z w ( x + y)+ ( w + x)+ ( x + z) x C C C x CC C C U \ + + Exerise I a i. ii a. No. If it was fair we would expet aroud or ourrees of eah umer. The spier appears to e iased towards.. a,,,,,, (,,,,,, ) a,,,,,,,,,,9,, + 99 Oxford Uiversity ress : this may e reprodued for lass use solely for the purhaser s istitute Worked solutios: Chapter

6 Exerise J a x x x T x x x x x x T\ ( T ) a T + + T ( ) ( ).. U ( ) ( ) ( ).. (roller skateoard) a (eve ot mult. of ) roller skateoard skateoard.9. (eve ot mult. of ) ot mult. of, x x x ad x x d x x x ad x x ( laptop ad desktop ). desktop 9. (laptop desktop) 9 Spaish Teh (Spaish Teh) ( ad ). ( Teh). a ( U adv) ( U V) (mutually exlusive) U V V U orv U V. U V (passed passed ). ad...% of those who passed the first also passed the seod. 9 (white o d lak o st) ( st lak ad d white ) lak ( o st).. a (male left haded) (right haded) (right haded female) (right haded ad female) female (other is male oe is male) ( oth male ) oe ( is male) ot ( females) Exerise K... ad idepedet C. C ad C ot idepedet RedQuee ad idepedet C red fae ard C adc are idepedet C Quee ad fae ard Quee C ad C are ot idepedet Oxford Uiversity ress : this may e reprodued for lass use solely for the purhaser s istitute Worked solutios: Chapter

7 U a ( ) ( ) ( ( )) ( ) ( ) ad are ot idepedet ( ) ad are idepedet ( ) ( ) + ( ) ( ) + ( ) ad are idepedet ad idepedet a... ( ) a a a a a a a a a or a Sie () ad must e a T, H,eve H,or a x, x, x,eve x, x, x, or,, (divisile y ) (last digits divisile y ) (x, x,, ) + (x, x,, ) + (x, x,, ) (x, x, 9, ) 9 Sie there is a large umer of itegers, a assume (odd) (eve) Suppose we selet. The (at least oe odd) Need.9. log. > log. eve >. eed to selet itegers (Julia fails to sore a wier). (Julia fails times i a row) (.) (at least oe wier i shots) log. log.. Julia eeds to hit 9 shots Exerise L (Rai, ot late)... (Corret diagosis) ( Sore out of ) a + Oxford Uiversity ress : this may e reprodued for lass use solely for the purhaser s istitute Worked solutios: Chapter

8 + a (orage, orage, orage) 9 9 (at least oe purple) (all orage) (more orage tha purple) o,o,o o,p,o o,o,p p,o, o a R,R,R H,H,H (all same suit) d (faes i same suit) Exerise M a (eve) firstoxeve eve (first ox eve) 9 9 a (defetive) (first mahie defetive) f d d a (V) V G. a ( from d ox) ( st ox W oth from d ox W) ( WWW) ( WWd ox) a i a ' ' ' 9 ( ) ii ( ) ( ) ( ) iii ( ) ( uiased ad ot ) ot (uiased ot ) a (o-smoker) a. (lug prolems heavy smoker) lug prolems ad heavy smoker heavy smoker R C R R C R 9 a (o time) ( o time ) ( late). ad o time otime..9. ad late.. late.. Oxford Uiversity ress : this may e reprodued for lass use solely for the purhaser s istitute Worked solutios: Chapter

9 a 9 9 Jar Jar, (Jar Jar ) (Jar Jar ) Jar a (male) (maagemet male) male ad maagemet male (marketig female) (seod mahie D) seod mahie ad D D (, S ),, S female ad marketig female S.9... (vowel) Review exerise 9 Mode, so smallest set is,, x, y Media, so set is,,, y Mee, so y y set is,,, WORKED SOLUTIONS ad idepedet Let x : x x x x x From graph: a a a Media kg Middle % kg There are studets roaility (oth o ommittee) ( girls, oy) + ( girls) 9 9 X Y does ot ivest (divided) ( Y divided) divided o divided divided o divided Y ad divided 9 divided Oxford Uiversity ress : this may e reprodued for lass use solely for the purhaser s istitute Worked solutios: Chapter 9

10 (st G d G) GG dg a (prime) (eve) (multiple of ) d 9 (divisile y eve) a i mea 9 a mi ii Variae mi 999 divisile y ad eve eve 9 mi 9 SD No, as 9% of the studets should get marks withi stadard deviatios of the mea, i.e. etwee ad, ad 99.% withi stadard deviatios of the mea, i.e etwee 9 ad. There are umer of the form k as k rus from to. ut every other oe is eve, so umer of odd umers is Hee (divisile y ) Review exerise Mea height a!!!!.m S!!! ( osoat) 9! 9!!!!!! 9! a (a divisile y ) (a perfet square) x, x roaility.9.9. a Mea. m Stadard deviatio. km a Mea.9 Stadard deviatio. RC CF Media th oservatio. Numer of hildre with RC >. 9 a (all stats ooks i first plaes) (all alulus ooks together) a (Keith wis et i.e. wiig).... ( played a higher rak lost)!!!!!!!!!!!!!!!!!!!!!!!!... Oxford Uiversity ress : this may e reprodued for lass use solely for the purhaser s istitute Worked solutios: Chapter

CHAPTER 2. Mean This is the usual arithmetic mean or average and is equal to the sum of the measurements divided by number of measurements.

CHAPTER 2. Mean This is the usual arithmetic mean or average and is equal to the sum of the measurements divided by number of measurements. CHAPTER 2 umerical Measures Graphical method may ot always be sufficiet for describig data. You ca use the data to calculate a set of umbers that will covey a good metal picture of the frequecy distributio.