ENGI 4421 Probability and Statistics Faculty of Engineering and Applied Science Problem Set 1 Descriptive Statistics

Transcription

1 ENGI 44 Probability ad Statistics Faculty of Egieerig ad Applied Sciece Problem Set Descriptive Statistics. If, i the set of values {,, 3, 4, 5, 6, 7 } a error causes the value 5 to be replaced by 50, (a) what effect will this chage have o the media value? (b) what effect will this chage have o the mea value? (c) what effect will this chage have o the mode? (d) which of mea ad media is the better measure of locatio for this chaged data set ad why?. The total scores obtaied o a pair of biased ( loaded ) dice whe they were throw 00 times are summarized i the frequecy table below: Score x Frequecy f Score x Total: 00 Frequecy f (a) Display this iformatio o a bar chart. (b) Idetify the mode. (c) Costruct the cumulative frequecy table ad hece fid the media. (d) Fid the arithmetic mea. (e) Fid the sample variace. (f) Commet o ay evidece for skew i these data.

2 ENGI 44 Problem Set Page of 5 3. The grades received by a egieerig class i a certai course are as show i the frequecy table below: Grade Frequecy A 34 B 47 C 50 D 8 F 6 Display this iformatio graphically i the form of (a) a bar chart (b) a pie chart Show the calculatio for the agle of ay two segmets of the pie chart. I questios 4 to 7 below, use Miitab (or some other software package) to aswer the questios. If you do ot use Miitab, the state what software package you have used. 4. For the followig data set, (also available as a plai text file here), (a) create a pritout of Miitab s stadard Descriptive Statistics output, icludig the default bar chart with superimposed ormal graph ad the default boxplot, (as was demostrated i the Miitab tutorial), (or provide equivalet iformatio from some other software package). (b) What evidece do you see for skewess i these data?

3 ENGI 44 Problem Set Page 3 of 5 5. For the followig data set of 00 values, (also available as a plai text file here), (a) create a pritout of Miitab s stadard Descriptive Statistics output, (or provide equivalet iformatio from some other software package). (b) costruct a stadard boxplot, orieted horizotally, with gridlies at itervals of 0.5 uits. (c) idetify ay outliers (list their values). (d) costruct a histogram, usig as class boudaries the cosecutive itegers, from 0 to the ext iteger above the largest observed value. (e) What evidece do you see for skewess i these data? 6. For the followig data set of 30 values, (also available as a plai text file here), (a) create a pritout of Miitab s stadard Descriptive Statistics output, (or provide equivalet iformatio from some other software package). (b) costruct a stadard boxplot ad add a symbol to idicate the locatio of the arithmetic mea. (c) idetify ay outliers (list their values). (d) costruct a histogram, class widths of 0., from 0 to. (e) What evidece do you see for skewess i these data?

4 ENGI 44 Problem Set Page 4 of 5 7. For the followig data set of 60 values, (also available as a plai text file here), (a) costruct a frequecy bar chart, with classes of width 5 ad cetres at { 3, 37, 4, 47,..., 67, 7 }. (b) create a pritout of Miitab s stadard Descriptive Statistics output, but display oly the umber cout, mea, stadard deviatio, media ad quartiles, (or provide equivalet iformatio from some other software package). (c) idetify the modal class ad the media class from your bar chart. (d) use the grouped data (from the bar chart) to calculate the mea, the populatio stadard deviatio ad the sample stadard deviatio (you may fid this easier to do i a spreadsheet program such as Microsoft Excel ). (e) Why are the mea ad stadard deviatio that you calculated i part (d) differet from the Miitab values? 8. Problem Set Bous Questio, Descriptive Statistics Prove that, for ay real costat a Hit: Use the idetities k x, i i ) i= i= ( x x) < ( x a = k (for ay costat k ) ad x i = x.

5 ENGI 44 Problem Set Page 5 of 5 Additioal Note for Questio 8: It the follows that, for ay radom sample of size draw from a populatio of true mea µ, ( x x ) ( x μ ) i i= i= (with equality oly i the very ulikely evet that x = μ ). N Recall that σ = ( x N i μ ) (where there are N members i the etire populatio). Oe ca the speculate [correctly] that, o average, ( ) xi x σ ( xi x ) is said to be a biased estimate of σ o average. variace s. i σ, i that it uderestimates the true value of The bias disappears whe this variace formula is replaced by the sample I the sectio o estimators we shall see a proof that s is a ubiased estimate of σ. Retur to the idex of questios O to the solutios to this problem set