Question

Question 2 [12 marks Let X be the maximum temperature in the month of January in Melbourne and let Y be the maximum temperature in the same month in Sydney. Historical records show that the distribution of X and Y have probability density functions fx(x), fy(y), respectively, shown in Figure 1 (a) By eye-balling the plot, explain why the population mean and variances of the maximum temperatures in Melbourne and Sydney are Hx 25°C, Hy28°C σ -1°C, and σ-2.25°C (b) Assume now that you have 10 independent measurements of maximum temperature X1o, and 5 independent measurements of maximum temp in Melbourne Λι, er ature in Sydney Yi,... . Ys (i) What are the distributions of TO 25) andY -28)? (ii) What is the probability that the sample average of the measured temperatures in Melbourne, denoted as X, will lie between 24.37°C and 25.32°C? (iii) What is the distribution of Y-X, where Υ is the sample average of the measured temperatures in Sydney? (iv) What critical temperature T, will yield PO-x T) 0.05? (X-25) and s30-28), where 陨and S루 and Y, respectively? (v) What are the distributions of the sample variances of X

Answer 1

Grdlen by temperatune in the Let xbe the maximum in Melbouoge & ld Y be month tempardloe Sthe Same month the makimuM Sylnay Hidooico Tecse stocs thd the dictabaion of Rantions PC, espadiva pabaleliy dersly x cnd Y have Byballing plot, paedaton mean the -the -tomperalunes in celbousna Narianas marimum mean fo X is 25 because Peah f f(x) ML =25 OccuYS at Pa y is a8 Sind baly ean od of mean 957. hes belween fox 951.lies belween 33 and a Sd - 2a5 =a So, Stomilarly fay a Sal = 31- 28-3 Sal= 1.5 Yaviante of Y= Naniante ofY 235 1.542-2.25

(b) (ndependeid neatoemds teperatine 5 independd naunats of temperabne L0 measurenut waxinum in mdbarno K- Xo and maximum in ci) The distibdiors oCx-25) - gA-sqrtn) folla stotnd X(xbar-mu) nomal distribelion Spt(lo ba-5) Stavdad naal dithibdig (Ci-s/spt (s) STt Caolat yban-) stardy So, CHbar-28) n8ma =Sant slishibuton Seuple aseroge oft C The probablity wrasd teperaluns conded Hhe thd th in Malboinno, as , hil lie bdwen a4.37 ( 25.30c P Cay.374bar25.32) = P C-1.1944l.ol) O.8205 The dichibdon of T-X, is the saue siy avenage of the measured teupesatines Ybar Y-NCag-25,1/10t 15/ EN C3, 0.55

C3) Citical tempeakie Tall ydld p-e T.< 0.05 PCY-XTC) o.05 PCR) 05 =1.645 so, Te= 3+ 1.645 sqat Co.55) Te - 4.21996 Critial tempetakue Te-4.a196 The distribalions of to -a5) &52) SK S1 the Sample vaiancu f where S aue These follow frelo ry of