Question

Please show ALL steps for each part

Suppose that, in a certain population, heights of males are normally distributed with mean M1 = 178cm and standard deviation 01 = 4cm while heights of females are normally dis- tributed with M2 = 170cm and 01 = 3cm. Now suppose that a random sample of 25 males and 25 females are selected from the population. a) What is the probability that the average height of the males is greater than 180cm? b) What is the probability that the average height of the females is between 169cm and 171cm? c) What is the probability that the average height of the males is more than 10cm greater than the average height of the females? d) Suppose that we take the average of all 50 heights. What is the probability that this average height is between 172cm and 175cm?

Answer 1

x= height of male Y= height of Female Given that X n Normal (198, 4²) Yn Normal (170, 3²) We know that - X., X2, Yz oo. No nN (4,0²) then X= Exit n Normal (e, at Now, suppose that a random sample of 25 males are selected from the population then their Average Ž is also follows Normal diskristibution with Mi, ar i Xn Normal (û, 02) X - Normal (178, 42 Ics Scanned with C#mScanner

Therefore, je in Normal (178, 4 ) û = 1782 M = 178 San And also a random sample of as females ase Selected ther their Average T 2180 follows normal distribale With Me on the in Normal (uz 1 / 2 Here [Mr = 170, a 12 3 3 = = 0.36 Therefore 7 - Nosmax (130 , ) Sanned with CamScanner

(a) . we have tn Normal (178, 0.8²) ī n Normal (170, 0.6) X = Avg. height of mate : Ta Aug, height of female The Pobability that the average height of the maleg is greater than 180cm is PCT > 180) = P( > 100-4) =P( 2 > 1905) = P(Z > 2.5) = 1 - P(Z <2.5) = 1-0.99379 1PC82180) -0.006211 1987 Therefore the probability height of male is is 0.00621 that the Average greater than 180 cm ICS Scanilled with 1 Camcanner

(6) The Probability that the Average height of the females is between 169 cm and 171 con is P(169 <ř <17) = P(169-4 < z < 171 M) = P(14.470 << < 17.1070 = P(1.67<7<1.67) = P(251.62) - P(Z < -1.67) = 0.95254 - 0.04746 IP (169 <Ý <171) = 0.90508 There fore, the probability that the Average height of females is between 169 cm I and 171 cm is 0.90509 a Shanned with CamScanner

c) The Probability that the average height of the males is more than 10cm greater than the Average height of the females is Let X- 9=G P(*_9>10) = P(G>10) - PCG- > o since G=X-fn Normal (mi-te al 22 =) G=xqn Normal (8,1 ::P(8-7>10) = P(Z > 10- = P(Z > 10-8) = P(Z >2) = k P(zaz) -1-0.97725 P(V->10) -0.02275 2 Scanned with Cant Scanner

Here X, X, Xã - - - x = 4 ( 179 ) - Y,,Year. Yes NN (170,32) Lek K x, + X₂ 4 - .Xast Y, A Yet - Y25 50 F(x) = 25 (118) +25 (170) = 174 SO VCK) = Susus ante 39 st- 20.25 .: X. + X, * -- -+ X29+ Y, +Yu Y O, 50 height The is Probability between 172 that and this 175 Average cm o is P(1925*3175)= P(17244 < < < !75) =P(12-174 <7 < 1795 pray) =P(43752) Ics Scanned with CamScanner

P(1726 x< 175) = P(-4ę732) ap(222) - P(z sah) -0.97725 - 0.00003 P (172<x< 195) = 0.97722! CS Scanned with CamScanner