Question

1. Suppose that 81% of the people in Houston live in the city and 19% of the people live in the suburbs. If the 21,000 undergraduate students from UH represent a random sample of the population, what is the probability that the number of UH undergraduates are from suburbs will be fewer than 4,000? (Hint: Apply normal approximation to binomial probability via CLT.) 2. Suppose that the body mass index (BMI) measure for adults is normally distributed with mean 21.7 and standard deviation 3.6. (a) One is considered as over-weighted with a BMI higher than 95% of the population. What is the cutoff value for this definition of overweight? (b) Obesity is defined as a BMI above 30. What is the proportion of obese adults? (c) For a randomly selected group of n = 10 adults, let X "bar denote the average BMI and s2 denote the sample variance. i. What is the likelihood that the average BMI of these 10 adults is above 25? ii. What is the probability that the sample variance s2 is above 2? Express this proba- bility in terms of an appropriate Chi-squared X^2 distribution.

Answer 1

1 Lut X denotas numher of peopla Livning in Hu aulurks out of he Yandomly slocteod 21,000 under g7a duate to n Thun X Binomial ( 2 ',000,0.19) now, by uaing CLT N ( 21,00x aja, J2iaoxo 14 x0 henu, p(xc4 000) 21000x J Ja X 4000-21000 xo.19 21000x0-1a XO-8i N 2 1000x0 19 x 0 *g Co.n5A) O.S698 fouws Nai) ut x denote BmI fa in olividlual ,Kun X~M2L7,3-6) to ind X at ai we need PCx<x)=0.9s wsing 2-s607e, COTYPonding to K-21-7- (.G6 -) 3. 6 we know O,GS is 2Seore 1 27-62 b) PCx 30) = F P( X3°)= 36 t(2-3056) = 0. 0106 Pro porhion

3.6 N(21.7,3) C iher 25-21-7 3-6/STo hence PC >25) -l- p( 217 O-00187 (i) We kn gw test statisrc n)S uus a chi-squor 2 degreu of freelom dastoiburon wi Compute We heed ho PS2) P(4xs 29 3-62 362 2. < 1-3४४६१) 0.979 folous with dof a

Here I've attached the R code also:

Console Terminal pnorm(4000, 21000*0.19,sqrt(21000*0.19*0. 81)) #1 [1] 0. 5698146 > qnorm(0. 95,21.7,3.6) [1] 27.62147 1-pnorm(30, 21.7,3.6) [1] 0.01056774 1-pnorm(25,21.7,3.6/sqrt(10)) [1] 0.00187324 1-pchisq(2*9/(3. 6*3.6),9) [1] 0.9978902 A #2(a) #2 (b) #2 (c)i #2(c)ii