Consider a sequence of IID random samples, Y_1, Y_2, a) If the sample mean and v
ID: 3254271 • Letter: C
Question
Consider a sequence of IID random samples, Y_1, Y_2, a) If the sample mean and variance of the first n = 10^4 random variables are mu_10000 = 47.29 and S^2_10000 = 13.56, respectively, construct an approximate Central Limit Theorem 99% confidence interval for the true (population) mean of Y. b) For the situation in part a), how large a sample size would be required to make the half-width of the confidence interval no greater than 0.01? c) Suppose that 1123 of the first 10000 Y_i are at least its large as 50. Based on the Central Limit Theorem, construct an approximate 99% confidence interval for P(Y greaterthanorequalto 50)Explanation / Answer
a)here as sample size is very large we can use z (standard normal) distribution
std error of mean =std deviation/(n)1/2 =0.1356
for 99% CI, z=2.5758
hence 99% confidence interval =sample mean -/+ z*std error =46.9407 ; 47.6393
b)here margin of error E =2*0.01=0.02
for 99% CI ; z=2.5758
std deviation s=(13.56)1/2
hence sample size n=(z*std deviation/E)2 =~224923
c)here estimated proportion p=4123/10000=0.4123
for std error =(p(!-p)/n)1/2 =0.0049
for 99% CI, z=2.5758
hence 99% confidence interval =p -/+ z*std error =0.3996 ; 0.4250