Let X be the population that consists of all integer numbers from 100 to 200 and
ID: 2931238 • Letter: L
Question
Let X be the population that consists of all integer numbers from 100 to 200 and each number appears only once (i.e., X = {100, 101, 102, … , 200}). Calculate the population mean, variance, and standard deviation. If a random sample of size 64 is taken with replacement from X, what is (approximately) the probability that the sample mean will be less than or equal to 145? That is, calculate P(x 145)? Explain briefly how you arrive at your answer. If a random sample of size 100 is taken and the sample mean (x ) is used to estimate the true population mean () calculated in part a, what is the probability that the estimation error will be within ± 3? Explain briefly how you arrive at your answer.
Explanation / Answer
population mean, = 150
variance, = 850
standard deviation = 29.15476
sample mean follow normal distribution with mean = 150 , sd = s/sqrt(n) = 29.15476/sqrt(64) = 3.644345
P(Xbar < 145 )
= P(Z < (145 -150)/ 3.644345)
= P (Z<1.37)
=0.0853
if n = 100 , then sd(xbar) = s/sqrt(n) = 29.15476/sqrt(100) = 2.915476
P(-3 < (Xbar - 150) < 3)
= P(-3/2.915476 < Z< 3/2.915476)
= P(-1.028991 <Z<1.028991)
=0.697
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