Let X1,X2,...,X12 be independent identically distributed N(3.6,0.70) random variables, with an average X¯. (a) Calculate P(3.2≤X¯≤3.9) (b) Find the value of c for which P(3.6−c≤X¯≤3.6+c)=0.95

Answer 1

z = (xbar - mu)/(std dev/sqrt(sample size))

a) P(3.2 < xbar <3.9) =

P( (3.2 - 3.6(/(0,7/sqrt(12) < z < (3.9 - 3.6)(0.7/sqrt(12)) =

P(-1.98 < z < 1.48) =

P(z < 1.48) - P(z < -1.98) =

0.9306 - 0.0694 = 0.8612

b) For middle 95%, is between 2.5 and 97.5 percentile. z-score for 97.5 percentile is 1.96

1.96 = (3.6 + c - 3.6)/(0.7/sqrt(12)

1.96 = c * 0.202

c = 9.7