Question

(1 point) You have information to suggest that a certain continuous variable of a population has a mean of 12.56 and a standard deviation of o= 6.15. You are to randomly pick n = 68 individuals from this population and observe the value of the population variable on each. This value is to measured as X. After the random sample of n 68 has been taken, you are asked to consider the behavior of the statistic X. (a) Complete the statement below. Use at least two decimals in each numeric answer. The distribution of X ? VW with a mean fx and a standard deviation (b) Find the probability that mean of the sample of n 68 is between 11.76 and 12.96.

Answer 1

Given : u = 12.56,0= 6.15 and n=68 By Central Limit Theorem, ă follows a Normal distribution with mean = u = 12.56 standard error = 0/Vn= (6.15/V68) = 0.7458 Then z=(x-mean)/SD = (x-12.56)/0.7458 follows standard Normal. When 7=11.76, z=(11.76-12.56)/0.7458 = -1.07 P(Z<-1.07) =0.1423 [using Normal table or Excel : NORM.S.DIST(-1.07, TRUE)] When 7=12.96, z=(12.96-12.56)/0.7458 = 0.54 P(Z<0.54) =0.7054 [using Normal table or Excel : NORM.S.DIST(0.54, TRUE)] P(11.76<3 <12.96 ) = P(-1.07 <Z < 0.54 ) = P(Z < 0.54) - P(Z < -1.07 ) = 0.7054 - 0.1423 = 0.5631