The Common- wealth of Pennsylvania is concerned about the dwindling number of fa

Question

The Common- wealth of Pennsylvania is concerned about the dwindling number of family-owned farms and the number of smaller, less efficient farms. For a random sample, the total acreage of each farm was recorded. The mean was 1125 acres, with a standard deviation of 250. The shape of the distribution of areas is not normal. Approximately what proportion of areas is between 625 and 1625 acres? Approximately what proportion of areas is between 375 and 1875 acres? Approximately what proportion of areas is less than 375 acres? Approximately what proportion of areas is between 750 and 1500 acres?

Explanation / Answer

MEAN = 1125

STD DEV = 250

A) P(625<X<1625) =

For x = 625 , z = (625 - 1125) /250 = -2 and for x = 1625, z = (1625 - 1125) / 250 = 2

Hence P(625 < x < 1625) = P(-2 < z < 2) = [area to the left of z = 2] - [area to the left of -2]

= 0.9772 - 0.0228 = 0.9544

B) P(375<X<1875) =

For x = 375 , z = (375 - 1125) / 250 = -3 and for x = 1875, z = (1875 - 1125) / 250 = 3

Hence P(375 < x < 1875) = P(-3 < z < 3) = [area to the left of z = 3] - [area to the left of -3]

= 0.9987 - 0.0013 = 0.9974

C)P(X<375)

For x = 375 , z = (375 - 1125) / 250 = -3

P(Z<-3) = - AREA LEFT OF -3

= 0.0013

D) P(750<X<1500) =

For x = 750 , z = (750 - 1125) / 250 = -1.5 and for x = 1500, z = (1500 - 1125)/250 = 1.5

Hence P(750 < x < 1500) = P(-1.5 < z < 1.5) = [area to the left of z = 1.5] - [area to the left of -1.5]

= 0.9332 - 0.0688 = 0.8644

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