#16 The mean value of land and buildings per acre for farms in Colorado is $1170

Question

#16

The mean value of land and buildings per acre for farms in Colorado is $1170 with standard deviation $220. Assume that the values are normally distributed. (a) Find the probability that a randomly selected farm has a value of land and buildings per acre of more than $1500. (b) What percentage of Colorado farms have a value of land and buildings per acre between $1000 and $1500? (c) What is the value of land and buildings per acre that corresponds to the fifteenth percentile? (d) What is the value of land and buildings per acre that corresponds to the third quartile? (e) What is the lowest value of land and buildings per acre that would still place a farm in the top 15%? (f) What is the highest value of land and buildings per acre that would still place a farm in the bottom 35% if the values? A random sample of 32 Colorado farms was selected. What is the probability that the mean value of land and buildings per acre for the sample is between $1100 and $1250?

Explanation / Answer

1) here z=(X-mean)/std deviation

hence a) P(X>1500)=1-P(X<1500)=1-P(Z<(1500-1170)/220)=1-P(Z<1.5)=1-0.9332=0.0668

b)P(1000<X<1500)=P(-0.7727<Z<1.5)=0.9332-0.2198=0.7134

c)at 15th %, z=-1.0364

hence corresponding value =mean +z*std deviation =941.98

d)for 3rd quartile ; z=0.6745

Value =1318.39

e)for 85% ; z=1.0364

value=1398.015

f)for bottom 35%; z=-0.3853

hence value=1085.23

g) for 32 fields std error =std deviation/(n)1/2=38.89

hence P(X>1200)=1-P(Z<0.7714)=1-0.7798=0.2202

h)

for 60 fields std error =std deviation/(n)1/2=28.40

hence P(1100<X<1250)=P(-2.4646<Z<2.8167)=0.9976-0.0069=0.9907

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