The annual rainfall X in Sacramento, California, is a normal random variable wit

Question

The annual rainfall X in Sacramento, California, is a normal random variable with a mean mu = 20.27 inches and a standard deviation sigma = 4.75 inches. Calculate the following probabilities (a) P(16 lessthanorequalto X lessthanorequalto 21), (b) P(X greaterthanorequalto 15), (c) P(12 lessthanorequalto X lessthanorequalto 17), and (d) P(X lessthanorequalto 22). If the rainfall in Sacramento in a given year is below 10 inches, it is considered to be a drought year, and special water conservation measures are set in place. (e) What is the probability year that next year will be a drought year?

Explanation / Answer

mean is 20.27 and sd is 4.75

a) P(16<x<21)=P((16-20.27)/4.75<z<(21-20.27)/4.75)=P(-0.89<z<0.15)=P(z<0.15)-(1-P(z<0.89)). From normal distribution table we get 0.5596-(1-0.8133) = 0.3729

b) P(x>15)= P(z>(15-20.27)/4.75)=1-P(z<1.11). from normal table 1-0.8665= 0.1335

c) P(12<x<17)= P((12-20.27)/4.75<z<(17-20.27)/4.75)=P(-1.74<z<-0.69) =P(z<1.74)-P(z<0.69) =0.9591-0.7549 = 0.2042

d) P(x<22)= P(z<(22-20.27)/4.75)=P(z<0.36) . from normal table we get 0.6404

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