High temperatures in a certain city for the month of August follow a uniform dis

Question

High temperatures in a certain city for the month of August follow a uniform distribution over the interval 77F to 101F.

a) What is the probability that the high temperature on a day in August exceeds 85F? [Round to four decimal places] 0.6667

b) What is the probability that the high temperature on a day in August is within 1.8 standard deviation of mean temperature? [Round to four decimal places]

c) Above what temperature are 35% of the daily high temperatures of August? [Round to two decimal places]

Explanation / Answer

for unifrom distribution CDf =P(X<x)=(X-a)/(b-a) =(X-77)/(101-77)=(X-77)/24

a)probability that the high temperature on a day in August exceeds 85F =P(X>85)=1-P(X<85)

=1-(85-77)/24=0.6667

b)here mean =(a+b)/2 =(101+77)/2 =89

and std deviation=(b-a)/(12)1/2 =6.9282

probability that the high temperature on a day in August is within 1.8 standard deviation of mean temperature

=P(89-1.8*6.9282<X<89+1.8*6.9282)=P(76.53<X<101.47) =1 (as that interval contain all values)

for 35% highest will be at 65th percentile

hence (x-77)/24 =0.65

x=92.6 F

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