A weather forecaster predicts that the May rainfall in a local area will be betw

Question

A weather forecaster predicts that the May rainfall in a local area will be between 1 and 7 inches but has no idea where within the interval the amount will be. Let x be the amount of May rainfall in the local area, and assume that x is uniformly distributed over the interval 1 to 7 inches.

What is the probability that the observed May rainfall will fall within two standard deviations of the mean? Within one standard deviation of the mean? (Round all intermediate and final answers to 4 decimal places.)

A weather forecaster predicts that the May rainfall in a local area will be between 1 and 7 inches but has no idea where within the interval the amount will be. Let x be the amount of May rainfall in the local area, and assume that x is uniformly distributed over the interval 1 to 7 inches.

Explanation / Answer

A)

Note that here,          
          
a = lower fence of the distribution =    1      
b = upper fence of the distribution =    7      
          
Thus, the mean, variance, and standard deviations are          
          
u = mean = (b + a)/2 =    4 in   [ANSWER]

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b)

Note that here,          
          
a = lower fence of the distribution =    1      
b = upper fence of the distribution =    7      
          
Thus, the mean, variance, and standard deviations are          
          
u = mean = (b + a)/2 =    4      

sigma^2 = variance = (b -a)^2 / 12 =    3      

sigma = standard deviation = sqrt(s^2) =    1.732050808      

Half of the distribution is just 3 units in length, and 2*sigma = 3.46. Hence, this covers the whole distirbution,

P(within 2 SD) = 1 [ANSWER]

For within 1 SD,

P(within 1 SD) = 2*sigma/(b-a) = 2*1.7320508/(7-1) = 0.5774 [ANSWER]

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