The utility bill Mike pays per month is known to be normally

The utility bill Mike pays per month is known to be normally distributed with a mean of 250 dollars and a standard deviation of 60.

A) The probability is 0.2 that the utility bill will be less than what amount?

B) Find two numbers that bound the shortest range such that the probability is 0.95 that the utility bill will fall in this range?

Solution

a)

First, we get the z score from the given left tailed area. As          
          
Left tailed area =    0.2      
          
Then, using table or technology,          
          
z =    -0.841621234      
          
As x = u + z * s,          
          
where          
          
u = mean =    250      
z = the critical z score =    -0.841621234      
s = standard deviation =    60      
          
Then          
          
x = critical value =    199.502726   [ANSWER]

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b)

As the middle area is          
          
Middle Area = P(x1<x<x2) =    0.95      
          
Then the left tailed area of the left endpoint is          
          
P(x<x1) = (1-P(x1<x<x2))/2 =    0.025      
          
Thus, the z score corresponding to the left endpoint, by table/technology, is          
          
z1 =    -1.959963985      
By symmetry,          
z2 =    1.959963985      
          
As          
          
u = mean =    250      
s = standard deviation =    60      
          
Then          
          
x1 = u + z1*s =    132.4021609      
x2 = u + z2*s =    367.5978391      
  
The numbers are 132.4021609 and 367.5978391. [ANSWER]