Question

The monthly utility bills in a certain city are normally distributed with a mean of $100 and standard deviation of $12.

a) A utility bill is randomly selected. Find the probability that it is: i) less than $80

ii) between $75 and $ 115

b) What percentage of the utility bills are for more than $125?

c) If 300 utility bills were selected at random, how many would be less than $90?

d) The utility company wants to give a small prize to those customers who do the most to conserve energy. They will give this only to the 15% of customers with the lowest utility bills. What is the highest utility bill one can have and still receive the prize?

e) Forty-one utility bills are selected at random. Determine the probability that the average bill amount is greater than $107

Answer 1

a) i)

µ =    100          
σ =    12          
              
P( X ≤    80   ) = P( (X-µ)/σ ≤ (80-100) /12)      
=P(Z ≤   -1.667   ) =   0.04779   (answer)
.

ii)

µ =    100                              
σ =    12                              
we need to calculate probability for ,                                  
P (   75   < X <   115   )                  
=P( (75-100)/12 < (X-µ)/σ < (115-100)/12 )                                  
                                  
P (    -2.083   < Z <    1.250   )                   
= P ( Z <    1.250   ) - P ( Z <   -2.083   ) =    0.8944   -    0.0186   =    0.8757
...........

b)

µ =    100                  
σ =    12                  
                      
P ( X ≥   125.00   ) = P( (X-µ)/σ ≥ (125-100) / 12)              
= P(Z ≥   2.083   ) = P( Z <   -2.083   ) =    0.0186 = 1.86% (answer)

.........

c)

µ =    100      
σ =    12      
          
P( X ≤    90   ) = P( (X-µ)/σ ≤ (90-100) /12)  
=P(Z ≤   -0.833   ) =   0.20233
the number of bills less than 90 will be = 300*0.20233

= 60.699 ~ 61 (round up)

d)

µ=   100                  
σ =    12                  
proportion=   0.15                  
                      
Z value at    0.15   =   -1.04   (excel formula =NORMSINV(   0.15   ) )
z=(x-µ)/σ                      
so, X=zσ+µ=   -1.04   *   12   +   100  
X   =   87.56   (answer)          

thanks

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