Question

The monthly utility bills in a city are normally distributed, with a mean of $100 and a standard deviation of $14. Find the probability that a randomly selected utility bill is (a) less than $67. (b) between $82 and 5100, and (c) more than $120. 

(a) The probability that a randomly selected utility bill is less than $67 is _______ 

(b) The probability that a randomly selected utility bill is between $82 and $100 is _______ 

(c) The probability that a randomly selected utility bill is more than $120 is  _______ 

Answer 1

Given that,

mean =  μ = 100

standard deviation = σ = 14

a ) P( x < 67 )

P ( x - μ  / σ  ) < ( 67 - 100 / 14)

P ( z < -33 /14 )

P ( z < -2.36)

= 0.0091

Probability = 0.0091

b ) P (82 < x < 100 )

P ( 82- 100 / 14) < ( x -  μ  /  σ  ) < ( 100 - 100 / 14 )

P ( - 18 / 14 < z < 0 /14 )

P (-1.28 < z < 0)

P ( z < 0 ) - P ( z < -1.28)

Using z table

= 0.5000 - 0.1003

= 0.3997

Probability = 0.3997

c ) P (x > 120)

= 1 - P (x < 120 )

= 1 - P ( x -   μ  /  σ  ) < ( 120 - 100 / 14)

= 1 - P ( z < 20 / 14 )

= 1 - P ( z < 1.43)

Using z table

= 1 -0.9236

= 0.0764

Probability = 0.0764