Question

The annual per capita consumption of bottled water was 30.7 gallons. Assume that the per capita consumption of bottled water is approximately normally distributed with a mean of 30.7 and a standard deviation of 13 gallons.

a. What is the probability that someone consumed more than 31 gallons of bottled​ water?

b. What is the probability that someone consumed between 25 and 35 gallons of bottled​ water?

c. What is the probability that someone consumed less than 25 gallons of bottled​ water?

d. 99.5% of people consumed less than how many gallons of bottled​ water?

Answer 1

Solution :

Given that,

mean = = 30.7

standard deviation = =13

a ) P (x > 31 )

= 1 - P (x < 31 )

= 1 - P ( x -  / ) < ( 31 - 30.7 / 13)

= 1 - P ( z < 0.30 / 13 )

= 1 - P ( z < 0.02 )

Using z table

= 1 - 0.5092

= 0.4908

Probability = 0.4908

b ) P (25 < x < 35 )

P ( 25 - 30.7 / 13) < ( x -  / ) < ( 35 - 30.7 / 13)

P ( -5.7 / 13 < z < 4.3 / 13 )

P (-0.44 < z < 0.33 )

P ( z < 0.33 ) - P ( z < -0.44)

Using z table

= 0.6293 - 0.3300

= 2993

Probability = 2993

c ) P( x < 25 )

P ( x - / ) < ( 25- 30.7 / 13)

P ( z < -5.7 / 13)

P ( z < -0.44)

= 0.3300

Probability = 0.3300

d ) P( Z < z) = 99.5%
P(Z < z) = 0.995

z = 2.50

Using z-score formula,

x = z * +

x =2.50 * 13 + 30.7

= 63.2

x = 63.2