Question

The annual per capita consumption of bottled water was 30.5 gallons. Assume that the per capita consumption of bottled water is approximately normally distributed with a mean of 30.5 and a standard deviation of 10 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 20 and 30 gallons of bottled​ water?

c. What is the probability that someone consumed less than 20 gallons of bottled​ water?

d. 90​% of people consumed less than how many gallons of bottled​ water?

Answer 1

Solution :

Given that ,

mean = = 30.5

standard deviation = = 10

a.

P(x > 31) = 1 - P(x < 31)

= 1 - P((x - ) / < (31 - 30.5) / 10)

= 1 - P(z < 0.05)

= 1 - 0.5199

= 0.4801

Probability = 0.4801

b.

P(20 < x < 30) = P((20 - 30.5)/ 10) < (x - ) / < (30 - 30.5) / 10) )

= P(-1.05 < z < -0.05)

= P(z < -0.05) - P(z < -1.05)

= 0.4801 - 0.1469

= 0.3332

Probability = 0.3332

c.

P(x < 20) = P[(x - ) / < (20 - 30.5) / 10]

= P(z < -1.05)

= 0.1469

Probability = 0.1469

d.

Using standard normal table,

P(Z < z) = 90%

P(Z < 1.28) = 0.9

z = 1.28

Using z-score formula,

x = z * +

x = 1.28 * 10 + 30.5 = 43.3

gallons of bottled​ water = 43.3