Question

The annual per capita consumption of bottled water was

31.531.5

gallons. Assume that the per capita consumption of bottled water is approximately normally distributed with a mean of

31.531.5

and a standard deviation of

1212

gallons.a. What is the probability that someone consumed more than

3232

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

2525

and

3535

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

2525

gallons of bottled​ water?d.

9999​%

of people consumed less than how many gallons of bottled​ water?

a. The probability that someone consumed more than

3232

gallons of bottled water is

nothing.

​(Round to four decimal places as​ needed.)

b. The probability that someone consumed between

2525

and

3535

gallons of bottled water is

nothing.

​(Round to four decimal places as​ needed.)

c. The probability that someone consumed less than

2525

gallons of bottled water is

nothing.

​(Round to four decimal places as​ needed.)

d.

9999​%

of people consumed less than

nothing

gallons of bottled water.

​(Round to two decimal places as​ needed.)

Answer 1

Solution :

Let X be a random variable which represents the annual per capita consumption of bottled water.

Given that, X ~ N(31.5, 12²)

Mean (μ) = 31.5

SD (σ) = 12

a) We have to find P(X > 32).

We know that, if X ~ N(μ, σ²) then,

Using "pnorm" function of R we get, P(Z > 0.04167) = 0.4834

Hence, the probability that someone consumed more than 32 gallons of bottled water is 0.4834.

b) We have to find P(25 < X < 35).

P(25 < X < 35) = P(X < 35) - P(X ≤ 25)

We know that, if X ~ N(μ, σ²) then,

Using "pnorm" function of R we get,

P(Z < 0.2917) = 0.6147 and P(Z ≤ -0.5417) = 0.2940

Hence, the probability that someone consumed between 25 and 35 gallons of bottled water is 0.3247.

c) We have to find P(X < 25).

We know that, if X ~ N(μ, σ²) then,

Using "pnorm" function of R we get, P(Z ≤ -0.5417) = 0.2940

Hence, the probability that someone consumed less than 25 gallons of bottled water is 0.2940.

d) Let 99% of people consumed less than k gallons of water.

Hence,

P(X < k) = 0.99

........................(1)

Using "qnorm" function of R we get, P(Z < 2.3263) = 0.99

Comparing, P(Z < 2.3263) = 0.99 and (1) we get,

Hence, 99% of people consumed less than 59.42 gallons of water.

Please rate the answer. Thank you.