The annual per capita consumption of bottled water was 32.1 gallons. Assume that

ID: 3253324 • Letter: T

Question

The annual per capita consumption of bottled water was

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

32.1 and a standard deviation of 13 gallons.

a. What is the probability that someone consumed more than 32 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 % of people consumed less than how many gallons of bottled water?

Solution

Back-up Theory

If a random variable X ~ N(µ, 2), i.e., X has Normal Distribution with mean µ and variance 2, then,

Z = (X - µ)/ ~ N(0, 1), i.e., Standard Normal Distribution ………………………..(1)

P(X or t) = P[{(X - µ)/ } or {(t - µ)/ }] = P[Z or {(t - µ)/ }] .………(2)

Now to work out the solution,

Let X = consumption of bottled water by a person. Then, we are given X ~ N(32.1, 132)

NOTE: All probabilities are obtained using Excel Function

Part (a)

Probability that someone consumed more than 32 gallons of bottled water = P(X > 32)

= P[Z > {(32 – 32.1)/13}] [by (2) of Back-up Theory]

= P(Z > - 0.0077) = 0.5031 ANSWER

Part (b)

Probability that someone consumed between 25 and 35 gallons of bottled water

= P(25 < X < 35)

= P[{(25 – 32.1)/13} < Z > {(35 – 32.1)/13}] [by (2) of Back-up Theory]

= P(- 0.5462 < Z < 0.2231)

= P(Z < 0.2231) - P(Z < - 0.5462)

= 0.5883 – 0.2925

= 0.2958 ANSWER

Part (c)

Probability that someone consumed less than 25 gallons of bottled water = P(X < 25)

= P[Z < {(25 – 32.1)/13}] [by (2) of Back-up Theory]

= P(Z < - 0.5462)

= 0.2925 ANSWER

Part (d)

Suppose 99 % of people consumed less than t gallons of bottled water. This, when translated in probability language means: P(X < t) = 0.99

=> P[Z < {(t – 32.1)/13}] = 0.99[by (2) of Back-up Theory]

=> {(t – 32.1)/13} = 2.3263 or t = 62.34 gallons ANSWER

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