Question

The annual per capita consumption of bottled water was 32.7 gallons. Assume that the per capita consumption of bottled water is approximately normally distributed with a mean of 32.7 and a standard deviation of 13 gallons.

a. What is the probability that someone consumed more than 43 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. 95​% of people consumed less than how many gallons of bottled​ water?

Answer 1

a)

Here, μ = 32.7, σ = 13 and x = 43. We need to compute P(X >= 43). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (43 - 32.7)/13 = 0.79

Therefore,
P(X >= 43) = P(z <= (43 - 32.7)/13)
= P(z >= 0.79)
= 1 - 0.7852 = 0.2148

b)

Here, μ = 32.7, σ = 13, x1 = 25 and x2 = 35. We need to compute P(25<= X <= 35). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z1 = (25 - 32.7)/13 = -0.59
z2 = (35 - 32.7)/13 = 0.18

Therefore, we get
P(25 <= X <= 35) = P((35 - 32.7)/13) <= z <= (35 - 32.7)/13)
= P(-0.59 <= z <= 0.18) = P(z <= 0.18) - P(z <= -0.59)
= 0.5714 - 0.2776
= 0.2938

c)
Here, μ = 32.7, σ = 13 and x = 25. We need to compute P(X <= 25). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (25 - 32.7)/13 = -0.59

Therefore,
P(X <= 25) = P(z <= (25 - 32.7)/13)
= P(z <= -0.59)
= 0.2776

d)

z value at 95% = -1.64
-1.64 = (x - 32.7)/13
x = -1.64 * 13 + 32.7
x = 11.38

x = 12 rounding to whole number