Question

In: Statistics and Probability

The annual per capita consumption of bottled water was 32.7 gallons. Assume that the per capita...

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?

Solutions

Expert Solution

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


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