Question

One environmental group did a study of recycling habits in a California community. It found that 73% of the aluminum cans sold in the area were recycled. (Use the normal approximation. Round your answers to four decimal places.)

(a) If 394 cans are sold today, what is the probability that 300 or more will be recycled?


(b) Of the 394 cans sold, what is the probability that between 260 and 300 will be recycled?

Answer 1

a)
Here, μ = 287.62, σ = 8.8123 and x = 300. We need to compute P(X >= 300). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (300 - 287.62)/8.8123 = 1.4

Therefore,
P(X >= 300) = P(z <= (300 - 287.62)/8.8123)
= P(z >= 1.4)
= 1 - 0.9192
= 0.0808

b)
z = (x - μ)/σ
z1 = (260 - 287.62)/8.8123 = -3.13
z2 = (300 - 287.62)/8.8123 = 1.4

Therefore, we get
P(260 <= X <= 300) = P((300 - 287.62)/8.8123) <= z <= (300 - 287.62)/8.8123)
= P(-3.13 <= z <= 1.4) = P(z <= 1.4) - P(z <= -3.13)
= 0.9192 - 0.0009
= 0.9183