Question

One environmental group did a study of recycling habits in a California community. It found that 75% of the aluminum cans sold in the area were recycled. (Use the normal approximation. Round your answers to four decimal places.)

(a) If 398 cans are sold today, what is the probability that 300 or more will be recycled?


(b) Of the 398 cans sold, what is the probability that between 260 and 300 will be recycled?

Answer 1

X ~ Bin(n,p)

Where n = 398 , p = 0.75

Mean = np = 398 * 0.75 = 298.5

Standard deviation = sqrt [ np(1-p)]

= sqrt (398 * 0.75 * 0.25)

= 8.6386

Using normal approximation,

P(X < x) = P(Z < ( x - mean ) / SD)

a)

With continuity correction,

P(X >= 300) = P(Z > (299.5 - 298.5) / 8.6386)

= P( Z > 0.12)

= 0.4522

b)

P(260 < X < 300) = P(259.5 < X < 300.5)

= P(X < 300.5) - P(X < 259.5)

= P(Z < (300.5 - 298.5) / 8.6386) - P(Z < (259.5 - 298.5) / 8.6386)

= P(Z < 0.23) - P(Z < -4.51)

= 0.5910 - 0.00000

= 0.5910