Question

A food processor claims that at most 10% their jars of instant coffee contain less coffee than stated on the label. To test the food processor’s claim, 20 jars of their instant coffee are randomly selected and the contents are weighed. The food processor’s claim will be accepted if 3 or fewer of the 20 jars are under-filled. Otherwise, the claim will be rejected.

10. What is the probability that the food processors claim will be accepted when the actual percentage of under-filled jars of instant coffee is 25%?

11. What is the probability that the food processors claim will be rejected when the actual percentage of under-filled jars of instant coffee is 10%?

Answer 1

Binomial distribution: P(X) = nCx p^x q^n-x

10. P(a jar in under-filled), p = 0.25

q = 1 - p = 0.75

Sample size, n = 20

P(the food processors claim will be accepted) = P(X 3)

= P(0) + P(1) + P(2) + P(3)

= 0.75²⁰ + 20x0.25x0.75¹⁹ + 20C2x0.25²x0.75¹⁸ + 20C3x0.25³x0.75¹⁷

= 0.0032 + 0.0211 + 0.0670 + 0.1339

= 0.2252

10. P(a jar in under-filled), p = 0.1

q = 1 - p = 0.9

Sample size, n = 20

P(the food processors claim will be accepted) = P(X 3)

= P(0) + P(1) + P(2) + P(3)

= 0.9²⁰ + 20x0.1x0.9¹⁹ + 20C2x0.1²x0.9¹⁸ + 20C3x0.1³x0.9¹⁷

= 0.1216 + 0.2702 + 0.2851 + 0.1901

= 0.8670