Question

The probability density function of the net weight in pounds of a packaged chemical herbicide is fx) 2.0 for 49.75x < 50.25 pounds.

(a) Determine the probability that a package weighs more than 50 pounds.
(b) How much chemical is contained in 90%ofall packages?

Answer 1

The probability that a package weighs more than 50 pounds is 0.5, and the weight corresponding to the 90th percentile of the packages is 50.20 pounds.

Step-by-step explanation:

The given probability density function is constant for a package's net weight between 49.75 and 50.25 pounds, which indicates a uniform distribution. To find the probability that a package weighs more than 50 pounds, we consider the total probability for the weight to be between 49.75 and 50.25 and subtract the probability that it is less than or equal to 50 pounds.

(a) The probability that a package weighs more than 50 pounds is therefore:

• P(X > 50) = P(49.75 < X < 50.25) - P(49.75 < X ≤ 50).
• P(X > 50) = 1 - P(X ≤ 50).
• P(X ≤ 50) = (50 - 49.75) × 2.0.
• P(X ≤ 50) = 0.25 × 2.0 = 0.5.
• P(X > 50) = 1 - 0.5 = 0.5.

(b) To find the amount of chemical in 90% of all packages, we need to find the weight that corresponds to the 90th percentile under the uniform distribution:

• This is 0.90 × (50.25 - 49.75) + 49.75.
• 0.90 × 0.5 + 49.75 = 0.45 + 49.75.
• Therefore, the 90th percentile weight of the packages is 50.20 pounds.