Question

boxes are labeled as containing 400 g of cereal. the machine filling the boxes produces weights that are normally distributed with standard deviation 13 g. (a) if the target weight is 400 g, what is the probability that the machine produces a box with less than 380 g of cereal? (round your answer to three decimal places.) (b) suppose a law states that no more than 9% of a manufacturer's cereal boxes can contain less than the stated weight of 400 g. at what target weight should the manufacturer set its filling machine? (round your answer to the nearest gram.) g

Answer 1

The probability that a machine fills a box with less than 380 g of cereal is approximately 0.062. To comply with a law stating that no more than 9% of cereal boxes can weigh less than the marked 400 g, the target weight should be adjusted to approximately 417 g.

Step-by-step explanation:

Calculating Probability for Machine Filling in Cereal Boxes

In the context of a machine filling cereal boxes normally distributed with a standard deviation of 13 g, to find the probability that a box weighs less than 380 g with the target mean of 400 g:

1. First, determine the z-score which is Z = (X - μ) / σ, where X is the weight of interest (380 g), μ is the mean weight (400 g), and σ is the standard deviation (13 g).
2. Z = (380 - 400) / 13 = -20 / 13 = -1.54 approximately.
3. Next, look up the z-score in a standard normal distribution table or use a calculator to find the corresponding probability. The probability of Z < -1.54 is about 0.0618 or 6.18%.

To answer part b, we need to find the target mean that fits the given condition that no more than 9% of boxes contain less than 400 g:

1. Find the z-score corresponding to the 9th percentile in the standard normal distribution, which is approximately -1.34.
2. Solve for the new mean (μ') using the z-score formula with the lower bound weight (400 g) and the standard deviation (13 g): μ' = X - Z σ = 400 g - (-1.34)(13 g).
3. The target weight to be set on the filling machine is roughly 417.42 g, and rounding to the nearest gram gives us 417 g.

Thus, to stay within the legal requirements, the manufacturer should set the target weight for its filling machine to 417 g.