Question

The amount of cereal dispensed into "16-ounce" boxes of Captain Crisp cereal was normally distributed with mean 16.18 ounces and standard deviation 0.25 ounces.

a. What proportion of boxes are "underfilled"? That is, what is the probability that the amount dispensed into a box is less than 16 ounces? Give your answer to 4 decimal places.

b. Find the probability that exactly 3 out of 9 randomly and independently selected boxes of cereal contain less than 16 ounces. Give your answer to 4 significant figures.

c. Find the probability that the sample mean amount of cereal for a random sample of 9 boxes is less than 16 ounces. Give your answer to 4 decimal places.

d. Suppose that the machine can be adjusted to change the mean while the standard deviation remains at 0.25 ounces. What must the mean be so that only 20% of all the boxes are "underfilled"? Give your answer to 3 decimal places. (Mean in ounces)

Showing work is much appreciated.

Answer 1

Let X is a random variable shows the amount of cereal. Here X has normal distribution with parameters as follows:

(a)

The z-score for X = 16 is

The probability that the amount dispensed into a box is less than 16 ounces is

P(X < 16) = P(z < -0.72) = 0.2358

b)

Let Y is a random variable shows the number of boxes of cereal contain less than 16 ounces out of 9. Here Y has binomial distribution with parameters n=9 and p=0.2358.

The probability that exactly 3 out of 9 randomly and independently selected boxes of cereal contain less than 16 ounces is

c)

The z-score for is

The probability that the sample mean amount of cereal for a random sample of 9 boxes is less than 16 ounces is

d)

Here we need z-score that has 0.20 area to its left. The z-score for -0.84 has 0.20 area to its left. The requried X is

Answer: 16.21 ounces