Question

A machine in a factory is responsible for filling boxes of cereal. The weight of cereal in each box has a mean of 500 g and a variance of 20 g. A box is picked at random. What is the probability that the box will weigh less than 490 g?

Answer 1

To calculate the probability, we determine the z-score for 490 grams with the given mean and standard deviation. Next, we look up the probability associated with that z-score using a standard normal distribution table or calculator.

Step-by-step explanation:

To find the probability that a box weighs less than 490 g given a mean of 500 g and a variance of 20 g, we need to standardize the value using the z-score formula and then find the corresponding probability from the standard normal distribution.

The z-score is calculated as follows:

Z = (X - μ) / σ

Where X is the value in question (490 g), μ is the mean (500 g), and σ is the standard deviation. Since variance is given as 20 g², the standard deviation will be the square root of variance, which is √20 ≈ 4.47 g.

Substituting the values into the z-score formula:

Z = (490 - 500) / 4.47 ≈ -2.24

Using a standard normal distribution table, or a calculator, we find the probability of Z being less than -2.24.

This represents the probability that a randomly selected box will weigh less than 490 g.