Question

A manufacturer knows that their items have a normally distributed length, with a mean of 5 inches, and standard deviation of 0.5 inches.

If one item is chosen at random, what is the probability that it is less than 4.6 inches long?

Answer 1

Explanation:

To solve this problem, we will use the properties of the normal distribution and standardize the given value using the formula:

z = (x - μ) / σ

where:

x = 4.6 inches (the given value)

μ = 5 inches (the mean)

σ = 0.5 inches (the standard deviation)

So we have:

z = (4.6 - 5) / 0.5

z = -0.8

Now we need to find the probability that the standardized value is less than -0.8. We can use a standard normal distribution table or a calculator to find this probability.

Using a standard normal distribution table, we can look up the probability for z = -0.8, which is 0.2119.

Therefore, the probability that a randomly chosen item is less than 4.6 inches long is 0.2119 or approximately 21.19%.