Question

A manufacturer knows that their items have a normally distributed length, with a mean of 16.9 inches, and standard deviation of 3.7 inches.

If one item is chosen at random, what is the probability that it is less than 21.9 inches long?

Answer 1

To find the probability that a randomly selected item has a length less than 21.9 inches, we can use the properties of the normal distribution. We will first standardize the value using the Z-score formula:

\[
Z = \frac{X - \mu}{\sigma}
\]

where:
- \( X \) is the value we are interested in (21.9 inches),
- \( \mu \) is the mean (16.9 inches),
- \( \sigma \) is the standard deviation (3.7 inches).

Substituting the values:

\[
Z = \frac{21.9 - 16.9}{3.7} = \frac{5.0}{3.7} \approx 1.3514
\]

Now that we have the Z-score, we can use the Z-table (or a calculator) to find the probability that corresponds to \( Z = 1.3514 \).

Looking at the Z-table or using a calculator, we find:

\[
P(Z < 1.3514) \approx 0.9115
\]

Thus, the probability that a randomly selected item is less than 21.9 inches long is approximately **0.9115** or **91.15%**.

So, the final answer is:

\[
\text{Probability} \approx 0.9115
\]