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