A manufacturer knows that their items have a normally distributed length, with a mean of 17.8 inches, and standard deviation of 3.1 inches.

If one item is chosen at random, what is the probability that it is less than 13.9 inches long?


To solve this problem, we need to find the probability that a randomly chosen item has a length less than 13.9 inches. Since the lengths are normally distributed, we will use the standard normal distribution (also called the Z-distribution) to calculate this probability.

### Step-by-Step Solution:

1. **Given Data:**
   - Mean (\(\mu\)) = 17.8 inches
   - Standard deviation (\(\sigma\)) = 3.1 inches
   - We want to find \( P(X < 13.9) \), where \( X \) is the length of the item.

2. **Standardizing the Value:**
   We convert the value 13.9 into a Z-score using the formula:
   \[
   Z = \frac{X - \mu}{\sigma}
   \]
   Substituting the given values:
   \[
   Z = \frac{13.9 - 17.8}{3.1} = \frac{-3.9}{3.1} \approx -1.258
   \]

3. **Finding the Probability:**
   Now, we use the Z-score to find the probability. We need to look up the Z-score of -1.258 in the standard normal distribution table or use a calculator.

   From standard normal tables or using a calculator, the probability corresponding to \( Z = -1.258 \) is approximately 0.104. 

4. **Conclusion:**
   The probability that a randomly chosen item is less than 13.9 inches long is approximately:
   \[
   P(X < 13.9) \approx 0.104 \text{ or } 10.4\%
   \]

Would you like further details or any clarifications?

### Related Questions:
1. What is the probability that an item is more than 20 inches long?
2. How do we find the probability that an item is between 13.9 inches and 17.8 inches long?
3. What percentage of items are longer than 15 inches?
4. If the standard deviation was larger, how would that affect the probability of lengths below 13.9 inches?
5. How can we calculate the probability for multiple items being below a certain length?

**Tip:** Z-scores allow us to compare different normal distributions by converting values to a common standard scale.

A manufacturer knows that their items have a normally distributed length, with a mean of 17.8 inches, and standard deviation of 3.1 inches. If one item is chosen at random, what is the probability that it is less than 13.9 inches long?

Learn how to calculate the probability that a randomly chosen item has a length less than 13.9 inches, given that the lengths are normally distributed with a mean of 17.8 inches and a standard deviation of 3.1 inches. This step-by-step solution uses the Z-score formula and standard normal distribution tables. Suitable for Grades 9-12.

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