A manufacturer knows that their items have a normally distributed length, with a mean of 13.8 inches, and standard deviation of 2 inches.

If one item is chosen at random, what is the probability that it is less than 17.2 inches long?

We can solve this problem using the properties of a normal distribution. Since we know the length follows a normal distribution with a mean \( \mu = 13.8 \) inches and standard deviation \( \sigma = 2 \) inches, we are tasked with finding the probability that a randomly chosen item is less than 17.2 inches long.

### Step-by-step solution:

1. **Standardize the value**:  
   We need to convert the length of 17.2 inches into a z-score, which represents how many standard deviations away from the mean this value is. The formula for the z-score is:
   \[
   z = \frac{x - \mu}{\sigma}
   \]
   where \( x = 17.2 \), \( \mu = 13.8 \), and \( \sigma = 2 \).

   Substituting the values:
   \[
   z = \frac{17.2 - 13.8}{2} = \frac{3.4}{2} = 1.7
   \]

2. **Find the probability**:  
   Next, we use the z-score to find the corresponding probability from the standard normal distribution table, or we can use a calculator to find the cumulative probability for \( z = 1.7 \).

   The cumulative probability for \( z = 1.7 \) is approximately:
   \[
   P(Z < 1.7) \approx 0.9554
   \]

### Conclusion:
The probability that a randomly chosen item is less than 17.2 inches long is approximately **0.9554**, or 95.54%.

Would you like further details or have any questions?

Here are five related questions:
1. What is the probability that the item is more than 17.2 inches long?
2. How would the probability change if the mean were 15 inches?
3. What is the probability of an item being between 13 and 17.2 inches?
4. How would you calculate the probability for a length of exactly 17.2 inches?
5. What is the probability that an item is shorter than 12 inches?

**Tip**: The total area under a normal distribution curve is always 1, representing the entire probability space.

A manufacturer knows that their items have a normally distributed length, with a mean of 13.8 inches, and standard deviation of 2 inches. If one item is chosen at random, what is the probability that it is less than 17.2 inches long?

This problem involves calculating the probability of a randomly chosen item being less than 17.2 inches long, given that the lengths are normally distributed with a mean of 13.8 inches and a standard deviation of 2 inches. We use the z-score formula and the standard normal distribution to solve the problem. Suitable for students in Grades 10-12.

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