A company produces steel rods. The lengths of the steel rods are normally distributed with a mean of 196.2 cm and a standard deviation of 1.5-cm.

Find the probability that the length of a randomly selected steel rod is between 196.7 cm and 198.3 cm.



To solve this problem, we will use the properties of the normal distribution. The lengths of the steel rods are normally distributed with a mean (\(\mu\)) of 196.2 cm and a standard deviation (\(\sigma\)) of 1.5 cm.

### Step 1: Standardize the values
We need to standardize the values of 196.7 cm and 198.3 cm to their corresponding z-scores using the formula:

\[
z = \frac{x - \mu}{\sigma}
\]

where:
- \(x\) is the value we are standardizing,
- \(\mu = 196.2 \, \text{cm}\) (mean),
- \(\sigma = 1.5 \, \text{cm}\) (standard deviation).

#### For \(x = 196.7 \, \text{cm}\):

\[
z_1 = \frac{196.7 - 196.2}{1.5} = \frac{0.5}{1.5} = 0.33
\]

#### For \(x = 198.3 \, \text{cm}\):

\[
z_2 = \frac{198.3 - 196.2}{1.5} = \frac{2.1}{1.5} = 1.40
\]

### Step 2: Find the corresponding probabilities
Now, using a standard normal distribution table or calculator, we find the probabilities corresponding to these z-scores.

- The probability for \(z_1 = 0.33\) is approximately 0.6293.
- The probability for \(z_2 = 1.40\) is approximately 0.9192.

### Step 3: Find the probability between the two z-scores
To find the probability that the steel rod length is between 196.7 cm and 198.3 cm, subtract the smaller probability from the larger one:

\[
P(196.7 \leq X \leq 198.3) = P(z_2) - P(z_1) = 0.9192 - 0.6293 = 0.2899
\]

Thus, the probability that a randomly selected steel rod is between 196.7 cm and 198.3 cm is approximately **0.2899**, or **28.99%**.

Would you like any more details or have further questions?

### Related questions:
1. How do you find z-scores for different values of \(x\) in a normal distribution?
2. What is the probability that a rod is shorter than 196.0 cm?
3. How can the normal distribution be used to assess product quality in manufacturing?
4. What are the key differences between the standard normal distribution and any other normal distribution?
5. How do you calculate probabilities for values more than two standard deviations from the mean?

### Tip:
When using the normal distribution, always remember that about 68% of data falls within one standard deviation of the mean, and about 95% within two standard deviations.

A company produces steel rods. The lengths of the steel rods are normally distributed with a mean of 196.2 cm and a standard deviation of 1.5 cm. Find the probability that the length of a randomly selected steel rod is between 196.7 cm and 198.3 cm.

Learn how to calculate the probability that the length of a steel rod is between 196.7 cm and 198.3 cm using normal distribution. This example covers standardization with z-scores and the use of a standard normal distribution table, perfect for high school statistics students.

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