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Let's solve the problem step by step using the given dataLet's solve the problem step by step using the given 

### Given:
- Population mean \(\mu = 120 \, \text{cm}\)
- Population standard deviation \(\sigma = 0.05 \, \text{cm}\)
- Sample size \(n = 100\)
- Sample mean we are interested in \(X = 119.985 \, \text{cm}\)

We need to find the probability that Claude's sample mean is less than 119.985 cm.

### Step 1: Calculate the standard error of the mean (SEM)

The standard error of the mean is given by:

\[
\text{SEM} = \frac{\sigma}{\sqrt{n}} = \frac{0.05}{\sqrt{100}} = \frac{0.05}{10} = 0.005
\]

### Step 2: Find the z-score

The z-score is calculated using the formula:

\[
z = \frac{X - \mu}{\text{SEM}} = \frac{119.985 - 120}{0.005} = \frac{-0.015}{0.005} = -3
\]

### Step 3: Find the probability

The z-score of -3 corresponds to a cumulative probability of approximately 0.0013 from the standard normal distribution.

### Answer:
The probability that Claude's sample has a mean less than 119.985 cm is approximately **0.0013**. Hence, the correct answer is **D. 0.0013**.

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Would you like more details on the steps involved? Here are some follow-up questions:

1. How is the standard error of the mean (SEM) derived?
2. Why do we use the z-score in this type of problem?
3. What does a z-score of -3 represent in terms of probability?
4. How would the result change if the sample size were larger or smaller?
5. How is the normal distribution used to approximate probabilities in large samples?

**Tip**: Always ensure your sample size is large enough to use the normal distribution approximation when applying z-scores.

A carload of steel rods has arrived at Cybermatic Construction Company. The car contains 50,000 rods. Claude Vider, Quality Assurance Manager, directs his crew to measure the lengths of 100 randomly selected rods. If the population of rods has a mean length of 120 cm and a standard deviation of 0.05 cm, the probability that Claude’s sample has a mean less than 119.985 cm is ________.

Solve a probability problem where Claude Vider, the Quality Assurance Manager, measures the length of rods. Learn how to calculate the probability that a sample mean is less than 119.985 cm using normal distribution, standard error, and z-scores. This problem involves population mean, standard deviation, and sample size.

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