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Let's work through this problem step by step.

The question is asking for the probability that the sample's mean length is greater than 17.5 inches. We are provided with the following information:

- Population mean \( \mu = 18.2 \) inches.
- Population standard deviation \( \sigma = 4.3 \) inches.
- Sample size \( n = 38 \).
- We need to find the probability that the sample mean is greater than 17.5 inches.

### Step 1: Standard Error of the Mean

The standard error of the mean (SE) is calculated using the formula:
\[
SE = \frac{\sigma}{\sqrt{n}}
\]
Substituting the given values:
\[
SE = \frac{4.3}{\sqrt{38}} \approx 0.6977
\]

### Step 2: Z-Score Calculation

To find the Z-score corresponding to 17.5 inches, we use the formula for the Z-score:
\[
Z = \frac{X - \mu}{SE}
\]
Where:
- \( X = 17.5 \) (the value we are comparing to the mean),
- \( \mu = 18.2 \) (population mean),
- \( SE \approx 0.6977 \) (standard error of the mean).

Substituting the values:
\[
Z = \frac{17.5 - 18.2}{0.6977} \approx -1.003
\]

### Step 3: Find the Probability

Now, we need to find the probability associated with this Z-score. Using a standard normal distribution table or a Z-score calculator, we find that the probability corresponding to a Z-score of -1.003 is approximately 0.1583.

Since the question asks for the probability that the sample mean is **greater than** 17.5 inches, we need to calculate the area to the right of this Z-score. The probability is:
\[
P(X > 17.5) = 1 - 0.1583 = 0.8417
\]

### Final Answer:
The probability that the sample mean is greater than 17.5 inches is approximately **0.8417** (rounded to four decimal places).

---

Would you like any more details on any of the steps? Here are some related questions to explore:

1. How do you interpret the standard error of the mean in statistics?
2. What is the difference between the Z-score for a population and for a sample?
3. Can you explain why we subtract the Z-score from 1 in this case?
4. What are the conditions for using the Z-distribution in sample problems?
5. How does sample size affect the standard error of the mean?

**Tip**: Always double-check your Z-scores and tables to ensure accuracy when working with normal distributions!

A manufacturer produces a commodity where the length of the commodity has approximately normal distribution with a mean of 18.2 inches and standard deviation of 4.3 inches. If a sample of 38 items are chosen at random, what is the probability the sample's mean length is greater than 17.5 inches? Round answer to four decimal places.

Learn how to calculate the probability that a sample mean is greater than 17.5 inches for a normally distributed commodity length with a mean of 18.2 inches and a standard deviation of 4.3 inches. Includes Z-Score and standard error calculations. Suitable for high school and undergraduate students.

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