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We are given that the heights of adult women are normally distributed with:

- Mean \(\mu = 62\) inches,
- Standard deviation \(\sigma = 4.1\) inches,
- We are asked to find the probability that a randomly chosen woman will have a height less than 65.2 inches.

### Step 1: Calculate the Z-score
The Z-score is given by the formula:

\[
Z = \frac{X - \mu}{\sigma}
\]

Where:
- \(X = 65.2\) inches is the value we are interested in,
- \(\mu = 62\) is the mean,
- \(\sigma = 4.1\) is the standard deviation.

Substituting the values:

\[
Z = \frac{65.2 - 62}{4.1} = \frac{3.2}{4.1} \approx 0.78
\]

### Step 2: Determine the probability
To find the probability corresponding to this Z-score, we consult the standard normal distribution table or use a calculator.

For \(Z = 0.78\), the cumulative probability from the Z-table is approximately 0.7823. This means that about 78.23% of women have a height less than 65.2 inches.

### Final Answers:
1. The Z-score needed is \(0.78\).
2. The probability that a randomly chosen woman will have a height less than 65.2 inches is approximately 0.7823 or 78.23%.

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Would you like more details on how to use the Z-table or further clarification? 

Here are some related questions:
1. What is the probability for a height greater than 65.2 inches?
2. How do you calculate the probability for heights between two values?
3. What if the height was 60 inches—how would we calculate that?
4. How does the standard deviation affect the spread of data in this scenario?
5. What does a Z-score of 0.78 imply in terms of standard deviations?

**Tip:** When calculating probabilities involving a normal distribution, always standardize using the Z-score formula to make use of the standard normal distribution.

The heights of adult women are normally distributed with a mean of 62 inches and a standard deviation of 4.1 inches. What is the probability that a randomly chosen adult woman will have a height of less than 65.2 inches?

Learn how to calculate the probability that a randomly chosen adult woman will have a height less than 65.2 inches using normal distribution and Z-scores. This problem covers key statistical concepts, including the normal distribution, Z-score calculation, and interpretation. Suitable for students in Grades 10-12.

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