The heights of adult women in a population are normally distributed with a mean of 166 cm and a standard deviation of 6 cm. What is the probability of a woman’s height being between 163 and 175 cm?

To solve this problem, we can use the properties of the normal distribution.

The height of adult women is normally distributed with:
- Mean \( \mu = 166 \) cm
- Standard deviation \( \sigma = 6 \) cm

We are asked to find the probability that a woman's height is between 163 cm and 175 cm, which can be represented as:

\[
P(163 \leq X \leq 175)
\]

### Step 1: Convert to the Standard Normal Distribution

We convert the values 163 and 175 to their respective z-scores using the z-score formula:

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

1. For \( X = 163 \):
   \[
   z_1 = \frac{163 - 166}{6} = \frac{-3}{6} = -0.5
   \]

2. For \( X = 175 \):
   \[
   z_2 = \frac{175 - 166}{6} = \frac{9}{6} = 1.5
   \]

### Step 2: Find the Probability Using the Z-Table

Next, we use the standard normal distribution table (z-table) to find the probabilities corresponding to these z-scores.

1. For \( z_1 = -0.5 \):
   \[
   P(Z \leq -0.5) = 0.3085
   \]

2. For \( z_2 = 1.5 \):
   \[
   P(Z \leq 1.5) = 0.9332
   \]

### Step 3: Calculate the Desired Probability

The probability of a woman's height being between 163 cm and 175 cm is:

\[
P(163 \leq X \leq 175) = P(Z \leq 1.5) - P(Z \leq -0.5)
\]

\[
P(163 \leq X \leq 175) = 0.9332 - 0.3085 = 0.6247
\]

### Final Answer:
The probability of a woman’s height being between 163 cm and 175 cm is approximately **0.6247** or **62.47%**.

Would you like further details or have any questions? 

Here are 5 related questions you might find interesting:
1. What is the probability of a woman being taller than 180 cm?
2. How would the probability change if the standard deviation was 4 cm instead of 6 cm?
3. What is the probability of a woman’s height being less than 160 cm?
4. What proportion of women are between 160 cm and 170 cm tall?
5. How would you find the height that corresponds to the 90th percentile of this population?

**Tip**: The z-score is a powerful tool to standardize any normal distribution problem into the standard normal distribution, making it easier to use tables or calculators.

Calculate the probability of a woman’s height being between 163 cm and 175 cm, given a mean height of 166 cm and a standard deviation of 6 cm. This problem uses the normal distribution and Z-scores, ideal for Grades 10-12.

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