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Let's go through each part of the problem step by step, summarizing the solutions we obtained:

### a) Normal Distribution Curve

To visualize the normal distribution curve for the ages of the members in the hiking club:

- **Mean (\(\mu\))**: 32 years
- **Standard Deviation (\(\sigma\))**: 6 years

The normal distribution curve is bell-shaped, centered at the mean. Here's a description of the curve:
- The peak of the curve is at the mean, 32 years.
- The spread of the curve is determined by the standard deviation.

### b) Percent of Members Between 32 and 44 Years

To find the percentage of members between 32 and 44 years:

1. Calculate the z-scores:
   \[
   z_{32} = \frac{32 - 32}{6} = 0
   \]
   \[
   z_{44} = \frac{44 - 32}{6} \approx 2
   \]

2. Using the z-table or cumulative distribution function:
   \[
   P(0 < Z < 2) = \Phi(2) - \Phi(0) \approx 0.9772 - 0.5 = 0.4772
   \]

3. Convert to percentage:
   \[
   0.4772 \times 100 \approx 47.72\%
   \]

So, approximately **47.72%** of the members are between 32 and 44 years old.

### c) Percent of Members Younger than 42 Years

To find the percentage of members younger than 42 years:

1. Calculate the z-score:
   \[
   z_{42} = \frac{42 - 32}{6} \approx 1.67
   \]

2. Using the z-table or cumulative distribution function:
   \[
   P(Z < 1.67) = \Phi(1.67) \approx 0.9522
   \]

3. Convert to percentage:
   \[
   0.9522 \times 100 \approx 95.22\%
   \]

So, approximately **95.22%** of the members are younger than 42 years old.

### Summary

- **a)** The normal distribution curve is centered at 32 years with a standard deviation of 6 years.
- **b)** Approximately 47.72% of the members are between 32 and 44 years old.
- **c)** Approximately 95.22% of the members are younger than 42 years old.

Would you like further details or have any questions about this solution? Here are some follow-up questions you might consider:

1. How do we interpret z-scores in the context of normal distribution?
2. What is the significance of the standard deviation in a normal distribution?
3. How can we use normal distribution in different real-life scenarios?
4. What other types of probability distributions are commonly used in statistics?
5. How do we use cumulative distribution functions to find probabilities?

**Tip:** When interpreting normal distributions, the z-score helps in standardizing values, making it easier to compare different datasets or understand probabilities associated with certain ranges.

Explore the normal distribution analysis of age data in a hiking club. Includes calculations for mean, standard deviation, z-scores, and percentage calculations. Suitable for college-level students studying statistics.

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