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To solve the given problem regarding baby weights, we need to use the properties of the normal distribution. The problem provides the mean and standard deviation of the baby weights:

- Mean (μ) = 12.2 pounds
- Standard deviation (σ) = 4.5 pounds

### (a) Finding the 86th percentile:
To find the 86th percentile, we use the inverse of the cumulative distribution function (CDF) for the normal distribution. Specifically, we need to find the z-score corresponding to the 86th percentile, and then use the formula:

\[
X = \mu + (z \times \sigma)
\]

The z-score for the 86th percentile can be found from standard normal distribution tables or by using a calculator. The z-score corresponding to the 86th percentile is approximately **1.08**.

So:

\[
X = 12.2 + (1.08 \times 4.5) = 12.2 + 4.86 = 17.06 \, \text{pounds}
\]

### (b) Finding the 15th percentile:
For the 15th percentile, we use the same method but with the z-score corresponding to the 15th percentile. The z-score is approximately **-1.04**.

So:

\[
X = 12.2 + (-1.04 \times 4.5) = 12.2 - 4.68 = 7.52 \, \text{pounds}
\]

### (c) Finding the third quartile (75th percentile):
The third quartile is the 75th percentile. The z-score for the 75th percentile is **0.674**.

So:

\[
X = 12.2 + (0.674 \times 4.5) = 12.2 + 3.03 = 15.23 \, \text{pounds}
\]

### Final Answers:
- (a) 86th percentile: **17.06 pounds**
- (b) 15th percentile: **7.52 pounds**
- (c) Third quartile (75th percentile): **15.23 pounds**

Would you like more details on how these values were computed, or do you have any questions?

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Here are 5 related questions to explore further:
1. How would you calculate the median weight of babies from the given data?
2. What is the probability that a randomly chosen baby weighs more than 17 pounds?
3. How would you find the interquartile range for the baby weights?
4. What is the z-score for a baby that weighs 10 pounds?
5. How do you interpret the 86th percentile in terms of baby weight distribution?

**Tip**: You can use Excel functions like `NORM.INV` to calculate percentiles in a normal distribution directly.

The weight of male babies less than 2 months old in the United States is normally distributed with a mean of 12.2 pounds and a standard deviation of 4.5 pounds. (a) Find the 86th percentile of the baby weights. (b) Find the 15th percentile of the baby weights. (c) Find the third quartile of the baby weights. Use Excel and round the answers to at least two decimal places.

Learn how to calculate the 86th and 15th percentiles and the third quartile for baby weights using normal distribution. This problem involves statistical concepts like mean, standard deviation, z-scores, and percentiles. The data provided includes a mean baby weight of 12.2 pounds and a standard deviation of 4.5 pounds.

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