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Let's analyze each part of the problem step-by-step.

### Part (a)
To construct a 95% confidence interval for the mean weight of all one-year-old baby boys in the United States, given:
- Sample mean (\(\bar{x}\)) = 25.5 pounds
- Population standard deviation (\(\sigma\)) = 5.3 pounds
- Sample size (\(n\)) = 360

The formula for a confidence interval when the population standard deviation is known is:
\[
\bar{x} \pm Z \cdot \frac{\sigma}{\sqrt{n}}
\]
where \(Z\) is the z-score corresponding to a 95% confidence level (approximately 1.96).

1. **Calculate the standard error**:
   \[
   \text{Standard Error} = \frac{\sigma}{\sqrt{n}} = \frac{5.3}{\sqrt{360}} \approx 0.279
   \]

2. **Determine the margin of error**:
   \[
   \text{Margin of Error} = Z \cdot \text{Standard Error} = 1.96 \cdot 0.279 \approx 0.546
   \]

3. **Construct the confidence interval**:
   \[
   25.5 \pm 0.546
   \]
   This gives us an interval of approximately:
   \[
   (24.9, 26)
   \]

So, the answer in part (a) is:
\[
24.9 < \mu < 26
\]

### Part (b)
This confidence interval was calculated specifically for one-year-old **baby boys**, not all one-year-old babies (which would include girls). Therefore, it **should not** be used to estimate the mean weight of all one-year-old babies in the United States, because it is only representative of boys.

### Part (c)
Since the confidence interval for the mean weight is \(24.9 < \mu < 26\), it is **not likely** that the mean weight of all one-year-old baby boys is less than 23 pounds, as 23 pounds falls outside the lower bound of the interval (24.9).

Would you like further details on any part of the solution, or do you have questions?

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### 5 Related Questions:
1. How would the confidence interval change with a 99% confidence level?
2. What is the effect of increasing the sample size on the margin of error?
3. How would this calculation change if we didn't know the population standard deviation?
4. Why is the z-score 1.96 for a 95% confidence interval?
5. How would the confidence interval change if the sample mean were 26 pounds instead?

### Tip:
Remember, the width of a confidence interval is directly affected by the sample size and the confidence level chosen. Larger sample sizes yield narrower intervals for the same confidence level.

According to a recent report, a sample of 360 one-year-old baby boys in the United States had a mean weight of 25.5 pounds. Assume the population standard deviation is σ = 5.3 pounds. Construct a 95% confidence interval for the mean weight of all one-year-old baby boys in the United States, and answer related questions about the interval.

Learn how to construct a 95% confidence interval for the mean weight of one-year-old baby boys in the United States. This problem covers core statistical concepts, including confidence intervals and the central limit theorem, with given population standard deviation and sample data.

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