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To construct a 95% confidence interval for the true population mean weight of textbooks, we can use the formula:

\[
\text{Confidence Interval} = \bar{x} \pm Z \cdot \frac{\sigma}{\sqrt{n}}
\]

where:
- \(\bar{x}\) is the sample mean (30 ounces),
- \(\sigma\) is the population standard deviation (3.5 ounces),
- \(n\) is the sample size (30),
- \(Z\) is the Z-score corresponding to a 95% confidence level (approximately 1.96 for a 95% confidence level).

### Step-by-Step Solution
1. **Calculate the standard error** of the mean:
   \[
   \frac{\sigma}{\sqrt{n}} = \frac{3.5}{\sqrt{30}}
   \]

2. **Compute the margin of error** using the Z-score:
   \[
   \text{Margin of Error} = Z \cdot \frac{\sigma}{\sqrt{n}} = 1.96 \cdot \frac{3.5}{\sqrt{30}}
   \]

3. **Calculate the confidence interval** by adding and subtracting the margin of error from the sample mean.

Let’s perform these calculations:

#### Step 1: Calculate Standard Error
\[
\frac{3.5}{\sqrt{30}} \approx 0.6397
\]

#### Step 2: Calculate Margin of Error
\[
1.96 \cdot 0.6397 \approx 1.2539
\]

#### Step 3: Confidence Interval
\[
\text{Lower Limit} = 30 - 1.2539 \approx 28.7461
\]
\[
\text{Upper Limit} = 30 + 1.2539 \approx 31.2539
\]

### Final Answer
The 95% confidence interval for the true population mean textbook weight is approximately:

\[
(28.7461, 31.2539)
\]

Would you like a deeper explanation of any steps, or do you have any specific questions?

---

1. What is the importance of a confidence interval in statistics?
2. How does sample size affect the width of a confidence interval?
3. Why do we use a Z-score for confidence intervals when the population standard deviation is known?
4. How would the interval change with a 99% confidence level instead?
5. What are the implications if the sample mean were significantly different from the population mean?

**Tip:** Remember that a wider confidence interval indicates greater uncertainty, while a narrower interval suggests more precision.

You measure 30 textbooks' weights, and find they have a mean weight of 30 ounces. Assume the population standard deviation is 3.5 ounces. Based on this, construct a 95% confidence interval for the true population mean textbook weight. Round answers to at least 4 decimal places.

Learn how to construct a 95% confidence interval for the mean weight of textbooks based on a sample mean of 30 ounces and a population standard deviation of 3.5 ounces. This guide covers confidence interval calculations with step-by-step instructions.

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