You measure 40 textbooks' weights, and find they have a mean weight of 31 ounces. Assume the population standard deviation is 10.8 ounces. Based on this, construct a 99% confidence interval for the true population mean textbook weight.

Round your answer to 3 decimal places.

To construct a 99% confidence interval for the true population mean weight of the textbooks, we can use the formula for the confidence interval (CI):

\[
\text{CI} = \bar{x} \pm z \left( \frac{\sigma}{\sqrt{n}} \right)
\]

Where:
- \(\bar{x}\) = sample mean = 31 ounces
- \(z\) = z-score corresponding to the confidence level (for 99%, \(z \approx 2.576\))
- \(\sigma\) = population standard deviation = 10.8 ounces
- \(n\) = sample size = 40

### Step 1: Calculate the standard error (SE)
\[
SE = \frac{\sigma}{\sqrt{n}} = \frac{10.8}{\sqrt{40}} \approx \frac{10.8}{6.3246} \approx 1.708
\]

### Step 2: Calculate the margin of error (ME)
\[
ME = z \cdot SE = 2.576 \cdot 1.708 \approx 4.404
\]

### Step 3: Construct the confidence interval
\[
\text{Lower limit} = \bar{x} - ME = 31 - 4.404 \approx 26.596
\]
\[
\text{Upper limit} = \bar{x} + ME = 31 + 4.404 \approx 35.404
\]

### Final Confidence Interval
The 99% confidence interval for the true population mean textbook weight is:

\[
(26.596, 35.404)
\]

Rounded to three decimal places, the final answer is:

\[
(26.596, 35.404)
\]

Would you like any further details or explanations? Here are some related questions you might consider:
1. What is the significance of the z-score in this calculation?
2. How would the confidence interval change if the sample size were larger?
3. What assumptions must be met to use this method for constructing a confidence interval?
4. How does the choice of confidence level affect the width of the interval?
5. What would happen to the confidence interval if the population standard deviation were unknown? 

**Tip:** When interpreting confidence intervals, remember that it does not mean there's a 99% chance the true mean falls within the interval; rather, if you were to take many samples, 99% of those intervals would capture the true mean.

You measure 40 textbooks' weights, and find they have a mean weight of 31 ounces. Assume the population standard deviation is 10.8 ounces. Based on this, construct a 99% confidence interval for the true population mean textbook weight.

Explore how to construct a 99% confidence interval for the mean weight of textbooks based on a sample of 40 with a mean weight of 31 ounces and a population standard deviation of 10.8 ounces. This problem illustrates key statistical concepts suitable for advanced high school students.

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