https://file.aiquickdraw.com/mathbot/file/1727314915863556uwbm6.jpg

To solve the problem in the image, we need to calculate the 95% confidence interval for the population mean when the population standard deviation (\(\sigma\)) is known.

### Given:
- Population standard deviation (\(\sigma\)) = 5.6
- Sample mean (\(\bar{x}\)) = 32
- Sample size (\(n\)) = 40
- Confidence level = 95%

### Step-by-Step Solution:

1. **Find the Z-score for a 95% Confidence Level:**
   - The Z-score corresponding to a 95% confidence interval is approximately **1.96** (since we are using the standard normal distribution).

2. **Calculate the Standard Error (SE):**
   The standard error of the mean is given by:
   \[
   SE = \frac{\sigma}{\sqrt{n}}
   \]
   Substituting the given values:
   \[
   SE = \frac{5.6}{\sqrt{40}} = \frac{5.6}{6.3246} \approx 0.885
   \]

3. **Calculate the Margin of Error (ME):**
   The margin of error is calculated using the Z-score:
   \[
   ME = Z \times SE = 1.96 \times 0.885 = 1.7346
   \]

4. **Calculate the Confidence Interval:**
   The confidence interval for the population mean is given by:
   \[
   \text{CI} = \bar{x} \pm ME
   \]
   Substituting the values:
   \[
   \text{CI} = 32 \pm 1.7346
   \]
   So, the 95% confidence interval is:
   \[
   (32 - 1.7346, 32 + 1.7346) = (30.27, 33.73)
   \]

### Conclusion:
The 95% confidence interval for the population mean is approximately **(30.27, 33.73)**.

---

Would you like more details or have any questions on this calculation?

Here are 5 related questions:
1. How is the Z-score determined for different confidence levels?
2. What happens to the confidence interval if the sample size increases?
3. Why is the standard error smaller when the sample size is larger?
4. What would the confidence interval look like if a 99% confidence level was used instead?
5. How is a t-distribution used for confidence intervals when the population standard deviation is unknown?

**Tip:** As the sample size increases, the standard error decreases, leading to a narrower confidence interval, meaning greater precision.

Scores on an exam are normally distributed with a population standard deviation of 5.6. A random sample of 40 scores on the exam has a mean of 32. Find the 95% confidence interval of the population mean.

Find the 95% confidence interval for the population mean with a known standard deviation of 5.6, using a sample size of 40 and a sample mean of 32. Learn step-by-step how to calculate using the standard error, Z-score, and margin of error.

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