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To construct a 95% confidence interval for the mean mathematics SAT score, we will use the following formula for a confidence interval when the population standard deviation (σ) is known:

\[
\text{Confidence interval} = \bar{x} \pm z_{\alpha/2} \left(\frac{\sigma}{\sqrt{n}}\right)
\]

Where:
- \(\bar{x} = 455\) (sample mean)
- \(\sigma = 113\) (population standard deviation)
- \(n = 100\) (sample size)
- \(z_{\alpha/2} = 1.96\) for a 95% confidence level

### Step-by-step solution:
1. **Calculate the standard error (SE)**:
\[
SE = \frac{\sigma}{\sqrt{n}} = \frac{113}{\sqrt{100}} = \frac{113}{10} = 11.3
\]

2. **Find the margin of error (MOE)**:
\[
MOE = z_{\alpha/2} \times SE = 1.96 \times 11.3 = 22.148
\]

3. **Construct the confidence interval**:
\[
\bar{x} \pm MOE = 455 \pm 22.148
\]
So, the confidence interval is approximately:
\[
455 - 22.148 = 432.85 \quad \text{and} \quad 455 + 22.148 = 477.15
\]
Rounded to the nearest whole number, the 95% confidence interval is:
\[
433 < \mu < 477
\]

Would you like more details or have any questions?

### Relative questions:
1. How does the confidence level affect the width of a confidence interval?
2. What is the difference between using a t-distribution and z-distribution for confidence intervals?
3. How does increasing the sample size impact the confidence interval?
4. Why is the population standard deviation needed for this confidence interval?
5. How would the confidence interval change if the sample mean were different?

**Tip:** Always check if you're using the correct distribution (z vs t) based on whether the population standard deviation is known and sample size.

A college admissions officer takes a simple random sample of 100 entering freshmen and computes their mean mathematics SAT score to be 455. Assume the population standard deviation is 113. Construct a 95% confidence interval for the mean mathematics SAT score for the entering freshman class.

Learn how to construct a 95% confidence interval for the mean SAT score of a sample of 100 students. The problem involves population standard deviation, sample mean, and z-scores. Suitable for college-level introductory statistics courses.

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