A random sample of 23 items is drawn from a population whose standard deviation is unknown. The sample mean is  x¯
  = 770 and the sample standard deviation is s = 25. Use Appendix D to find the values of Student’s t. Construct an interval estimate of μ with 95% confidence, assuming that s = 50.

To construct a 95% confidence interval for the population mean \(\mu\) using the given data, we follow these steps:

### Given Data:
- Sample size (\(n\)) = 23
- Sample mean (\(\bar{x}\)) = 770
- Sample standard deviation (\(s\)) = 25
- Population standard deviation (\(\sigma\)) is unknown
- We are asked to use \(s = 50\) instead.

### Step 1: Determine the Degrees of Freedom
The degrees of freedom (\(df\)) is calculated as:
\[
df = n - 1 = 23 - 1 = 22
\]

### Step 2: Find the Critical Value (\(t_{\alpha/2}\))
For a 95% confidence interval and 22 degrees of freedom, the critical value \(t_{\alpha/2}\) is obtained from the t-distribution table.

Looking at Appendix D (or typical t-distribution tables):
\[
t_{\alpha/2, 22} \approx 2.074
\]

### Step 3: Compute the Margin of Error
The margin of error (ME) is calculated using the formula:
\[
\text{ME} = t_{\alpha/2} \times \frac{s}{\sqrt{n}}
\]
Substituting the values:
\[
\text{ME} = 2.074 \times \frac{50}{\sqrt{23}} \approx 2.074 \times 10.42 \approx 21.61
\]

### Step 4: Construct the Confidence Interval
The 95% confidence interval for \(\mu\) is given by:
\[
\bar{x} \pm \text{ME}
\]
Substituting the values:
\[
770 \pm 21.61
\]
So the interval is:
\[
\text{95% Confidence Interval} = [748.39, 791.61]
\]

### Conclusion:
The 95% confidence interval for the population mean \(\mu\) is \([748.39, 791.61]\).

Would you like further details or have any questions?

---

Here are 8 questions you might consider next:

1. How would the confidence interval change if the sample size were larger?
2. What if the confidence level was 99% instead of 95%?
3. Why is the t-distribution used instead of the normal distribution in this scenario?
4. How does the margin of error relate to sample size?
5. What effect does increasing the sample standard deviation have on the confidence interval?
6. Can you explain how degrees of freedom impact the t-distribution?
7. What assumptions are necessary for the confidence interval to be valid?
8. How would the confidence interval change if the sample mean were different?

**Tip:** Always check if your data meets the assumptions for using the t-distribution, especially when the sample size is small.

Learn how to construct a 95% confidence interval for the population mean μ using sample data. Includes step-by-step calculation with a sample size of 23, sample mean of 770, and sample standard deviation of 25. Understand the use of Student’s t-distribution and margin of error formula.

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