Find the critical value for a 90% confidence interval for the mean for a sample size of 75. Assume the population standard deviation is unknown.

Round to four decimal places. 

To find the critical value for a 90% confidence interval when the population standard deviation is unknown, we use the *t-distribution* rather than the *z-distribution* since we’re dealing with an unknown population standard deviation. 

### Step-by-Step Solution
1. **Determine the Confidence Level and Degrees of Freedom:**
   - Confidence Level, \( C = 0.90 \), so the significance level \( \alpha = 1 - C = 0.10 \).
   - Since we want the middle 90%, we need 5% in each tail of the distribution.
   - Degrees of Freedom \( df = n - 1 = 75 - 1 = 74 \).

2. **Locate the Critical t-Value:**
   - Using a *t-distribution table* or calculator, find the t-value that corresponds to 0.05 in the upper tail with 74 degrees of freedom.

The **critical t-value** for a 90% confidence interval with 74 degrees of freedom is approximately:

\[
t_{0.05, 74} \approx 1.6652
\]

Thus, the critical value for this confidence interval is **1.6652**.

Would you like further details on how to use the t-table or any additional questions on this process?

### Related Questions
1. How would the critical value change if the confidence level was 95% instead of 90%?
2. What happens to the critical value as the sample size increases to 100?
3. How is the calculation affected if the population standard deviation was known?
4. Why is the t-distribution used here instead of the normal distribution?
5. How would you construct the confidence interval using this critical value?

### Tip:
The t-distribution approaches the normal distribution as the sample size increases, making them nearly identical for very large samples.

Find the critical value for a 90% confidence interval for the mean for a sample size of 75. Assume the population standard deviation is unknown. Round to four decimal places.

Learn how to find the critical value for a 90% confidence interval when the population standard deviation is unknown, given a sample size of 75. This guide uses the t-distribution and includes a step-by-step solution with key mathematical concepts, such as confidence intervals and degrees of freedom. Suitable for college-level students.

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