You intend to estimate a population mean with a confidence interval. You believe the population to have a normal distribution. Your sample size is 19.

While it is an uncommon confidence level, find the critical value that corresponds to a confidence level of 89.1%.

To find the critical value for an 89.1% confidence level with a sample size of 19 (meaning 18 degrees of freedom), we’ll use the t-distribution since the sample size is relatively small.

The 89.1% confidence level implies that the area in the tails (combined) is \(1 - 0.891 = 0.109\). Thus, the area in each tail is \( \frac{0.109}{2} = 0.0545 \).

### Steps:
1. Determine the area to the left of the critical value in the t-distribution table. Since we are dealing with the right tail:
   \[
   \text{Area to the left} = 1 - 0.0545 = 0.9455
   \]

2. Using this area and 18 degrees of freedom, we find the critical value \( t_{\alpha/2} \) that leaves 5.45% in the upper tail.

Using a t-distribution table or calculator:
\[
t_{0.0545, 18} \approx 1.769
\]

### Answer:
The critical value for an 89.1% confidence level with 18 degrees of freedom is approximately:
\[
t \approx 1.769
\]

Would you like a further breakdown or details on the steps?

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1. What is the purpose of using a critical value in confidence intervals?
2. How does sample size affect the choice of distribution (t or z) for critical values?
3. Why is it necessary to adjust for degrees of freedom in small sample sizes?
4. What changes when we use a t-distribution instead of a z-distribution for critical values?
5. How would the critical value change if the confidence level were 95% instead of 89.1%?

**Tip:** For confidence intervals with smaller sample sizes (n < 30), using the t-distribution provides a more accurate critical value than the z-distribution.

You intend to estimate a population mean with a confidence interval. You believe the population to have a normal distribution. Your sample size is 19. While it is an uncommon confidence level, find the critical value that corresponds to a confidence level of 89.1%.

Discover how to find the critical value for an 89.1% confidence level using a sample size of 19. This statistical problem explores confidence intervals, the t-distribution, and adjustments for degrees of freedom. Suitable for college-level statistics students.

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