You intend to estimate a population mean with a confidence interval. You believe the population to have a normal distribution. Your sample size is 20.

While it is an uncommon confidence level, find the critical value that corresponds to a confidence level of 95.5%.
*(Report answer accurate to three decimal places with appropriate rounding.)*

*t*a/2 = ±

To find the critical value \( t_{\alpha/2} \) for a 95.5% confidence level, we can break down the problem as follows:

1. **Confidence Level (CL)**: 95.5%, which implies a significance level \( \alpha = 1 - 0.955 = 0.045 \).
2. **Degrees of Freedom (df)**: Since the sample size is 20, we have \( df = n - 1 = 19 \).
3. **Two-Tailed Test**: For a two-tailed confidence interval, we are interested in \( \alpha/2 = 0.045 / 2 = 0.0225 \) for each tail.

Now, we need the \( t \)-value that leaves 2.25% in each tail with 19 degrees of freedom. Using a \( t \)-table or statistical software:

\[
t_{\alpha/2} \approx \pm 2.197
\]

**Answer:** The critical value \( t_{\alpha/2} = \pm 2.197 \).

Would you like a deeper explanation or have further questions?

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### Additional Related Questions
1. How does the choice of sample size affect the confidence interval's width?
2. What is the interpretation of a 95.5% confidence level?
3. How would the critical value change if the confidence level were set to 99%?
4. What if we assumed a population that was not normally distributed?
5. How does the sample size influence the accuracy of the t-distribution approximation?

**Tip:** Remember, as sample sizes increase, the t-distribution approaches the standard normal distribution (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 20. While it is an uncommon confidence level, find the critical value that corresponds to a confidence level of 95.5%. (Report answer accurate to three decimal places with appropriate rounding.) tα/2 = ±

Calculate the critical t-value for a 95.5% confidence level with a sample size of 20. Explore how to determine the degrees of freedom and the critical value for uncommon confidence levels using the t-distribution table or statistical software.

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