You intend to estimate a population mean with a confidence interval. You believe the population to have a normal distribution. Your sample size is 4.

While it is an uncommon confidence level, find the critical value that corresponds to a confidence level of 96.3%.
(Report answer accurate to three decimal places with appropriate rounding.)

ta/2 =
±

To find the critical value corresponding to a 96.3% confidence level for a normally distributed population, we will follow these steps:

1. **Calculate the area in the tails**:
   - The confidence level is 96.3%, so the area in the tails is \(1 - 0.963 = 0.037\).
   - Since the normal distribution is symmetric, the area in one tail is \(0.037 / 2 = 0.0185\).

2. **Find the z-value corresponding to the area in the tail**:
   - We need to find the z-value that leaves 0.0185 in the upper tail of the standard normal distribution.

The critical value \( z_{\alpha/2} \) can be found using standard normal distribution tables or a calculator with inverse normal function.

Let's calculate this value.The critical value that corresponds to a 96.3% confidence level is approximately \( \pm 2.086 \).

Would you like more details on this process or have any further questions?

Here are 8 related questions you might find useful:

1. What is the confidence interval formula for estimating a population mean?
2. How does sample size affect the width of a confidence interval?
3. What is the significance of the z-value in constructing confidence intervals?
4. How would you calculate a 95% confidence interval with the same sample size?
5. What role does the standard deviation play in confidence intervals?
6. What is the difference between z-distribution and t-distribution in constructing confidence intervals?
7. How does changing the confidence level affect the critical value?
8. What assumptions are required for the confidence interval calculation to be valid?

**Tip:** Remember, as the confidence level increases, the critical value increases, leading to a wider confidence interval.

Learn how to find the critical value for a 96.3% confidence level in a normal distribution. This problem involves statistical concepts such as normal distribution, confidence intervals, and standard normal distribution tables. Suitable for advanced college-level students and statisticians.

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