Find the critical value z Subscript c necessary to form a confidence interval at the level of confidence shown below.
cequals0.80 

To find the critical value \( z_c \) necessary to form a confidence interval at a confidence level of 80% (\( c = 0.80 \)), follow these steps:

1. The confidence level \( c = 0.80 \) corresponds to the middle 80% of the standard normal distribution.
2. To find the critical value, we need to determine the z-scores that leave 10% in each tail of the distribution. This is because the remaining 20% is split evenly between the two tails, with 10% on each side.
3. Using a standard normal distribution table or a calculator, find the z-score that corresponds to an area of \( 1 - \frac{1 - c}{2} = 1 - 0.1 = 0.90 \).

The z-score for an area of 0.90 is approximately:

\[
z_c \approx 1.28
\]

Thus, the critical value \( z_c \) is approximately **1.28**.

Would you like more details or explanations? Here are some related questions to expand your understanding:

1. How is the critical value related to the margin of error in a confidence interval?
2. What is the difference between a 95% confidence interval and an 80% confidence interval in terms of z-scores?
3. How can you calculate the margin of error if you know the critical value and standard deviation?
4. What is the significance of using z-scores in hypothesis testing and confidence intervals?
5. How would the confidence interval change if we increased the confidence level to 90%?

**Tip:** The critical value decreases as the confidence level decreases because the area in the tails of the normal distribution becomes larger.

Find the critical value z Subscript c necessary to form a confidence interval at the level of confidence shown below. c equals 0.80

This page explains how to find the critical value z Subscript c for forming an 80% confidence interval. Learn how to calculate the z-score and understand its role in confidence intervals using the standard normal distribution. Suitable for advanced high school or college-level statistics students.

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