When constructing a confidence interval for the mean height of students at CBC, the critical value is utilized to help determine the margin of error. Here’s the step-by-step solution to determine the critical value for a 99% confidence interval:
1. Sample Size (n):
- The sample size given is 25.
2. Confidence Level:
- The problem specifies a 99% confidence interval. This means that we want our interval to capture the true mean height with 99% certainty.
3. Critical Value (Z-score):
- To construct a 99% confidence interval, we need to find the Z-score that corresponds to the middle 99% of the standard normal distribution.
4. Distribution of the Tails:
- The 99% confidence interval leaves out 1% of the distribution in the tails.
- This 1% is split equally between the two tails of the distribution, placing 0.5% (0.005) in each tail.
5. Finding the Z-score:
- We look up the Z-score that corresponds to the cumulative probability of 0.995 (since 1 - 0.005 = 0.995). This is because the Z-score for the right tail critical value up to 99.5% cumulative probability will give us our desired number.
6. Result:
- The Z-score that corresponds to a cumulative probability of 0.995 is approximately 2.5758293035489004.
Therefore, the appropriate value of the critical value for constructing the 99% confidence interval to estimate the true mean height of students at CBC is 2.5758293035489004.
