1. What is a Confidence Interval?

A confidence interval (CI) is a range of values that's likely to include a population parameter with a certain degree of confidence. It provides an estimated range of values which is likely to include an unknown population parameter.

2. How Does the Calculator Work?

The calculator uses the confidence interval formula:

Where:

• \( \bar{x} \) — Sample mean
• \( z \) — Z-score for desired confidence level
• \( \sigma \) — Standard deviation
• \( n \) — Sample size

Explanation: The formula calculates the margin of error (z × standard error) and adds/subtracts it from the sample mean to create the interval.

3. Importance of Confidence Intervals

Details: Confidence intervals provide more information than point estimates alone. They indicate the precision of an estimate and the uncertainty around it, helping in statistical inference and decision making.

4. Using the Calculator

Tips: Enter the sample mean, standard deviation, and sample size. Select the appropriate z-score for your desired confidence level (or enter a custom z-score). All values must be valid (n > 0, σ ≥ 0).

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between 90%, 95%, and 99% CIs?
A: Higher confidence levels create wider intervals. A 95% CI means if we repeated the study 100 times, we'd expect the interval to contain the true parameter 95 times.

Q2: When should I use a t-score instead of z-score?
A: Use t-scores when sample size is small (typically n < 30) and population standard deviation is unknown.

Q3: What does a narrow vs wide CI indicate?
A: Narrow CIs suggest more precise estimates (usually from larger samples or less variability). Wide CIs indicate more uncertainty.

Q4: Can CIs be used for hypothesis testing?
A: Yes, if a CI doesn't contain the null value (often 0), you can reject the null hypothesis at that confidence level.

Q5: What are common z-score values?
A: 1.645 (90% CI), 1.96 (95% CI), and 2.576 (99% CI) are most common for normal distributions.