1. What is a Confidence Interval?

A confidence interval is a range of values that's likely to contain 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
• \( \sigma \) — Standard deviation
• \( n \) — Sample size
• \( z \) — Z-score corresponding to the desired confidence level

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, which is crucial for statistical inference.

4. Using the Calculator

Tips: Enter the sample mean, standard deviation, sample size, and z-score (common values: 1.96 for 95% CI, 2.576 for 99% CI). All values must be valid (n > 0, σ ≥ 0).

5. Frequently Asked Questions (FAQ)

Q1: What does a 95% confidence interval mean?
A: It means that if we were to take 100 different samples and compute a confidence interval for each, we would expect about 95 of them to contain the true population parameter.

Q2: How do I choose the right z-score?
A: The z-score depends on your desired confidence level. Common values are 1.645 (90%), 1.96 (95%), and 2.576 (99%).

Q3: When should I use t-scores instead of z-scores?
A: Use t-scores when the population standard deviation is unknown and the sample size is small (typically n < 30).

Q4: What affects the width of a confidence interval?
A: Interval width increases with higher confidence levels, larger standard deviations, and smaller sample sizes.

Q5: Can I use this for proportions?
A: Yes, but for proportions you might want to use the specific proportion formula: \( \hat{p} \pm z \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \).