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. The most common is the 95% confidence interval, which means if the same population is sampled multiple times, approximately 95% of the confidence intervals would contain the true population parameter.

2. How Does the Calculator Work?

The calculator uses the basic confidence interval formula:

Where:

• \( \bar{x} \) — Sample mean (point estimate)
• \( MOE \) — Margin of error (half the width of the confidence interval)

Explanation: The margin of error represents how much we expect our estimate to vary from the true population parameter. A smaller margin of error indicates greater precision.

3. Importance of Confidence Intervals

Details: Confidence intervals provide more information than point estimates alone. They indicate both the size of the effect and the uncertainty around the estimate, helping researchers and analysts understand the precision of their measurements.

4. Using the Calculator

Tips: Enter the sample mean and margin of error in the same units. The calculator will output the confidence interval range. Both values must be valid numbers (MOE ≥ 0).

5. Frequently Asked Questions (FAQ)

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

Q2: How is margin of error determined?
A: MOE typically depends on the standard error of the estimate and the critical value from the appropriate distribution (often z or t distribution).

Q3: What affects the width of a confidence interval?
A: Three main factors: sample size (larger samples → narrower intervals), variability in the data (less variability → narrower intervals), and confidence level (higher confidence → wider intervals).

Q4: When should I use z-score vs t-score for MOE?
A: Use z-scores when population standard deviation is known (rare) or sample size is large (n > 30). Use t-scores for small samples with unknown population standard deviation.

Q5: Can confidence intervals be used for hypothesis testing?
A: Yes, if a 95% CI doesn't contain the null hypothesis value (often zero), you can reject the null hypothesis at the 5% significance level.