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:

• \( estimate \) — The sample estimate of the parameter
• \( critical\ value \) — Value from z or t distribution based on confidence level
• \( standard\ error \) — Standard error of the estimate

Explanation: The interval gives a range within which we can be confident (usually 95%) that the true population parameter lies.

3. Importance of Confidence Intervals

Details: Confidence intervals provide more information than point estimates alone by indicating the precision of the estimate and the uncertainty around it.

4. Using the Calculator

Tips: Enter the point estimate, critical value (e.g., 1.96 for 95% CI with normal distribution), and standard error. All values must be valid numbers.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between 90%, 95%, and 99% CIs?
A: Higher confidence levels produce wider intervals. 95% is most common, meaning if we repeated the study many times, 95% of the CIs would contain the true parameter.

Q2: How do I find the critical value?
A: For large samples with normal distribution, use z-scores (1.96 for 95%). For small samples, use t-distribution values.

Q3: What affects the width of a confidence interval?
A: Interval width depends on sample size (larger n → narrower CI), variability (more variation → wider CI), and confidence level (higher % → wider CI).

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 significance level.

Q5: What's the relationship between CIs and p-values?
A: Both provide information about statistical significance. A 95% CI that excludes the null value corresponds to p < 0.05.