Find Confidence Interval Calculator

Confidence Interval Formula:

\[ \text{CI} = \text{mean} \pm (\text{critical value} \times \text{standard error}) \] \[ \text{Lower bound} = \text{mean} - \text{crit} \times \text{se} \] \[ \text{Upper bound} = \text{mean} + \text{crit} \times \text{se} \]

Confidence Interval:

Lower Bound:

Upper Bound:

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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 following formulas:

\[ \text{Lower bound} = \text{mean} - (\text{critical value} \times \text{standard error}) \] \[ \text{Upper bound} = \text{mean} + (\text{critical value} \times \text{standard error}) \]

Where:

\( \text{mean} \) — Sample mean or point estimate
\( \text{critical value} \) — Value from z or t distribution based on confidence level
\( \text{standard error} \) — Standard error of the estimate

Explanation: The confidence interval gives a range around the sample mean that likely contains the true population mean.

3. Importance of Confidence Intervals

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

4. Using the Calculator

Tips: Enter the sample mean, critical value (from z or t table), and standard error. The calculator will compute the lower and upper bounds of the confidence interval.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between 90%, 95%, and 99% CIs?
A: The percentage indicates how confident we are that the interval contains the true parameter. Higher confidence means wider intervals.

Q2: How do I find the critical value?
A: Critical values come from statistical tables (z-table for normal distribution, t-table for t-distribution) based on your confidence level and degrees of freedom.

Q3: What affects the width of a confidence interval?
A: Interval width depends on sample size (n), variability in data (σ), and confidence level (higher confidence = wider interval).

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

Q5: Can CI be used for hypothesis testing?
A: Yes, if a CI doesn't contain the null hypothesis value, you can reject the null hypothesis at your chosen significance level.