# Confidence Interval Calculator

> Calculate confidence intervals for population mean or proportion. Supports z-intervals and t-intervals at various confidence levels.

**Category:** Statistics
**Keywords:** confidence interval, margin of error, z-interval, t-interval, statistics, sample size, confidence level
**URL:** https://complete.tools/confidence-interval-calculator

## How it calculates

To calculate a confidence interval, different formulas are used depending on whether the population standard deviation is known or unknown. For a z-interval, the confidence interval is calculated as: CI = µ ± z*(σ/√n), where CI is the confidence interval, µ is the sample mean, z is the z-score corresponding to the desired confidence level, σ is the population standard deviation, and n is the sample size. For a t-interval, the formula is: CI = µ ± t*(s/√n), where t is the t-score from the t-distribution, and s is the sample standard deviation. For a proportion interval, the formula is: CI = p ± z*(√(p(1-p)/n)), where p is the sample proportion. These calculations provide a range within which the true population parameter is expected to lie, taking into account sample variability.

## Who should use this

Biostatisticians analyzing clinical trial data to estimate treatment effects. Market researchers assessing consumer preferences based on survey results. Quality control engineers determining acceptable limits for product specifications based on sample production data.

## Worked examples

Example 1: A biostatistician wants to estimate the average effect of a drug. From a sample of 30 patients, the average recovery time is 15 days with a known population standard deviation of 4 days. For a 95% confidence interval, the z-score is 1.96. Calculating the interval: CI = 15 ± 1.96*(4/√30) = 15 ± 1.44. Thus, the confidence interval is (13.56, 16.44). This means the true average recovery time is likely between 13.56 and 16.44 days. 

Example 2: A market researcher surveys 100 consumers, finding that 60 prefer Product A, yielding a sample proportion (p) of 0.6. For a 95% confidence interval, using a z-score of 1.96: CI = 0.6 ± 1.96*(√(0.6(1-0.6)/100)) = 0.6 ± 0.098. The confidence interval is (0.502, 0.698), indicating that the true proportion of consumers who prefer Product A is likely between 50.2% and 69.8%.

## Limitations

The tool assumes that the sample data is randomly selected and normally distributed, which may not always be the case, especially for small sample sizes. It also assumes that the population standard deviation is known for z-intervals, which may lead to inaccurate results if this assumption does not hold. Additionally, the calculator does not account for non-response bias in survey data, potentially skewing proportion estimates. Finally, it may not perform well with very small sample sizes where the Central Limit Theorem does not apply, leading to less reliable interval estimates.

## FAQs

**Q:** How does the choice between z-interval and t-interval affect the results?


**A:** The z-interval is used when the population standard deviation is known, providing narrower intervals due to a fixed z-score. The t-interval accounts for additional uncertainty when the standard deviation is estimated from the sample, resulting in wider intervals as the t-score increases with smaller sample sizes.


**Q:** What impact does sample size have on confidence intervals?


**A:** Larger sample sizes yield narrower confidence intervals, enhancing precision, as the standard error (s/√n) decreases. Conversely, smaller samples produce wider intervals, reflecting higher uncertainty in the estimation of the population parameter.


**Q:** Can confidence intervals be interpreted as probabilities?


**A:** No, confidence intervals are not probabilities regarding the parameter itself. Instead, they represent a range that, based on repeated sampling, would contain the true parameter a specified percentage of the time. 


**Q:** How does the confidence level affect the width of the interval?


**A:** Higher confidence levels (e.g., 99% vs. 95%) produce wider intervals, as they require capturing more of the distribution's area, which increases the margin of error around the estimate.

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*Generated from [complete.tools/confidence-interval-calculator](https://complete.tools/confidence-interval-calculator)*