A confidence interval calculator is a tool that helps calculate a range of values that is likely to contain a population parameter, such as a mean or proportion, with a certain degree of confidence.

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## What is confidence interval?

A simple definition of the confidence interval is a range of values that has the inclusion of a population parameter. The value of this parameter is unknown. When it comes to the best calculation option, using a confidence interval calculator is the finest alternative.

## Confidence Interval Formula

A confidence interval is a range of values that is likely to contain the true value of a population parameter, such as the mean or proportion. The formula for a confidence interval depends on the type of data and the parameter of interest. Here are the most common formulas:

- Confidence interval for the population mean (when the population standard deviation is known):CI = x̄ ± zα/2 * σ / sqrt(n)Where:
- CI is the confidence interval
- x̄ is the sample mean
- zα/2 is the critical value from the standard normal distribution based on the desired confidence level (for example, zα/2 = 1.96 for a 95% confidence level)
- σ is the population standard deviation
- n is the sample size

- Confidence interval for the population mean (when the population standard deviation is unknown):CI = x̄ ± tα/2 * s / sqrt(n)Where:
- CI is the confidence interval
- x̄ is the sample mean
- tα/2 is the critical value from the t-distribution based on the desired confidence level and the degrees of freedom (for example, tα/2 = 2.093 for a 95% confidence level and 24 degrees of freedom)
- s is the sample standard deviation
- n is the sample size

- Confidence interval for the population proportion:CI = p̂ ± zα/2 * sqrt(p̂(1-p̂) / n)Where:
- CI is the confidence interval
- p̂ is the sample proportion
- zα/2 is the critical value from the standard normal distribution based on the desired confidence level (for example, zα/2 = 1.96 for a 95% confidence level)
- n is the sample size

Note that the confidence interval formula assumes that the sample is representative of the population, and that the data is normally distributed. Also, the confidence level, which is typically 95% or 99%, represents the probability that the true population parameter falls within the calculated confidence interval.

## How to calculate Confidence interval?

To calculate a confidence interval, you need to follow these general steps:

- Determine the type of data and the population parameter you want to estimate. The most common parameters are the population mean and proportion.
- Collect a random sample from the population and calculate the sample mean or proportion and standard deviation.
- Choose a confidence level, which is typically 95% or 99%. This represents the probability that the true population parameter falls within the calculated confidence interval.
- Determine the appropriate formula to use based on the type of data and parameter of interest. See the formulas in the previous answer.
- Calculate the critical value based on the desired confidence level and degrees of freedom. For the normal distribution, you can use a z-table or a calculator to find the critical value. For the t-distribution, you need to use a t-table or a calculator.
- Plug in the values from steps 2-5 into the formula and calculate the confidence interval.
- Interpret the confidence interval by saying “we are ___% confident that the true population parameter is between ___ and ___.”

Here is an example of calculating a confidence interval for the population mean:

Suppose you want to estimate the mean height of adult men in a certain country. You randomly sample 50 men and find that their average height is 175 cm with a standard deviation of 6 cm.

You choose a 95% confidence level and determine the critical value from the z-table to be 1.96.

Using the formula for the confidence interval for the population mean when the population standard deviation is unknown, you get:

CI = 175 ± 1.96 * 6 / sqrt(50) = (172.04, 177.96)

Interpretation: We are 95% confident that the true mean height of adult men in the country is between 172.04 and 177.96 cm.

## Example

Here is an example of how to calculate a confidence interval:

Suppose you want to estimate the proportion of adults in a city who support a particular policy. You randomly sample 500 adults and find that 280 of them support the policy.

You want to calculate a 95% confidence interval for the true proportion of adults who support the policy.

Using the formula for the confidence interval for the population proportion, you get:

CI = 0.56 ± 1.96 * sqrt(0.56 * 0.44 / 500) = (0.516, 0.604)

Interpretation: We are 95% confident that the true proportion of adults in the city who support the policy is between 0.516 and 0.604. This means that if we were to repeat the sampling process many times and calculate the confidence interval each time, 95% of those intervals would contain the true proportion.