how to calculate confidence interval
How to Calculate Confidence Interval
Answer:
Calculating a confidence interval involves determining a range of values within which the true population parameter is likely to fall. In the context of statistics, the formula for calculating a confidence interval for a population mean, when the population standard deviation is known, is as follows:
\bar{x} \pm z \left( \frac{\sigma}{\sqrt{n}} \right)
Where:
- \bar{x} = sample mean
- z = z-score corresponding to the chosen confidence level
- \sigma = population standard deviation
- n = sample size
If the population standard deviation is not known and the sample size is large enough, we can use the t-distribution and the following formula:
\bar{x} \pm t \left( \frac{s}{\sqrt{n}} \right)
Where:
- \bar{x} = sample mean
- t = t-score corresponding to the chosen confidence level and degrees of freedom
- s = sample standard deviation
- n = sample size
To calculate a confidence interval for a population proportion, we use the formula:
\hat{p} \pm z \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}
Where:
- \hat{p} = sample proportion
- z = z-score corresponding to the chosen confidence level
- n = sample size
These formulas are commonly used to calculate confidence intervals for different types of data, providing a range of values that are likely to contain the true population parameter with a specified level of confidence.