Confidence Interval Simulation

Simulation

Interactive Confidence Interval Simulation

Population Mean

Population Standard Deviation

Sample Size

Confidence Level

Number of Simulations

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Understanding Confidence Intervals

Overview

A confidence interval is a range of values that provides an estimate of an unknown population parameter with a specified level of confidence. It helps quantify the uncertainty in sample estimates and provides a range of plausible values for the true population parameter.

Common Misconception

Misconception: A 95% confidence interval means there is a 95% probability that the true parameter lies within the interval.

Reality: The confidence level (95%) refers to the procedure's reliability, not the probability of the parameter being in any specific interval. If we were to repeat the sampling process many times, about 95% of the resulting intervals would contain the true parameter.

Key Concepts

1. Confidence Level

The probability (typically expressed as a percentage) that the confidence interval contains the true population parameter. Common levels are 90%, 95%, and 99%.

2. Margin of Error

The distance between the point estimate and the confidence interval bounds, calculated as: $MOE = z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}$

3. Coverage Rate

The proportion of intervals that contain the true population parameter when the process is repeated many times. Should match the confidence level.

4. Sample Size Effect

Larger sample sizes ( $n$ ) lead to narrower confidence intervals, providing more precise estimates of the population parameter.

Mathematical Foundation

For a population mean with known standard deviation:

CI = \bar{x} \pm z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}

Where:

$\bar{x}$ is the sample mean
$z_{\alpha/2}$ is the critical value
$\sigma$ is the population standard deviation
$n$ is the sample size