Confidence Intervals Cheat Sheet

Confidence intervals provide a range of plausible values for an unknown population parameter, based on sample data. They quantify the precision of an estimate and help in making more meaningful interpretations of research findings.

Core Principles

A confidence interval (CI) is a range of values that is likely to contain the true population parameter.
The width of the CI reflects the precision of the estimate; narrower intervals indicate greater precision.
Confidence level (e.g., 95%) indicates the probability that the interval contains the true population parameter if the sampling process were repeated many times.
The margin of error is half the width of the confidence interval and represents the maximum expected difference between the sample statistic and the population parameter.
Confidence intervals can be calculated for means and proportions.
The choice between using a z-statistic or a t-statistic depends on whether you are estimating a proportion or a mean, respectively, and sample size considerations.
The t-distribution is used for means when the population standard deviation is unknown and sample sizes are small, adjusting for sample size through degrees of freedom.
Degrees of freedom (df) are typically N-1 for a single sample mean and influence the shape of the t-distribution.

Action Steps

Identify whether you are estimating a mean or a proportion.
Determine the sample size (N), sample mean (Y-bar) or proportion (p), and sample standard deviation (S_Y).
Choose the appropriate statistic (z or t) based on the parameter being estimated and sample characteristics.
Find the critical value (z* or t*) corresponding to your desired confidence level and degrees of freedom (if using t).
Calculate the standard error (S_Y-bar or S_p).
Calculate the margin of error (critical value * standard error).
Construct the confidence interval: (sample statistic - margin of error, sample statistic + margin of error).
Interpret the confidence interval in the context of the problem, stating the confidence level and the plausible range for the population parameter.

Formulas

For a mean: $ \bar{Y} \pm z^* S_{\bar{Y}} $ or $ \bar{Y} \pm t^* S_{\bar{Y}} $
Standard Error of the Mean (population SD known): $ \sigma_{\bar{Y}} = \frac{\sigma_Y}{\sqrt{N}} $
Standard Error of the Mean (sample SD used): $ S_{\bar{Y}} = \frac{S_Y}{\sqrt{N}} $
For a proportion: $ p \pm z^* S_p $
Standard Error of a Proportion: $ S_p = \sqrt{\frac{p(1-p)}{N}} $
Degrees of Freedom: $ df = N - 1 $
Confidence Level Calculation: $ confidence = (1 - \alpha) * 100 $

Key Terms

Confidence Interval (CI): A range of values, calculated from sample data, that is likely to contain the true value of a population parameter.
Confidence Level: The probability, expressed as a percentage, that a confidence interval will contain the true population parameter if the sampling process is repeated many times (e.g., 95%).
Margin of Error (MOE): The amount added and subtracted from the sample statistic to create the confidence interval. It represents the uncertainty in the estimate.
Standard Error (SE): The standard deviation of the sampling distribution of a statistic. It measures the typical error or variability of the statistic across different samples.
Sampling Distribution: The probability distribution of a statistic (e.g., sample mean, sample proportion) obtained from all possible samples of a given size from a population.
Z-statistic: A standardized score used when the population standard deviation is known or for proportions with large sample sizes. It measures how many standard errors a sample statistic is from the population parameter.
T-statistic: A standardized score used when the population standard deviation is unknown and estimated from the sample, especially with smaller sample sizes. It accounts for the extra uncertainty introduced by estimating the standard deviation.
Degrees of Freedom (df): A parameter that characterizes the shape of the t-distribution, typically calculated as sample size minus one (N-1) for a single sample mean.
Alpha Level (α): The probability of making a Type I error (rejecting a true null hypothesis), which is equal to 1 minus the confidence level.

Real World Examples

Estimating average IQ: A 95% CI for IQ scores might be [100.6, 103.4], suggesting that we are 95% confident the true average IQ of the population is within this range.
Assessing support for a policy: A 95% CI for support of a new subway line might be [69%, 75%]. If the policy requires 70% support, this interval indicates uncertainty about meeting the threshold.
Measuring GPA: A 95% CI for GPA might be [2.70, 2.90], providing a range for the true average GPA of students.