chapter 7 confidence intervals

confidence interval (CI)

a range of plausible values for an unknown population parameter, based on sample data

interpretation of 95% CI

if you repeated the study many times, 95% of the intervals would contain the true population parameter

point estimate

the sample-based value used as the best guess of the true population parameter (e.g., sample mean)

margin of error

the amount added and subtracted from the point estimate to create the confidence interval; calculated as critical value × standard error

standard error (SE)

estimates how much a point estimate would vary across repeated samples; SE = SD / √n

critical value

the t- or z-score corresponding to the desired confidence level (e.g., 1.96 for 95% CI with normal distribution)

formula for CI

CI = estimate ± (critical value × standard error)

wide confidence interval

indicates greater uncertainty or less precision

narrow confidence interval

indicates more precision and confidence in the estimate

values inside the CI

considered plausible and consistent with the data

values outside the CI

considered implausible or inconsistent with the data

relationship between CI and p-value

a two-sided 95% CI excludes 0 if and only if the p-value is < 0.05

CI crosses 0

the result is not statistically significant at the 0.05 level

CI does not cross 0

the result is statistically significant at the 0.05 level

how sample size affects CI

larger sample size → smaller SE → narrower CI → more precision

confidence level and CI width

higher confidence level (e.g., 99%) → wider CI

CI vs. p-value

p-values test whether an effect exists; CIs show how large the effect might be and how uncertain the estimate is

CI around effect size

provides a range of plausible values for the effect size, showing both magnitude and uncertainty

visualizing CI

confidence intervals can be shown with error bars to indicate precision and significance in plots

parameter

the true but unknown value in the population (e.g., true mean, true difference)

statistic

a value calculated from sample data used to estimate a population parameter

null value (usually 0)

represents “no effect” or “no difference”; if it’s inside the CI, the effect is not statistically significant