4.2 Estimates, standard errors, and confidence intervals

Jargon

• Parameters are the unknown numbers that determine the statistical model

• Coefficients are, for example, the slope and intercept

• scale or variance is the measurement error

• estimand or quantity of interest is some summary of parameters or data of interest

We use data to contruct estimates of parameters or other quantities of interest.

• standard error is the estimated standard deviation of an estimate.

• Confidence interval represents a range of values of a parameter or quantity of interest that are roughly consistent with the data, given the assumed sampling distribution. If the model is correct, then in repeated applications the 50% and 95% confidence intervals will include the true value 50% and 95% of the time.

When the sampling distribution is a normal distribution with mean \(\mu\) and standard deviation \(\sigma\), and \(n\) draws (data) are made from this distribution, then the estimate for \(\mu\) is just the mean(data), the standard error is the sd(data)/sqrt(n), and confidence intervals can be estimated using quantiles.

If the normal distribution is a good approximation:

• 2 standard errors ~ 95% quantile
• 2/3 standard errors ~ 50% quantile