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