Confidence Intervals
This page features confidence intervals’ definition, interpretation, and programming.
Definition
There are different versions of how to compute a confidence interval (CI) based on sample size and the shape of the parent distribution.
The general form of a CI: \(\text{statistic} \pm \text{critical value} * \text{standard error}\),
where critical value and standard error will depend on the statistic.
CI for Means
The following example is for a \(100 * (1 - \alpha) \%\) confidence interval for mean \(\mu\) when the value of \(\sigma\) is known:
Large Sample CI for a Mean (Z-test)
\([\bar{x} - z_{\alpha/2} \frac{\sigma}{\sqrt{n}}, \bar{x} + z_{\alpha/2} \frac{\sigma}{\sqrt{n}}]\)
\(= \bar{x} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}}\)
Small Sample CI for a Mean (t-test)
\(= \bar{x} \pm t_{n-1, \alpha/2} \frac{\sigma}{\sqrt{n}}\)
CI for Difference in Means
Large Sample CI for a Mean (Z-test)
\(\bar{x} - \bar{y} \pm z_{\alpha/2} \sqrt{\frac{\sigma_{x}^{2}}{n_x} + \frac{\sigma_{y}^{2}}{n_y}}\)
Small Sample CI for a Mean (t-test)
\(\bar{x} - \bar{y} \pm t \sqrt{\frac{\sigma_{x}^{2}}{n_x} + \frac{\sigma_{y}^{2}}{n_y}}\)
where
\(t = t_{n_x + n_y - 2, \alpha/2}\)
CI for Proportions
\(p \pm \pm z_{\alpha/2} \sqrt{\frac{p(1-p)}{n}}\)
CI for Difference in Proportions
\((\hat{p_1} - \hat{p_2}) \pm z_{\alpha/2} \sqrt{\frac{\hat{p_1}(1-\hat{p_1})}{n_1} + \frac{\hat{p_2}(1-\hat{p_2})}{n_2}}\)
CI for Slope Distribution (\(\beta\)s in Regression Models)
\(\hat{\beta} \pm t_{\alpha / 2, df = n-2} SE(\hat{\beta})\)
where
\(SE(\hat{\beta}) = \frac{\hat{\sigma}}{\sqrt{\sum_{i=1}^n (x_i - \bar{x})^2}}\)
Interpretation
Question: What does it mean to say that we are 95% confident that the true population mean is in this interval?
Answer: In repeated sampling, 95% of all CIs obtained from sampling will actually contain the true population mean. The other 5% of CIs will not.
R Programming
Most of the work in calculating CIs in R are in knowing which statistic and that statistic’s standard error formula to use.
These statistics are our main concern in the calculations, as the other values are normally algebraically calculated.
- z-stat:
qnorm(p, mean, sd)p: quantilemean: meansd: standard deviation
- t-stat:
qt(x, df)x: quantiledf: degrees of freedom