Personal notes on confidence intervals.

Based on notes taken during a course [1].

## Confidence interval

Confidence interval for mean \(\mu\) is \((a,b)\) with confidence \(p\)

means the same thing as

The procedure from which the interval \((a,b)\) was sampled returns an interval which contains mean \(\mu\) with probability \(p\)

## For normal distributions

### With known variance

When \(X \sim N(\mu, \sigma^2)\) and variance \(\sigma^2\) is known, the \(1-\alpha\) confidence interval for \(\mu\) is

\[ \overline{X} \pm z_\frac{\alpha}{2} \frac{\sigma}{\sqrt{n}} \]

### With unknown variance

For sample mean \(\overline{X}\), sample variance \(S^2\), and sample count \(n\), \(\frac{\overline{X} - \mu}{\frac{S}{\sqrt{n}}}\) has a \(t\)-distribution.

\[ \frac{\overline{X} - \mu}{\frac{S}{\sqrt{n}}} \sim t(n-1) \]

The \(1-\alpha\) confidence interval is

\[ \overline{X} \pm t_{\frac{\alpha}{2}, n-1} \frac{S}{\sqrt{n}} \]

## For any distribution with large population

For any distribution, when sample count is large enough for the distribution of \(\overline{X}\) to approximate a normal distribution as per the central limit theorem, the confidence interval for normal distributions is a reasonable approximation.

## Difference of means

### For normal distributions

#### With known variances

\[ \left( \overline{X_1} - \overline{X_2} \right) \pm z_\frac{\alpha}{2} \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}} \]

#### With same unknown variance

\[ \left( \overline{X_1} - \overline{X_2} \right) \pm t_{\frac{\alpha}{2}, n_1 + n_2 - 2} \sqrt{S_p^2 \left( \frac{1}{n_1} + \frac{1}{n_2} \right)} \]

##### Pooled variance

\[ S_p^2 = \frac{(n_1 - 1) S_1^2 + (n_2 - 1) S_2^2}{n_1 + n_2 - 2} \]

#### With unknown variances

Obtaining a confidence interval for the difference of means from normal distributions with separate unknown variances is known as the Behrens-Fisher problem. It can be approximated with Welchâ€™s approximation:

\[ T = \frac{(\overline{X_1} - \overline{X_2}) - (\mu_1 - \mu_2)} {\sqrt{\frac{S_1^2}{n_1} + \frac{S_2^2}{n_2}}} \approx t(\nu) \]

\[ \nu = \frac{\left( \frac{S_1^2}{n_1} + \frac{S_2^2}{n_2} \right)^2} {\frac{\left( \frac{S_1^2}{n_1} \right)^2}{n_1 - 1} + \frac{\left( \frac{S_2^2}{n_2} \right)^2}{n_2 - 1} } \]

## Proportion of variances

If \(X_1 \sim N(\mu_1,\sigma_1^2)\) and \(X_2 \sim N(\mu_2,\sigma_2^2)\), then the following has the \(F\)-distribution.

\[ \frac{\sigma_2^2}{\sigma_1^2} \cdot \frac{S_1^2}{S_2^2} \sim F(n_1-1, n_2-1) \]

\(\sigma_1\) and \(\sigma_2\) are the true variances, while \(S_1\) and \(S_2\) are the sample variances.