STA 209 – Exam 3 Formula Sheet
Single Proportion HT
\[\begin{equation*} Z := \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} \sim \textbf{N(0,1)} \end{equation*}\]
Conditions:
- Representative sample
- \(n \times p_0 \geq 10\)
- \(n \times (1-p_0) \geq 10\)
Difference in Proportions
\[\begin{equation*} Z := = \frac{(\hat{p}_1 - \hat{p}_2)}{\sqrt{\widehat{p}_{pool}(1-\widehat{p}_{pool})(\frac{1}{n_1}+\frac{1}{n_2})}} \sim \textbf{N(0,1)} \end{equation*}\],
\[\begin{equation*} \text{where} \hspace{3mm} \widehat{p}_{pool} = \frac{x_1 + x_2}{n_1 + n_2} = \frac{n_1\hat{p}_1 + n_2\hat{p}_2}{n_1 + n_2} \end{equation*}\]
Conditions:
- Representative, independent samples
- \(n_1 \times \widehat{p}_1 \geq 10\) and \(n_1 \times (1-\widehat{p}_1) \geq 10\)
- \(n_2 \times \widehat{p}_2 \geq 10\) and \(n_2 \times (1-\widehat{p}_2) \geq 10\)
Single Mean
\[\begin{equation*} T := \frac{\bar{x}-\mu_0}{s / \sqrt{n}} \sim \textbf{t(df = n-1)} \end{equation*}\]
Conditions:
- Representative sample
- Normal population n large enough for CLT to work
Difference in Means
\[\begin{equation*} T := \frac{\bar{x}_1-\bar{x}_2}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}} \sim \textbf{t(df = min($n_1$, $n_2$) - 1)} \end{equation*}\]
Conditions:
- Representative, independent samples
- Normal populations \(n_1\), \(n_2\) large enough for CLT to work
Strength of Evidence Chart
