| Inference for | Relevant Variables | Population Parameters | Sample Statistics | Theoretical Conditions | Degrees of Freedom | Standard Error (Hypothesis Testing) |
Test Statistic | Standard Error (Confidence Interval) |
Margin of Error | Confidence Interval |
|---|---|---|---|---|---|---|---|---|---|---|
| One proportion | One two-level categorical | \(p\) | \(\hat{p}\) | 1. Independent samples 2. Success-failure (\(np \ge 10\) and \(n(1-p) \ge 10\)) |
- | \(SE = \sqrt{\frac{p_0(1-p_0)}{n}}\) | \(Z = \frac{\hat{p} - p_0}{SE}\) | \(SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\) | \(ME = z^* SE\) | \(\hat{p} \pm ME\) |
| Difference of two proportions | Two two-level categorical | \(p_{A} - p_{B}\) | \(\hat{p}_{A} - \hat{p}_{B}\) | 1. Independent samples 2. Success-failure (\(n_A p_A \ge 10\) and \(n_A(1-p_A) \ge 10\) \(n_B p_B \ge 10\) and \(n_B(1-p_B) \ge 10\)) |
- | \(\hat{p}_{pool} = \frac{\hat{p}_A n_A + \hat{p}_B n_B}{n_A + n_B}\) \(SE = \sqrt{\hat{p}_{pool}(1-\hat{p}_{pool})\left(\frac{1}{n_A} + \frac{1}{n_B}\right)}\) |
\(Z = \frac{\hat{p}_A - \hat{p}_B - p_0}{SE}\) | \(SE = \sqrt{\frac{\hat{p}_A (1-\hat{p}_A)}{n_A} + \frac{\hat{p}_B (1-\hat{p}_B)}{n_B}}\) | \(ME = z^* SE\) | \(\hat{p}_A - \hat{p}_B \pm ME\) |
| Two-way table | Two categorical with two or more levels | - | - | 1. Independent samples 2. Normality (each cell has \(\ge 10\) samples) |
\(k = (C-1)(R-1)\) (where \(C\) is the number of columns and \(R\) is the number of rows) |
- | \(\chi_k^2 = \sum_{i=1}^n \frac{(O_i - E_i)^2}{E_i}\) (where \(n\) is the number of cells) |
- | - | - |
| One mean | Response: One numerical | \(\mu \longrightarrow \text{mean}\) \(\sigma \longrightarrow \text{standard deviation}\) |
\(\bar{x} \longrightarrow \text{mean}\) \(s \longrightarrow \text{standard deviation}\) |
1. Independent samples 2. Normality (\(n \ge 30\)) |
\(df = n-1\) | \(SE = \frac{s}{\sqrt{n}}\) (use \(\sigma\) if known) |
\(\frac{\bar{x} - \mu_0}{SE}\) | \(SE = \frac{s}{\sqrt{n}}\) (use \(\sigma\) if known) |
\(ME = t^*_{df} SE\) (use \(z^*\) if \(\sigma\) is known) |
\(\bar{x} \pm ME\) |
| Comparing two means | Response: One numerical Explanatory: One two-level categorical |
\(\mu_{A} - \mu_{B}\) | \(\bar{x}_{A} - \bar{x}_{B}\) | 1. Independent samples 2. Normality (\(n_A \ge 30\), \(n_B \ge 30\)) |
\(df = \min{(n_A - 1, n_B - 1)}\) | \(SE = \sqrt{\frac{s^2_A}{n_A} + \frac{s^2_B}{n_B}}\) (use \(\sigma_A\) and \(\sigma_B\) if known) |
\(T = \frac{\bar{x}_A - \bar{x}_B - \mu_0}{SE}\) | \(SE = \sqrt{\frac{s^2_A}{n_A} + \frac{s^2_B}{n_B}}\) (use \(\sigma_A\) and \(\sigma_B\) if known) |
\(ME = t^*_{df} SE\) (use \(z^*\) SE if \(\sigma_A\) and \(\sigma_B\) is known) |
\(\bar{x}_{A} - \bar{x}_{B} \pm ME\) |
| Comparing paired means | Response: One numerical (difference of two paired numerical) |
\(\mu_{diff} \longrightarrow \text{mean}\) \(\sigma_{diff} \longrightarrow \text{standard deviation}\) |
\(\bar{x}_{diff} \longrightarrow \text{mean}\) \(\sigma_{diff} \longrightarrow \text{standard deviation}\) |
1. Independent samples 2. Normality (\(n \ge 30\)) |
\(df = n_{diff}-1\) | \(SE = \frac{s_{diff}}{\sqrt{n_{diff}}}\) (use \(\sigma_{diff}\) if known) |
\(T = \frac{\bar{x}_{diff} - \mu_0}{SE}\) | \(SE = \frac{s_{diff}}{\sqrt{n_{diff}}}\) (use \(\sigma_{diff}\) if known) |
\(ME = t^*+{df} SE\) (use \(z^* SE\) if \(\sigma_{diff}\) is known) |
\(\bar{x}_{diff} \pm ME\) |
| Simple linear regression | Response: One numerical Explanatory: One numerical or one two-level categorical |
\(y = \beta_0 + \beta_1 x\) \(\beta_0 \longrightarrow \text{intercept}\) \(\beta_1 \longrightarrow \text{slope}\) \(\sigma_1 \longrightarrow \text{slope standard deviations}\) |
\(\hat{y} = b_0 + b_1 x\) \(b_0 \longrightarrow \text{intercept}\) \(b_1 \longrightarrow \text{slope}\) \(s_1 \longrightarrow \text{slope standard deviation}\) |
1. Linearity 2. Independent samples 3. Normal residuals 4. Homoscedasticity |
\(df = n-k-1\) | lm output \(SE =\) std.error (sample SE) |
\(T = \frac{b_1 - \text{null-value}}{SE}\) | lm output \(SE =\) std.error (sample SE) |
\(ME = t^*_{df} SE\) (use \(z^* SE\) if \(\sigma_1\) is known) |
\(b_1 \pm ME\) |
This cheat sheet - a comprehensive summary of R commands and procedures we did during lab - will help you on your R coding tasks throughout the course. The R coding cheat sheet below was made and provided by your lab assistant, Robin.
Equal sign: $=$ will render as \(=\).
Not equal sign: $\ne$ will render as \(\ne\).
Lesser than sign: $<$ will render as \(<\).
Greater than sign: $<$ will render as \(>\).
Lesser than or equal to sign: $\le$ will render as \(\le\).
Greater than or equal to sign: $\ge$ will render as \(\ge\).
Using hats: $\hat{p}$ will render as \(\hat{p}\).
Using bars: $\bar{x}$ will render as \(\bar{x}\).
Greek lowercase letter mu: $\mu$ will render as \(\mu\).
Greek lowercase letter beta: $\beta$ will render as \(\beta\).
The letter p: $p$ will render as \(p\).
Subscripting: $p_{A}$ will render as \(p_{A}\).
Set union: $\cup$ will render as \(\cup\).
Set intersection: $\cap$ will render as \(\cap\).
Summation Notation: $\sum_{i=0}^k$ will render as \(\sum_{i=0}^k\).
Infinity: $\infty$ will render as \(\infty\).