Coursekata
Theoretical Inference Summary
How to use this page
This page summarizes the assumptions/conditions for theoretical approach to confidence intervals and hypothesis tests in common settings.
Two big ideas appear throughout:
- A confidence interval are based on \(\text{point estimate} \pm ME\) where \(ME=m \times SE\).
- \(ME\) can be computed directly from the sampling distribution of the \(\text{point estimate}\).
- A hypothesis test uses a null distribution
The sampling distribution of the \(\text{point estimate}\) and the null distribution of the test statistic are determined based on some assumptions or conditions.
For a confidence interval:
- Compute the observed sample statistic.
- Compute the \(SE\) from mathematic formulas.
- Find the multiplier \(m\) from an appropriate distribution and desired confidence level.
- Calculate \(ME = m\times SE\)1.
- Compute C.I as \((\text{point estimate}-ME, \text{point estimate}+ME)\).
For a hypothesis test:
- State the null and alternative hypotheses.
- Compute the observed test statistic2.
- Using the null distribution:
- Determine Rejection Region; the extreme test statistic values under a significance level \(\alpha\).
- Compute \(p\)-value; the probability of values at least as extreme as the observed test statistic.
- Make a decision: Reject the null if
- observed test statistic is in Rejection Region.
- \(p\)-value is less than \(\alpha\).
- Write a conclusion in context.
Theoretical Inference
Parameter and Statistic
- Parameter: \(p\)
- Statistic: \(\hat{p}=x/n\)
- Sample size: \(n\)
- Number of success: \(x\)
Hypotheses
- \(H_0: p = p_0\)
- \(H_A: p \ne p_0\) (or one-sided: \(H_A: p < p_0\) or \(H_A: p > p_0\))
Test Statistic
\[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} \]
Distribution
- Standard Normal (\(z\))
Confidence Interval
\[ \hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \]
Conditions
- \(np_0 \ge 10\), \(n(1-p_0) \ge 10\)
- Expected number of successes and failures are 10 or more.
Parameter and Statistic
- Parameters: \(p_1, p_2\)
- Statistic: \(\hat{p}_1 - \hat{p}_2\)
- Sample size: \(n_1, n_2\)
- Number of success: \(x_1, x_2\)
Hypotheses
- \(H_0: p_1 = p_2\)
- \(H_A: p_1 \ne p_2\)
Test Statistic
\[ z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}} \]
- \(\hat{p}\) = pooled proportion \((x_1+x_2)/(n_1+n_2)\)
Distribution
- \(z\)
Confidence Interval
\[ (\hat{p}_1 - \hat{p}_2) \pm z^* \cdot SE \]
Conditions
- Expected counts ≥ 5
Parameter and Statistic
- Parameters: \(p_{1|g}, p_{2|j}, ..., p_{r|j}\), for groups \(j=1, 2, ..., c\)
- Statistics:
- Observed counts (\(O\)): \(n_{ij}, i=1,2,...,r; j=1,2,...,c\)
- Expected counts (\(E\)): \(e_{ij}=\frac{n_{i+}n_{+j}}{n_{++}}\)
Test Statistic
\[\chi^2=\sum \frac{(O - E)^2}{E}\]
Distribution
- \(\chi^2\) with \(df = (r-1)(c-1)\)
Conditions
- Expected counts (\(e_{ij}\)) ≥ 5
Parameter and Statistic
- Parameter: \(\mu\)
- Statistic: \(\bar{x}\)
- Sample size: \(n\)
- Sample mean: \(\bar x\)
- Sample std. dev.: \(s\)
Hypotheses
- \(H_0: \mu = \mu_0\)
- \(H_A: \mu \ne \mu_0\)
Test Statistic
\[ t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} \]
Distribution
- \(t\) with \(df = n - 1\)
Confidence Interval
\[ \bar{x} \pm t^* \frac{s}{\sqrt{n}} \]
Conditions
- Approximately normal data OR large sample
Parameter and Statistic
- Parameters: \(\mu_1, \mu_2\)
- Statistic: \(\bar{x}_1 - \bar{x}_2\)
- Sample size: \(n_1, n_2\)
- Sample mean: \(\bar x_1, \bar x_2\)
- Sample std. dev.: \(s_1, s_2\)
Hypotheses
- \(H_0: \mu_1 = \mu_2\)
- \(H_A: \mu_1 \ne \mu_2\)
Test Statistic
\[ t = \frac{\bar{x}_1 - \bar{x}_2}{SE} \]
Assuming Unequal Variance
\(SE = \sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}\)
Assuming Equal Variance
\(s_p^2 = \frac{(n_1-1)s_1^2 + (n_2)s_2^2}{n_1+n_2-2}\) \(SE = s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}\)
Distribution
- \(t~T(df)\)
- Equal variance: \(df=n_1+n_2-2\)
- Unequal variance: approximate \(df\) using Welch method (software) or \(min(n_1-1, n_2-1)\)
Confidence Interval
\[ (\bar{x}_1 - \bar{x}_2) \pm t^* \cdot SE \]
Parameter and Statistic
- Parameters: \(\mu_1, \mu_2, \dots\)
- Statistic: \(F\)
Hypotheses
- \(H_0\): all means equal
- \(H_A\): at least one differs
Test Statistic
\[ F = \frac{\text{MSM}}{\text{MSE}} \] - MSM = Between-group variability - MSE = Within-group variability
Distribution
- \(F\) with \((df_1, df_2)\)
Interpretation
- Large \(F\) → groups differ (at least one mean is different)
Parameter and Statistic
- Parameter: \(\beta_1\)
- Statistic: \(b_1\)
Hypotheses
- \(H_0: \beta_1 = 0\)
- \(H_A: \beta_1 \ne 0\)
Test Statistic
\[ t = \frac{b_1}{SE(b_1)} \]
Distribution
- \(t\) with \(df = n - 2\)
Confidence Interval
\[ b_1 \pm t^* SE(b_1) \]
Parameter and Statistic
- Parameters: \(\beta_j\), \(j=0, 1, ..., p\) (\(p\) is the number of predictors)
- Statistics: \(b_j\)
Hypotheses (each predictor)
- \(H_0: \beta_j = 0\)
- \(H_A: \beta_j \ne 0\)
Test Statistic
\[ t = \frac{b_j}{SE(b_j)} \]
Distribution
- \(t\) with \(df = n - p - 1\)
Confidence Interval
\[ b_j \pm t^* SE(b_j) \]
Key Interpretation
👉 Effect of \(X_j\) holding all other predictors constant
Big Picture Comparison
| Scenario | Statistic | Distribution | Key Idea |
|---|---|---|---|
| Proportion | \(\hat{p}\) | Normal | Count successes |
| Means | \(\bar{x}\) | t | Estimate average |
| Groups | \(F\) | F | Compare variability |
| Regression | \(b\) | t | Slope = relationship |