Confidence Interval

Constructing Confidence Intervals - Reference Guide

Type	Formula	Critical Value	Conditions
Mean	$\bar{x} \pm t^{*} \frac{s}{\sqrt{n}}$	$t^{*}$ (df = $n - 1$ )	$n \geq 30$ or normal
Proportion	$\hat{p} \pm z^{*} \sqrt{\frac{\hat{p} (1 - \hat{p})}{n}}$	$z^{*}$	$n \hat{p} \geq 10$ and $n (1 - \hat{p}) \geq 10$
Difference in Means	$({\bar{x}}_{1} - {\bar{x}}_{2}) \pm t^{*} \sqrt{\frac{s_{1}^{2}}{n_{1}} + \frac{s_{2}^{2}}{n_{2}}}$	$t^{*}$ (df = min $(n_{1} - 1, n_{2} - 1)$ )	Independent groups
Difference in Proportions	$({\hat{p}}_{1} - {\hat{p}}_{2}) \pm z^{*} \sqrt{\frac{{\hat{p}}_{1} (1 - {\hat{p}}_{1})}{n_{1}} + \frac{{\hat{p}}_{2} (1 - {\hat{p}}_{2})}{n_{2}}}$	$z^{*}$	Independent groups, both satisfy conditions

Follow these key steps when constructing confidence intervals:

Parameter	Symbol	Use Case	Example
Population Mean	$μ$	Quantitative variables	Height, weight, scores
Population Proportion	$p$	Categorical variables	Success/failure, yes/no
Difference in Parameters	$μ_{1} - μ_{2}$ or $p_{1} - p_{2}$	Group comparisons	Treatment vs. control

Condition	Requirement	Notes
Sample Type	Random sample	Essential for inference
Success	$n \times \hat{p} \geq 10$	Must check BEFORE proceeding
Failure	$n \times (1 - \hat{p}) \geq 10$	Must check BEFORE proceeding

Condition	Requirement	Notes
Independence	Groups must be independent	No overlap between groups
Individual Means	Each group meets mean conditions	Size and normality
Individual Props	Each group meets proportion conditions	Success and failure conditions
Degrees of Freedom	Use df = min $(n_{1} - 1, n_{2} - 1)$ for difference in means	Conservative approach

Find formula of standard error here: standard error

For any parameter $θ$ :

Point Estimate \pm (Critical Value \times Standard Error)

Parameter Type	Point Estimate	Critical Value	Standard Error
Mean	$\bar{x}$	$t^{*}$	$\frac{s}{\sqrt{n}}$
Proportion	$\hat{p}$	$z^{*}$	$\sqrt{\frac{\hat{p} (1 - \hat{p})}{n}}$
Difference in Means	${\bar{x}}_{1} - {\bar{x}}_{2}$	$t^{*}$	$\sqrt{\frac{s_{1}^{2}}{n_{1}} + \frac{s_{2}^{2}}{n_{2}}}$
Difference in Proportions	${\hat{p}}_{1} - {\hat{p}}_{2}$	$z^{*}$	$\sqrt{\frac{{\hat{p}}_{1} (1 - {\hat{p}}_{1})}{n_{1}} + \frac{{\hat{p}}_{2} (1 - {\hat{p}}_{2})}{n_{2}}}$

Component	Formula	Notes
Margin of Error	Critical value × Standard Error	Use appropriate standard error
Standard Error	From formula Standard error	Depends on parameter type

Component	Formula	Example
Lower Bound	Point estimate - Margin of Error	$\bar{x} - t^{*} \times \frac{s}{\sqrt{n}}$
Upper Bound	Point estimate + Margin of Error	$\bar{x} + t^{*} \times \frac{s}{\sqrt{n}}$

Template:

"We are [confidence level]% confident that the true [parameter] lies between [lower bound] and [upper bound]."

Component	Example	Notes
Confidence Level	95%	Most common choice
Parameter	population mean	Be specific about what you measured
Bounds	Numerical values from calculation	Round appropriately