Confidence Interval for difference between groups

Constructing Confidence Intervals: General Guidelines

When asked to construct a confidence interval, you should generally follow these steps:

1. Identify what you're estimating

Parameter Type	Symbol	Description	Example
Single Mean	$μ$	Center of one population	Average student height
Single Proportion	$p$	Success rate in one population	Percentage of voters
Difference in Means	$μ_{1} - μ_{2}$	Difference between two population centers	Difference in test scores
Difference in Proportions	$p_{1} - p_{2}$	Difference between two population rates	Difference in success rates

2. Check Validity Conditions

Type	Conditions	Requirements
Single Mean	Random Sample	Independent observations
	Sample Size/Distribution	$n \geq 30$ OR population is normal
Single Proportion	Random Sample	Independent observations
	Success-Failure	$n \hat{p} \geq 10$ AND $n (1 - \hat{p}) \geq 10$
Difference in Means	Independence	Between and within groups
	Sample Size/Distribution	Both $n \geq 30$ OR both normal
Difference in Proportions	Independence	Between and within groups
	Success-Failure	Both groups meet success-failure condition

3. Formula Selection

Scenario	Formula	Standard Error	Notes
Single Mean	$\bar{x} \pm t^{*} \times \frac{s}{\sqrt{n}}$	$\frac{s}{\sqrt{n}}$	df = $n - 1$
Single Proportion	$\hat{p} \pm z^{*} \times \sqrt{\frac{\hat{p} (1 - \hat{p})}{n}}$	$\sqrt{\frac{\hat{p} (1 - \hat{p})}{n}}$	Use $z^{*}$
Difference in Means	$({\bar{x}}_{1} - {\bar{x}}_{2}) \pm t^{*} \times \sqrt{\frac{s_{1}^{2}}{n_{1}} + \frac{s_{2}^{2}}{n_{2}}}$	$\sqrt{\frac{s_{1}^{2}}{n_{1}} + \frac{s_{2}^{2}}{n_{2}}}$	df = $min (n_{1} - 1, n_{2} - 1)$
Difference in Proportions	$({\hat{p}}_{1} - {\hat{p}}_{2}) \pm z^{*} \times \sqrt{\frac{{\hat{p}}_{1} (1 - {\hat{p}}_{1})}{n_{1}} + \frac{{\hat{p}}_{2} (1 - {\hat{p}}_{2})}{n_{2}}}$	$\sqrt{\frac{{\hat{p}}_{1} (1 - {\hat{p}}_{1})}{n_{1}} + \frac{{\hat{p}}_{2} (1 - {\hat{p}}_{2})}{n_{2}}}$	Use $z^{*}$

4. Critical Value Selection

Scenario	Distribution	Critical Value	Degrees of Freedom
Means (unknown $σ$ )	t-distribution	$t^{*}$	$n - 1$
Proportions	Normal	$z^{*}$	N/A
Difference in Means	t-distribution	$t^{*}$	$min (n_{1} - 1, n_{2} - 1)$
Difference in Proportions	Normal	$z^{*}$	N/A

Common Critical Values:

$z^{*}$ (90%): 1.645
$z^{*}$ (95%): 1.96
$z^{*}$ (99%): 2.576
$t^{*}$ values vary by df

5. Margin of Error Calculation

ME = critical value $\times$ standard error

6. Interval Construction

CI = point estimate $\pm$ margin of error

7. Interpretation Guidelines

Scenario	Interpretation Template	Additional Notes
Single Parameter	"We are [C]% confident that the true [parameter] lies between [lower] and [upper]."	Specify parameter clearly
Difference	"We are [C]% confident that the difference ( ${group}_{1} - {group}_{2}$ ) lies between [lower] and [upper]."	Specify which group was subtracted
Contains Zero	"Since the interval contains 0, there is no significant difference between groups at the [C]% confidence level."	For difference intervals only

Key Points:

Always specify confidence level (C%)
Clearly identify groups being compared
State which group was subtracted from which
Note practical significance of results