1. Definition
- The parameter(s) of interest are the population characteristics (numerical values) that you want to estimate or test in your study.
- They are usually unknown population values (true mean, true proportion, difference in means, regression coefficients, etc.).
- Your experiment or survey is designed specifically to learn about these parameters.
In short: The parameter of interest = the thing you care about measuring or inferring.
2. Examples
(a) Population Mean
- You run a study on student exam scores.
- Parameter of interest: the true population mean exam score ($μ$).
- Statistic used: sample mean ($\bar{x}$) estimates $μ$.
(b) Conversion Rate in A/B Test
- Parameter of interest: true conversion rate of control ($p_A$) and treatment ($p_B$), or their difference $p_B – p_A$.
(c) Medical Trial
- Parameter of interest: difference in recovery probability between drug and placebo groups.
$ATE = P(\text{Recovery} \mid \text{Drug}) – P(\text{Recovery} \mid \text{Placebo})$
(d) Regression Analysis
- Parameters of interest: regression coefficients ($\beta_1, \beta_2, \dots$) that measure the relationship between predictors and the outcome.
3. Why It Matters
- Clarifying the parameter(s) of interest is the first step in designing a study.
- The whole framework of:
- Estimators (sample statistics),
- Hypothesis tests (H₀ vs H₁), and
- Confidence intervals
is built around making inferences about those parameters.
4. General Framework
| Context | Parameter(s) of Interest | Statistic Used to Estimate/Test |
|---|---|---|
| Population mean | $μ$ | Sample mean $\bar{x}$ |
| Population proportion | $p$ | Sample proportion $\hat{p}$ |
| Difference in means | $μ_1 – μ_2$ | Difference in sample means |
| Regression | Coefficients ($\beta$) | Estimated coefficients ($\hat{\beta}$) |
| A/B Test | $p_B – p_A$ | Sample difference $\hat{p}_B – \hat{p}_A$ |
5. Key Takeaways
- Parameter(s) of interest = unknown population values we want to learn about.
- They are the targets of estimation and hypothesis testing.
- Sample statistics ($\bar{x}, \hat{p}, \hat{\beta}$) are the tools we use to estimate them.
In short:
The parameter(s) of interest are the specific population values (mean, proportion, difference, regression coefficient, etc.) that a study is designed to estimate or test. They are unknown constants, and all statistical inference focuses on learning about them from sample data.