1. Definition
- A population parameter is a fixed (but usually unknown) numerical value that describes a characteristic of an entire population.
- The word true emphasizes that this value exists, even if we don’t know it exactly.
- Examples of population parameters:
- μ (mu) = true population mean
- σ (sigma) = true population standard deviation
- p = true population proportion (e.g., % of voters who support a candidate)
- ρ (rho) = true population correlation
2. Why We Care
- In most cases, we cannot observe the entire population (too large, expensive, or impossible).
- Instead, we collect a sample and use sample statistics (like $\bar{x}, s, \hat{p}$) to estimate the true population parameters.
3. Population Parameter vs Sample Statistic
| Concept | Population | Sample |
|---|---|---|
| Mean | μ (true mean) | $\bar{x}$ (sample mean) |
| Standard deviation | σ | s |
| Proportion | p | $\hat{p}$ |
| Correlation | ρ | r |
The sample statistic is random (depends on which sample you draw).
The population parameter is fixed but unknown.
4. Example
Population vs Sample
- Population: All voters in the U.S.
- True parameter (p) = the real % who support Candidate A.
- Sample: 1,000 surveyed voters.
- Sample statistic ($\hat{p}$) = % supporting Candidate A in the survey.
If $\hat{p} = 0.52$ (52%), it’s an estimate of the true population parameter p.
5. Confidence Intervals and Hypothesis Testing
- We never know the true parameter exactly (unless we measure the whole population).
- Instead, we build confidence intervals around the sample statistic to say:
- “We are 95% confident the true population parameter lies within this interval.”
- In hypothesis testing, we test claims about true parameters (e.g., H₀: μ = 100).
6. Key Takeaways
- A true population parameter is the actual value describing the population (mean, proportion, variance, etc.).
- It is usually unknown.
- We estimate it using sample statistics, and infer its possible value using confidence intervals and hypothesis tests.
In short:
The true population parameter is the fixed but unknown number that describes the whole population (like μ, σ, or p). Since we rarely observe the entire population, we estimate it from sample data using statistics and inference methods.