Today, we shall look at regression estimation. We will begin by looking at the usual & simple straight line regression model: $y = B_0 + B_1 x$. Let $\hat{B_1}$ and $\hat{B_0}$ by the ordinary least squares (OLS) regression coefficients of the slope and intercept.
$\hat{B_1} = \frac{\sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y})}{\sum_{i=1}^n (x_i - \bar{x})^2}$ = $\frac{s_{xy}}{s_x^2}$ = $\frac{r s_y}{s_x}$
$\hat{B_0} = \bar{y} - \hat{B_1} \bar{x}$

Precision is increase, that is $latex SE(\hat{\bar{y}}_{reg}) < SE(\bar{y})$ [caption id="attachment_2634" align="alignnone" width="300"] Different estimators for population total[/caption]

We conclude here by observing that ratio or regression estimators give greater precision that $\hat{t_y}$ when $\sum_{i=1}^n e_i^2$ for the method is smaller than $\sum_{i=1}^n (y_i - \bar{y})^2$

Sampling & Survey #1 – Introduction
Sampling & Survey #2 – Simple Probability Samples
Sampling & Survey #3 – Simple Random Sampling
Sampling & Survey #4 – Qualities of estimator in SRS
Sampling & Survey #5 – Sampling weight, Confidence Interval and sample size in SRS
Sampling & Survey #6 – Systematic Sampling
Sampling & Survey #7 – Stratified Sampling
Sampling & Survey # 8 – Ratio Estimation
Sampling & Survey # 9 – Regression Estimation
Sampling & Survey #10 – Cluster Sampling
Sampling & Survey #11 – Two – Stage Cluster Sampling
Sampling & Survey #12 – Sampling with unequal probabilities (Part 1)
Sampling & Survey #13 – Sampling with unequal probabilities (Part 2)
Sampling & Survey #14 – Nonresponse