Understanding Linear Regression
Linear regression is a statistical method used to model the relationship between a dependent variable ($Y$) and an independent variable ($X$). When we want to estimate Age ($X$) based on Weight ($Y$), we are looking for the line of regression of X on Y.
The Equation
The line of regression of $X$ on $Y$ is given by: $X - \bar{X} = b_{xy}(Y - \bar{Y})$
Where:
- $b_{xy}$ is the regression coefficient of $X$ on $Y$: $b_{xy} = \frac{n\sum XY - (\sum X)(\sum Y)}{n\sum Y^2 - (\sum Y)^2}$
- $\bar{X}$ and $\bar{Y}$ are the means of $X$ and $Y$ respectively.
Step-by-Step Computation
1. Tabulate the Data
| $X$ | $Y$ | $X^2$ | $Y^2$ | $XY$ |
|---|---|---|---|---|
| 5 | 10 | 25 | 100 | 50 |
| 15 | 35 | 225 | 1225 | 525 |
| 30 | 50 | 900 | 2500 | 1500 |
| 45 | 65 | 2025 | 4225 | 2925 |
| 50 | 55 | 2500 | 3025 | 2750 |
| 60 | 45 | 3600 | 2025 | 2700 |
| **Sum: 205** | **260** | **9275** | **13100** | **10450** |
2. Calculate Means
- $n = 6$
- $\bar{X} = \frac{205}{6} \approx 34.17$
- $\bar{Y} = \frac{260}{6} \approx 43.33$
3. Calculate $b_{xy}$
$b_{xy} = \frac{6(10450) - (205)(260)}{6(13100) - (260)^2} = \frac{62700 - 53300}{78600 - 67600} = \frac{9400}{11000} \approx 0.8545$
4. Find the Regression Equation
$X - 34.17 = 0.8545(Y - 43.33)$ $X = 0.8545Y - 37.03 + 34.17$ $X = 0.8545Y - 2.86$
5. Estimate Age for Weight $Y = 37$
$X = 0.8545(37) - 2.86 = 31.62 - 2.86 = 28.76$
The estimated age for a weight of 37 kg is approximately 28.76 years.