1. **Stating the problem:** We want to analyze the influence of motivation at work ($X_1$) and work spirit ($X_2$) on employee performance ($Y$) using multiple linear regression. 2. **Formula used:** The general multiple linear regression equation is: $$Y = a + b_1X_1 + b_2X_2$$ where $a$ is the intercept, $b_1$ and $b_2$ are regression coefficients. 3. **Step a: Create a helper table** Calculate sums and products needed: - $\sum X_1$, $\sum X_2$, $\sum Y$ - $\sum X_1^2$, $\sum X_2^2$ - $\sum X_1X_2$, $\sum X_1Y$, $\sum X_2Y$ | NO | $X_1$ | $X_2$ | $Y$ | $X_1^2$ | $X_2^2$ | $X_1X_2$ | $X_1Y$ | $X_2Y$ | |----|-------|-------|-----|---------|---------|---------|--------|--------| | 1 | 25 | 12 | 21 | 625 | 144 | 300 | 525 | 252 | | 2 | 35 | 21 | 15 | 1225 | 441 | 735 | 525 | 315 | | 3 | 33 | 13 | 12 | 1089 | 169 | 429 | 396 | 156 | | 4 | 34 | 23 | 22 | 1156 | 529 | 782 | 748 | 506 | | 5 | 34 | 19 | 22 | 1156 | 361 | 646 | 748 | 418 | | 6 | 34 | 25 | 23 | 1156 | 625 | 850 | 782 | 575 | | 7 | 22 | 12 | 20 | 484 | 144 | 264 | 440 | 240 | | 8 | 24 | 17 | 14 | 576 | 289 | 408 | 336 | 238 | | 9 | 24 | 25 | 15 | 576 | 625 | 600 | 360 | 375 | | 10 | 37 | 13 | 13 | 1369 | 169 | 481 | 481 | 169 | Calculate sums: $$\sum X_1 = 302, \quad \sum X_2 = 180, \quad \sum Y = 177$$ $$\sum X_1^2 = 8412, \quad \sum X_2^2 = 3496$$ $$\sum X_1X_2 = 5555, \quad \sum X_1Y = 5335, \quad \sum X_2Y = 3244$$ 4. **Step b: Summary of helper table** We have all sums needed for regression calculation. 5. **Step c: Calculate $a$, $b_1$, and $b_2$** Use formulas for multiple regression coefficients: $$b_1 = \frac{n\sum X_1Y - \sum X_1 \sum Y}{n\sum X_1^2 - (\sum X_1)^2}$$ $$b_2 = \frac{n\sum X_2Y - \sum X_2 \sum Y}{n\sum X_2^2 - (\sum X_2)^2}$$ $$a = \bar{Y} - b_1 \bar{X_1} - b_2 \bar{X_2}$$ where $n=10$, $\bar{X_1} = \frac{302}{10} = 30.2$, $\bar{X_2} = \frac{180}{10} = 18$, $\bar{Y} = \frac{177}{10} = 17.7$. Calculate $b_1$ numerator and denominator: $$10 \times 5335 - 302 \times 177 = 53350 - 53454 = -104$$ $$10 \times 8412 - 302^2 = 84120 - 91204 = -7084$$ Calculate $b_1$: $$b_1 = \frac{-104}{-7084} = 0.01468$$ Calculate $b_2$ numerator and denominator: $$10 \times 3244 - 180 \times 177 = 32440 - 31860 = 580$$ $$10 \times 3496 - 180^2 = 34960 - 32400 = 2560$$ Calculate $b_2$: $$b_2 = \frac{580}{2560} = 0.22656$$ Calculate $a$: $$a = 17.7 - 0.01468 \times 30.2 - 0.22656 \times 18$$ $$a = 17.7 - 0.4437 - 4.078 = 13.1783$$ 6. **Step d: Write the regression equation** $$Y = 13.1783 + 0.01468 X_1 + 0.22656 X_2$$ 7. **Step e: Interpretation** - The intercept $a=13.1783$ is the predicted performance when motivation and work spirit are zero. - $b_1=0.01468$ means each unit increase in motivation ($X_1$) increases performance ($Y$) by about 0.015 units, holding $X_2$ constant. - $b_2=0.22656$ means each unit increase in work spirit ($X_2$) increases performance ($Y$) by about 0.227 units, holding $X_1$ constant. - Work spirit ($X_2$) has a stronger effect on performance than motivation ($X_1$) in this model.