1. **Problem Statement:** We want to analyze the relationship between seniority (X, years with the company) and hourly wages (Y, dollars per hour) for 12 secretaries. 2. **Measure of Association:** We will calculate the Pearson correlation coefficient $r$ to measure the strength and direction of the linear relationship between X and Y. 3. **Regression Equation:** We find the regression line $Y = a + bX$ where $a$ is the intercept and $b$ is the slope. The slope $b$ represents the change in hourly wage for each additional year of seniority. 4. **Prediction:** Using the regression equation, we predict the hourly wage for someone with 8 years of seniority. 5. **Coefficient of Determination:** $R^2$ tells us how much of the variation in Y is explained by X. 6. **Unexplained Variation:** $1 - R^2$ represents the variation in Y explained by other factors. --- **Step 1: Calculate means** $$\bar{X} = \frac{0+2+3+6+5+3+4+1+1+2+6+4}{12} = \frac{37}{12} \approx 3.083$$ $$\bar{Y} = \frac{12+13+14+16+15+14+13+12+15+15+18+14}{12} = \frac{171}{12} = 14.25$$ **Step 2: Calculate slope $b$ and intercept $a$** Formula for slope: $$b = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sum (X_i - \bar{X})^2}$$ Calculate numerator and denominator: | $X_i$ | $Y_i$ | $X_i - \bar{X}$ | $Y_i - \bar{Y}$ | $(X_i - \bar{X})(Y_i - \bar{Y})$ | $(X_i - \bar{X})^2$ | |-------|-------|-----------------|-----------------|-------------------------------|-------------------| | 0 | 12 | -3.083 | -2.25 | 6.937 | 9.506 | | 2 | 13 | -1.083 | -1.25 | 1.354 | 1.173 | | 3 | 14 | -0.083 | -0.25 | 0.021 | 0.007 | | 6 | 16 | 2.917 | 1.75 | 5.104 | 8.511 | | 5 | 15 | 1.917 | 0.75 | 1.438 | 3.676 | | 3 | 14 | -0.083 | -0.25 | 0.021 | 0.007 | | 4 | 13 | 0.917 | -1.25 | -1.146 | 0.841 | | 1 | 12 | -2.083 | -2.25 | 4.687 | 4.340 | | 1 | 15 | -2.083 | 0.75 | -1.562 | 4.340 | | 2 | 15 | -1.083 | 0.75 | -0.812 | 1.173 | | 6 | 18 | 2.917 | 3.75 | 10.938 | 8.511 | | 4 | 14 | 0.917 | -0.25 | -0.229 | 0.841 | Sum numerator: $\sum (X_i - \bar{X})(Y_i - \bar{Y}) = 27.751$ Sum denominator: $\sum (X_i - \bar{X})^2 = 42.920$ Calculate slope: $$b = \frac{27.751}{42.920} \approx 0.646$$ Calculate intercept: $$a = \bar{Y} - b\bar{X} = 14.25 - 0.646 \times 3.083 = 14.25 - 1.993 = 12.257$$ **Regression equation:** $$Y = 12.257 + 0.646X$$ **Interpretation:** - The intercept 12.257 means the predicted hourly wage for someone with 0 years seniority is about 12.26. - The slope 0.646 means each additional year of seniority increases the hourly wage by about 0.65. **Step 3: Predict wage for 8 years seniority** $$Y = 12.257 + 0.646 \times 8 = 12.257 + 5.168 = 17.425$$ Predicted hourly wage is approximately 17.43. **Step 4: Calculate correlation coefficient $r$** Formula: $$r = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sqrt{\sum (X_i - \bar{X})^2 \sum (Y_i - \bar{Y})^2}}$$ Calculate $\sum (Y_i - \bar{Y})^2$: $$= (-2.25)^2 + (-1.25)^2 + (-0.25)^2 + 1.75^2 + 0.75^2 + (-0.25)^2 + (-1.25)^2 + (-2.25)^2 + 0.75^2 + 0.75^2 + 3.75^2 + (-0.25)^2 = 32.75$$ Calculate $r$: $$r = \frac{27.751}{\sqrt{42.920 \times 32.75}} = \frac{27.751}{\sqrt{1405.37}} = \frac{27.751}{37.5} = 0.74$$ This shows a strong positive linear relationship. **Step 5: Calculate $R^2$ (coefficient of determination):** $$R^2 = r^2 = 0.74^2 = 0.5476$$ This means about 54.76% of the variation in hourly wages is explained by seniority. **Step 6: Variation explained by other factors:** $$1 - R^2 = 1 - 0.5476 = 0.4524$$ About 45.24% of the variation in wages is due to other factors besides seniority. --- **Summary:** - The relationship between seniority and hourly wage is positive and moderately strong. - The regression equation is $Y = 12.257 + 0.646X$. - Each year of seniority increases wage by about 0.65. - Predicted wage for 8 years seniority is about 17.43. - Seniority explains about 54.76% of wage variation. - Other factors explain about 45.24% of wage variation.