1. **State the problem:** We want to determine if the screening scores (Screen) are significantly correlated with the achievement scores (Achieve) using simple linear regression. 2. **Formula and explanation:** The simple linear regression model is given by: $$y = \beta_0 + \beta_1 x + \epsilon$$ where $y$ is the dependent variable (Achieve), $x$ is the independent variable (Screen), $\beta_0$ is the intercept, $\beta_1$ is the slope, and $\epsilon$ is the error term. 3. **Calculate means:** $$\bar{x} = \frac{1+3+3+3+5+6.5+6.5+8+9+10}{10} = \frac{55}{10} = 5.5$$ $$\bar{y} = \frac{46+52+41+92+59+48+66+82+41+60}{10} = \frac{587}{10} = 58.7$$ 4. **Calculate slope $\beta_1$:** $$\beta_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}$$ Calculate numerator: $$\sum (x_i - 5.5)(y_i - 58.7) = (1-5.5)(46-58.7) + (3-5.5)(52-58.7) + (3-5.5)(41-58.7) + (3-5.5)(92-58.7) + (5-5.5)(59-58.7) + (6.5-5.5)(48-58.7) + (6.5-5.5)(66-58.7) + (8-5.5)(82-58.7) + (9-5.5)(41-58.7) + (10-5.5)(60-58.7)$$ Calculate each term: $$( -4.5)(-12.7) = 57.15$$ $$( -2.5)(-6.7) = 16.75$$ $$( -2.5)(-17.7) = 44.25$$ $$( -2.5)(33.3) = -83.25$$ $$( -0.5)(0.3) = -0.15$$ $$(1)(-10.7) = -10.7$$ $$(1)(7.3) = 7.3$$ $$(2.5)(23.3) = 58.25$$ $$(3.5)(-17.7) = -61.95$$ $$(4.5)(1.3) = 5.85$$ Sum numerator: $$57.15 + 16.75 + 44.25 - 83.25 - 0.15 - 10.7 + 7.3 + 58.25 - 61.95 + 5.85 = 33.5$$ Calculate denominator: $$\sum (x_i - 5.5)^2 = (1-5.5)^2 + (3-5.5)^2 + (3-5.5)^2 + (3-5.5)^2 + (5-5.5)^2 + (6.5-5.5)^2 + (6.5-5.5)^2 + (8-5.5)^2 + (9-5.5)^2 + (10-5.5)^2$$ $$= 20.25 + 6.25 + 6.25 + 6.25 + 0.25 + 1 + 1 + 6.25 + 12.25 + 20.25 = 80$$ 5. **Calculate slope:** $$\beta_1 = \frac{33.5}{80} = 0.41875$$ 6. **Calculate intercept $\beta_0$:** $$\beta_0 = \bar{y} - \beta_1 \bar{x} = 58.7 - 0.41875 \times 5.5 = 58.7 - 2.303125 = 56.396875$$ 7. **Regression equation:** $$\hat{y} = 56.396875 + 0.41875 x$$ 8. **Interpretation:** The positive slope indicates a positive correlation between Screen and Achieve scores. 9. **Significance test:** Calculate the correlation coefficient $r$ and test if it is significantly different from zero. Calculate $r$: $$r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}$$ Calculate $\sum (y_i - \bar{y})^2$: $$(46-58.7)^2 + (52-58.7)^2 + (41-58.7)^2 + (92-58.7)^2 + (59-58.7)^2 + (48-58.7)^2 + (66-58.7)^2 + (82-58.7)^2 + (41-58.7)^2 + (60-58.7)^2$$ $$= 161.29 + 44.89 + 313.29 + 1108.89 + 0.09 + 114.49 + 53.29 + 544.89 + 313.29 + 1.69 = 2709.1$$ Calculate $r$: $$r = \frac{33.5}{\sqrt{80 \times 2709.1}} = \frac{33.5}{\sqrt{216728}} = \frac{33.5}{465.56} = 0.072$$ 10. **Conclusion:** The correlation coefficient $r=0.072$ is very low, indicating a weak linear relationship. Statistical significance would require further hypothesis testing (e.g., t-test), but this low $r$ suggests Screen is not significantly correlated with Achieve. **Final answer:** The regression equation is: $$\hat{y} = 56.40 + 0.42 x$$ The correlation between Screen and Achieve is weak and likely not significant.