1. **Problem Statement:** We want to estimate the relationship between savings ($S$) and investment ($I$) using the given data. 2. **Specify the model:** The simple linear regression model is: $$I = \beta_0 + \beta_1 S + \epsilon$$ where $\beta_0$ is the intercept, $\beta_1$ is the slope coefficient, and $\epsilon$ is the error term. 3. **Estimate the model using OLS:** Calculate the slope $\hat{\beta}_1$ and intercept $\hat{\beta}_0$ using formulas: $$\hat{\beta}_1 = \frac{\sum (S_i - \bar{S})(I_i - \bar{I})}{\sum (S_i - \bar{S})^2}$$ $$\hat{\beta}_0 = \bar{I} - \hat{\beta}_1 \bar{S}$$ Calculate means: $$\bar{S} = \frac{100+200+300+400+500+600+700+700+800+1000}{10} = 530$$ $$\bar{I} = \frac{40+45+50+65+70+70+80+85+85+95}{10} = 68.5$$ Calculate sums: $$\sum (S_i - \bar{S})(I_i - \bar{I}) = 54450$$ $$\sum (S_i - \bar{S})^2 = 409000$$ Thus, $$\hat{\beta}_1 = \frac{54450}{409000} \approx 0.1331$$ $$\hat{\beta}_0 = 68.5 - 0.1331 \times 530 = 68.5 - 70.543 = -2.043$$ Estimated regression equation: $$\hat{I} = -2.043 + 0.1331 S$$ 4. **Interpretation:** The slope $0.1331$ means for each unit increase in savings, investment increases by approximately 0.1331 units. The intercept $-2.043$ is the estimated investment when savings are zero (may not be meaningful if $S=0$ is outside data range). 5. **Coefficient of determination ($R^2$) and correlation coefficient ($r$):** Calculate total sum of squares (SST), regression sum of squares (SSR), and residual sum of squares (SSE): $$SST = \sum (I_i - \bar{I})^2 = 2637.5$$ $$SSR = \sum (\hat{I}_i - \bar{I})^2 = 2420.5$$ $$SSE = SST - SSR = 217$$ Coefficient of determination: $$R^2 = \frac{SSR}{SST} = \frac{2420.5}{2637.5} \approx 0.918$$ Correlation coefficient: $$r = \sqrt{R^2} = \sqrt{0.918} \approx 0.958$$ Interpretation: About 91.8% of the variation in investment is explained by savings. The strong positive correlation (0.958) indicates a strong linear relationship. 6. **Standard error of slope coefficient:** Calculate variance of residuals: $$s^2 = \frac{SSE}{n-2} = \frac{217}{8} = 27.125$$ Standard error: $$SE_{\hat{\beta}_1} = \sqrt{\frac{s^2}{\sum (S_i - \bar{S})^2}} = \sqrt{\frac{27.125}{409000}} \approx 0.00814$$ 7. **Hypothesis test on slope at 5% significance:** Null hypothesis: $H_0: \beta_1 = 0$ Alternative: $H_a: \beta_1 \neq 0$ Test statistic: $$t = \frac{\hat{\beta}_1 - 0}{SE_{\hat{\beta}_1}} = \frac{0.1331}{0.00814} \approx 16.35$$ Degrees of freedom: $n-2=8$ Critical $t$ value (two-tailed, 5%): approximately 2.306 Since $16.35 > 2.306$, reject $H_0$. There is strong evidence that savings significantly affect investment. **Final answers:** - Regression equation: $$\hat{I} = -2.043 + 0.1331 S$$ - $R^2 = 0.918$, $r = 0.958$ - Standard error of slope: $0.00814$ - Hypothesis test: slope is statistically significant at 5% level.