1. **Problem Statement:** You want to test if the relationship between two variables is positive based on a given graph. 2. **Statistical Test:** Use the Pearson correlation test because it measures the strength and direction of a linear relationship between two continuous variables. 3. **Hypotheses:** - Null hypothesis ($H_0$): There is no positive correlation between the variables, i.e., $\rho \leq 0$. - Alternative hypothesis ($H_a$): There is a positive correlation, i.e., $\rho > 0$. 4. **Test Type:** This is a one-tailed test because you are specifically testing for a positive relationship. 5. **Degrees of Freedom (df):** $df = n - 2$, where $n$ is the number of data points. 6. **Critical r-value ($r_{crit}$):** Determined from the correlation table based on $df$ and alpha level (commonly 0.05). 7. **Obtained r-value ($r_{obt}$):** Calculated from the data; it measures the actual correlation. 8. **Coefficient of Determination ($r^2$):** $r^2$ tells the proportion of variance in the dependent variable explained by the independent variable. 9. **Regression Equation:** Typically $y = b_0 + b_1 x$, where $b_1 = r \times \frac{s_y}{s_x}$ and $b_0 = \bar{y} - b_1 \bar{x}$. 10. **Conclusion:** If $r_{obt} > r_{crit}$, reject $H_0$ and conclude a significant positive correlation. 11. **Effect of Alpha 0.01:** A smaller alpha makes it harder to reject $H_0$, requiring a higher $r_{crit}$. 12. **Effect of Alpha 0.05 with Two-tailed Test:** Testing for any change (positive or negative) doubles the critical region, increasing $r_{crit}$. 13. **Interpretations:** - $r_{obt}$ shows strength and direction of correlation. - $r^2$ shows explained variance. - Regression equation predicts $y$ from $x$. 14. **Predicting Test Score for 11 Hours:** Substitute $x=11$ into regression equation. 15. **Predicting Hours for Score 89:** Solve regression equation for $x$ when $y=89$. Note: Specific numeric answers require data values not provided.