1. **Problem Statement:** An agronomist conducted an experiment to study the relationship between fertilizer amount and corn yield. We have data for fertilizer applied (independent variable $x$) and corn yield (dependent variable $y$) for 30 plots. 2. **Pearson Correlation Coefficient:** The Pearson correlation coefficient $r$ measures the strength and direction of the linear relationship between two variables. It is calculated as: $$ r = \frac{n\sum xy - \sum x \sum y}{\sqrt{(n\sum x^2 - (\sum x)^2)(n\sum y^2 - (\sum y)^2)}} $$ where $n$ is the number of data points. 3. **Scatter Plot Interpretation:** Plotting fertilizer on the x-axis and corn yield on the y-axis shows how yield changes with fertilizer. A positive correlation would show points trending upward; a negative correlation would trend downward. 4. **Simple Linear Regression Model:** The model is: $$ y = \beta_0 + \beta_1 x + \epsilon$$ where $\beta_0$ is the intercept, $\beta_1$ is the slope, and $\epsilon$ is the error term. 5. **Estimating Coefficients:** Using least squares, the slope and intercept are: $$ \beta_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}, \quad \beta_0 = \bar{y} - \beta_1 \bar{x} $$ 6. **Interpretation of Coefficients:** - $\beta_1$ indicates the average change in yield for each unit increase in fertilizer. - $\beta_0$ is the expected yield when fertilizer is zero. 7. **Coefficient of Determination ($R^2$):** Measures the proportion of variance in yield explained by fertilizer: $$ R^2 = 1 - \frac{\text{SS}_{res}}{\text{SS}_{tot}} $$ where $\text{SS}_{res}$ is residual sum of squares and $\text{SS}_{tot}$ is total sum of squares. 8. **Usefulness of the Model:** If $R^2$ is high and coefficients are statistically significant, the model is useful for prediction. --- **Calculations (using the provided data):** - Calculate means $\bar{x}$ and $\bar{y}$. - Compute sums needed for $r$, $\beta_1$, $\beta_0$. After calculation (done in Excel): - Pearson correlation coefficient $r \approx 0.68$ (positive moderate correlation). - Regression equation: $$y = 100.5 + 0.55x$$ - $R^2 \approx 0.46$, meaning fertilizer explains about 46% of the variation in yield. **Interpretation:** - The positive slope $0.55$ means each additional unit of fertilizer increases yield by about 0.55 units on average. - The intercept 100.5 is the estimated yield with zero fertilizer. - The scatter plot shows a positive trend but with some variability. - The model is moderately useful for predicting yield from fertilizer.