A Biologist Assumes There Is A Linear Relationship Between

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A biologist assumes that there is a linear relationship between the amount of fertilizer supplied to tomato plants and the subsequent yield of tomatoes obtained. Eight tomato plants of the same variety were selected at random and treated weekly with a solution in which x grams of fertilizer was dissolved in a fixed quantity of water. The yield y kilograms of tomatoes was recorded as follows. The data points are:

• Plant A: x=1.0, y=3.9
• Plant B: x=1.5, y=4.4
• Plant C: x=2.0, y=5.8
• Plant D: x=2.5, y=6.6
• Plant E: x=3.0, y=7.0
• Plant F: x=3.5, y=7.1
• Plant G: x=4.0, y=7.3
• Plant H: x=4.5, y=7.7

Paper For Above instruction

The investigation into the relationship between fertilizer application and tomato yield exemplifies an essential aspect of agricultural research: understanding how input variables influence crop productivity. Assuming a linear relationship, the biologist endeavors to establish a mathematical model that can predict yield based on fertilizer amount, facilitating optimized resource allocation and maximizing crop output.

Introduction

The effectiveness of fertilizer application is a critical factor in agricultural productivity. By modeling the relationship between fertilizer quantity and tomato yield using linear regression, researchers can predict outcomes and determine optimal fertilizer levels. This study analyzes data from eight tomato plants to develop such a model, providing insights into the fertilizer-yield relationship and offering a predictive tool for farmers and agronomists.

Data Description and Graphical Representation

The data comprises paired measurements of fertilizer amount (x) and corresponding tomato yield (y) across eight individual plants. The x values range from 1.0 to 4.5 grams, while the y values span from 3.9 to 7.7 kilograms. Plotting these points on a scatterplot allows visualization of the potential linear trend, which is a prerequisite for regression analysis.

Creating a scatterplot reveals a positive correlation—an increase in fertilizer appears associated with an increase in yield—but further statistical analysis quantifies this relationship. Visual inspection suggests that the trend is approximately linear, thus justifying the application of linear regression techniques.

Calculation of the Least Squares Regression Line

The regression equation can be expressed as:

y = a + bx

where:

• b is the slope of the line
• a is the y-intercept

Step 1: Compute Means of x and y

Mean of x:

\(\bar{x} = \frac{1.0 + 1.5 + 2.0 + 2.5 + 3.0 + 3.5 + 4.0 + 4.5}{8} = \frac{22.5}{8} = 2.8125 \)

Mean of y:

\(\bar{y} = \frac{3.9 + 4.4 + 5.8 + 6.6 + 7.0 + 7.1 + 7.3 + 7.7}{8} = \frac{52.8}{8} = 6.6 \)

Step 2: Calculate the Slope (b)

Using the formula:

\(b = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}\)

Calculate for each data point:

Sum of numerator terms:

\(\sum (x_i - \bar{x})(y_i - \bar{y}) = 4.898 + 2.888 + 0.65 + 0 + 0.075 + 0.344 + 0.832 + 1.856 = 11.543\)

Sum of denominator terms:

\(\sum (x_i - \bar{x})^2 = 3.285 + 1.722 + 0.660 + 0.098 + 0.035 + 0.473 + 1.413 + 2.846 = 10.532\)

Therefore, the slope:

\(b = \frac{11.543}{10.532} \approx 1.096\)

Step 3: Calculate the Intercept (a)

Using the formula:

\(a = \bar{y} - b \cdot \bar{x} = 6.6 - 1.096 \times 2.8125 \approx 6.6 - 3.082 = 3.518\)

Thus, the least squares regression line is:

y = 3.518 + 1.096x

Prediction of Yield for a Fertilizer of 3.2 grams

Substitute x = 3.2 into the regression equation:

y = 3.518 + 1.096 \times 3.2 = 3.518 + 3.507 = 7.025 (kg)

Therefore, the estimated yield of a tomato plant treated weekly with 3.2 grams of fertilizer is approximately 7.03 kilograms.

Discussion

The regression analysis confirms a positive linear relationship between fertilizer amount and tomato yield. The slope indicates that each additional gram of fertilizer increases yield by approximately 1.096 kilograms. This model can assist farmers in optimizing fertilizer use, balancing the cost of fertilizer against expected yield gains.

The prediction at 3.2 grams aligns closely with the observed trend, demonstrating the model's utility. However, validation with larger datasets and consideration of other factors such as soil quality and weather conditions is essential for more accurate predictions in practical settings.

Conclusion

Through regression analysis, a simple yet effective model relating fertilizer quantity to tomato yield was developed, offering a practical tool for enhancing crop productivity. Further research should incorporate more variables and larger samples to refine prediction accuracy and inform sustainable agricultural practices.

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