High School

An agronomist undertook an experiment to investigate the factors affecting potato harvest. The agronomist divided the farm into 30 half-hectare plots and applied varying levels of fertilizer. Potatoes were then planted, and the harvest at the end of the season was recorded.

| Fertilizer (Kg) | Harvest (tons) |
|-----------------|----------------|
| 210 | 43.5 |
| 220 | 40.0 |
| 230 | 48.0 |
| 240 | 65.0 |
| 250 | 80.0 |
| 260 | 85.0 |
| 270 | 95.0 |
| 280 | 80.0 |
| 290 | 97.3 |

Note: No Excel ANOVA output allowed. Students need to show all the steps in calculations.

You are required to:

a. Find the simple regression line and interpret the coefficients. (3 marks)
- ANSWER:

b. Find the coefficient of determination and interpret its value. (2 marks)
- ANSWER:

c. Does the model appear to be a useful tool in predicting the potato harvest? If so, predict the harvest when 250 kg of fertilizer is applied. If not, explain why not. (2 marks)
- ANSWER:

Answer :

Final answer:

To find the simple regression line for potato harvest, calculate the slope and y-intercept using the given formulas. The coefficients of the regression line can be interpreted in terms of the relationship between fertilizer and harvest. The coefficient of determination is used to assess the model's usefulness in predicting potato harvest.

Explanation:

To find the simple regression line, we need to calculate the slope and y-intercept of the line. The formula for calculating the slope (b) is given by:

b = (∑(x₂)∑(y) - ∑(x)(xy)) / (n∑(x₂) - (∑(x))²)

The formula for calculating the y-intercept (a) is given by:

a = (∑(y) - b∑(x)) / n

Once we have calculated the slope and y-intercept, we can write the equation of the regression line as y = ax + b. The coefficients of the regression line can be interpreted as follows:

The slope (b) represents the change in the dependent variable (harvest) for every unit change in the independent variable (fertilizer). The y-intercept (a) represents the value of the dependent variable (harvest) when the independent variable (fertilizer) is zero.

The coefficient of determination (r²) is a measure of how well the regression line fits the data. It represents the proportion of the variation in the dependent variable (harvest) that can be explained by the independent variable (fertilizer). A higher value of r² indicates a stronger relationship between the variables.

To determine if the model is a useful tool in predicting potato harvest, we can use the coefficient of determination. If r² is close to 1, it indicates a strong relationship and the model is likely to be useful. If r² is close to 0, it indicates a weak relationship and the model may not be useful.

The predicted harvest when 250KG of fertilizer is applied can be found by substituting the value of 250 in the regression equation and solving for y.

Learn more about Regression analysis here:

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