Question

The international rice research institute in the Philippines conducted a study of the relationship between and the yield of rice and the amount of nitrogen fertilizer used when growing the rice. A number of small plots of land (all the same size) were used in the study with varying amounts of fertilizer from one plot to the next.

When the rice is harvested, the yield (in ounces of rice) was measured and related to the amount (in ounces) of nitrogen applied.

The correlation between rice yield and nitrogen was R=0.95. A linear regression analysis with X=nitrogen Y= rice yield. Regression line y=240+20x

1. How much rice would we expect an unfertilized plot to produce?

2. By how much would we expect the rice yield to increase for each extra ounce of nitrogen fertilizer used?

3. For the plot that received 8 ounces of nitrogen, the observed rice yield y did not lie exactly on the regression line. The error for this observation was 12. What must be the value for the observed rice yield y?

4. What proportion os the variation in rice yield is due to the relationship with nitrogen fertilizer?

Answer 1

Solution:

The regression equation is, y = 240 + 20x

1) We want to obtain the expected yield from an unfertilized plot.

The plot is unfertilized means X = 0, and we want to estimate Y when x = 0.

When X = 0, then y = 240

Hence, we expect an unfertilized plot to produce 240 ounces of rice.

2) Slope of regression equation is 20.

Hence, we would expect the rice yield to increase by 20 ounces for each extra ounce of nitrogen fertilizer used.

3) Error for an observation is obtained as follows :

Error = observed value - Predicted value

When the plot that received 8 ounces of nitrogen (i.e. when x = 8), the predicted value of y is,

y = 240 + (20×8)

y = 400

We have, error = 12

Hence, 12 = observed value - 400

Observed value = 400 + 12 = 412

The value for the observed rice yield y is 412.

4) The value of R^{​​​​​​2} tells us that what proportion of variation in dependent variable is explained by the independent variable.

In other words, R^{​​​​​​2} tells us that, what proportion of variation in dependent variable is due to the relationship with independent variable.

We have, R = 0.95

Hence, R^{​​​​​​2} = (0.95)² = 0.9025 = 90.25%

Hence, 90.25% of the variation in rice yield is due to the relationship with nitrogen fertilizer.

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