Question

To investigate the relationship between yield of potatoes, y, and the level of fertilizer application, x, an experimenter divides a field into eight plots of equal size and applies differing amounts of fertilizer to each. The yield of potatoes (in pounds) and the fertilizer application (in pounds) are recorded for each plot. The data are as follows: x 1 1.5 2 2.5 3 3.5 4 4.5 y 25 31 27 28 36 35 32 34 Test at the 5% significance level if there is a linear correlation. Find the least squares regression equation. What is the best estimate for the yield of potatoes if 3.2 pounds of fertilizer is applied?

Answer 1

Solution: In order to test the above question, we construct our null and alternative hypotheses as:

H0: rho = 0 vs Ha: rho not equal to 0 where rho is the population correlation coefficient
The test statistic to answer the given question is T=r*sqrt(n-2)/sqrt(1 -(r*r)) where r is the sample correlation coefficient and n is the sample size, sqrt refers to the square root function.

We reject H0 iff|T(observed)| > t(alpha/2,(n-2)) where t(Alpha/2,(n-2)) is the upper alpha/2 point of a Student's t distribution with (n-2) degrees of freedom.

Here, the linear correlation coefficient r = 0.729 (rounded to 3 decimal places) where r is calculated by the formula

r = n*sum(x*y) - sum(x)*sum(y) / sqrt{(n*sum(x2) - sum(x)2)(n*sum(y2) - sum(y)2)}

The value of the test statistic is T(observed) = 27.11139 and t(alpha/2,(n-2)) = 1.963632

Thus, |T(observed)| > t(alpha/2,(n-2)).

Hence we reject H0 at a 5% level of significance and conclude on the basis of the given sample that there is a significant linear correlation between yield of potatoes, y, and the level of fertilizer application, x.

The least squares regression equation is y_hat = b0 + b1*x

where y_hat is the predicted value of the yield of potatoes and x is the given pound of fertilizer applied, b1 is the slope and b0 is the y-intercept.

Also,   and  

Thus, the least squares regression equation is y_hat = 24.452 + 2.381 x

The best estimate for the yield of potatoes if 3.2 pounds of fertilizer is applied is obtained by putting x = 3.2 in the obtained linear regression equation.

It is found to be 32.0712 pounds.