A researcher is investigating the relationship between yield of potatoes \((y)\) and level of fertilizer \((x\).\() She divides a eld into eight plots of equal\) size and applied fertilizer at a di erent level to each plot. The level of fertilizer and yield for each plot is recorded below:

(a) Plot a scatterplot of yield versus fertilizer level.
(b) Calculate the parameters of the least squares line.
(c) Graph the least squares line on your scatterplot.
(d) Calculate the estimated variance about the least squares line.
(e) Suppose that we know that yield given the fertilizer level is emphnormal ( \(0_{0}+\) \(x^{2}\) ), where \({ }^{2}=30^{2}\) is known. Use a normal \(\left(22^{2}\right)\) prior for . What is the posterior distribution of ?
(f) Find a 95\% credible interval for .
(g) Perform a Bayesian test of
\[
H_{0}: \quad 0 \text { versus } H_{1}:>0
\]
at the \(5 \%\) level of significance.

Transcribed Image Text:

Fertilizer Level X Yield y 1 25 1.5 31 2 27 2.5 28 3 36 3.5 35 4 32 4.5 34