Toughness and Fibrousness of Asparagus Are Major Determinants of Quality

Essay

Toughness and fibrousness of asparagus are major determinants of quality. This was the focus of a study reported in "Post-Harvest Glyphosphate Application Reduces Toughening, Fiber Content, and Lignification of Stored Asparagus Spears" (J. of the Amer. Soc. Of Horticultural Science, 1988: 569-572). The article reported the accompanying data (read from a graph) on x = shear force (kg) and y = percent fiber dry weight. $$\begin{array} { l l l l l l l l l l } \hline x & 46 & 48 & 55 & 57 & 60 & 72 & 81 & 85 & 94 \\\hline y & 2.18 & 2.10 & 2.13 & 2.28 & 2.34 & 2.53 & 2.28 & 2.62 & 2.63 \\\hline & & & & & & & & & \\\hline x & 109 & 121 & 132 & 137 & 148 & 149 & 184 & 185 & 187 \\\hline y & 2.50 & 2.66 & 2.79 & 2.80 & 3.01 & 2.98 & 3.34 & 3.49 & 3.26 \\\hline\end{array}$$ $n = 18 \quad \sum x _ { i } = 1950 \quad \sum x _ { i} ^ { 2 } = 251,970$ $\sum y _ {i } = 47.92 \sum y _ { i} ^ {2 } = 130.6074 \quad \sum x _ { i } y _ { i } = 5530.92$
a. Calculate the value of the sample correlation coefficient. Based on this value, how would you describe the nature of the relationship between the two variables?
b. If a first specimen has a larger value of shear force than does a second specimen, what tends to be true of percent dry fiber weight for the two specimens.
c. If shear force is expressed in pounds, what happens to the value of r? Why?
d. If the simple linear regression model were fit to this data, what proportion of observed variation in percent fiber dry weight could be explained by the model relationship?
e. Carry out a test at significance level .01 to decide whether there is a positive linear association between the two variables.