Sketch a scatter diagram that might be expected for the case when \(x\) and \(y\) are related approximately by \(y=a+bx^2\), where \(a\) is positive and \(b\) is negative. Your diagram should include 5 points, approximately equally spaced with respect to \(x\), and with all \(x\)- and \(y\)-values positive.[1]
The table gives the values of seven observations of bivariate data, \(x\) and \(y\).
| \(x\) | \(2\) | \(2.5\) | \(3\) | \(3.5\) | \(4\) | \(4.5\) | \(5\) |
| \(y\) | \(18.8\) | \(16.9\) | \(14.5\) | \(11.7\) | \(8.6\) | \(4.9\) | \(0.8\) |
Calculate the value of the product moment correlation coefficient, and explain why its value does not necessarily mean that the best model for the relationship between \(x\) and \(y\) is \(y=c+dx\).[2]
Explain how to use the values obtained by calculating product moment correlation coefficients to decide, for this data, whether \(y=a+bx^2\) or \(y=c+dx\) is the better model.[1]
It is desired to use the data in the table to estimate the value of \(y\) for which \(x=3.2\). Find the equation of the least-squares regression line of \(y\) on \(x^2\). Use your equation to calculate the desired estimate.[3]