Explain least squares concept

7 The diagram shows the results of an experiment involving some bivariate data. The least squares regression line of \(y\) on \(x\) for these results is also shown. \includegraphics[max width=\textwidth, alt={}, center]{48ffcd44-d933-40e0-818a-20d6db607298-5_748_919_390_612}

1. Given that the least squares regression line of \(y\) on \(x\) is used for an estimation, state which of \(x\) or \(y\) is treated as the independent variable.
2. Use the diagram to explain what is meant by 'least squares'.
3. State, with a reason, the value of Spearman's rank correlation coefficient for these data.
4. What can be said about the value of the product moment correlation coefficient for these data?

3

1. Using the scatter diagram in the Printed Answer Booklet, explain what is meant by least squares in the context of a regression line of \(y\) on \(x\).
2. A set of bivariate data \(( t , u )\) is summarised as follows. \(n = 5 \quad \sum t = 35 \quad \sum u = 54\) \(\sum t ^ { 2 } = 285 \quad \sum u ^ { 2 } = 758 \quad \sum \mathrm { tu } = 460\)
   1. Calculate the equation of the regression line of \(u\) on \(t\).
   2. The variables \(t\) and \(u\) are now scaled using the following scaling. \(\mathrm { v } = 2 \mathrm { t } , \mathrm { w } = \mathrm { u } + 4\) Find the equation of the regression line of \(w\) on \(v\), giving your equation in the form \(w = f ( v )\).

1 At a seaside resort the number \(X\) of ice-creams sold and the temperature \(Y ^ { \circ } \mathrm { F }\) were recorded on 20 randomly chosen summer days. The data can be summarised as follows. \(\sum x = 1506 \quad \sum x ^ { 2 } = 127542 \quad \sum y = 1431 \quad \sum y ^ { 2 } = 104451 \quad \sum x y = 111297\)

1. Calculate the equation of the least squares regression line of \(y\) on \(x\), giving your answer in the form \(y = a + b x\).
2. Explain the significance for the regression line of the quantity \(\sum \left[ y _ { i } - \left( a x _ { i } + b \right) \right] ^ { 2 }\).
3. It is decided to measure the temperature in degrees Centigrade instead of degrees Fahrenheit. If the same temperature is measured both as \(f ^ { \circ }\) Fahrenheit and \(c ^ { \circ }\) Centigrade, the relationship between \(f\) and \(c\) is \(\mathrm { c } = \frac { 5 } { 9 } ( \mathrm { f } - 32 )\). Find the equation of the new regression line.

In an agricultural experiment, the relationship between the amount of water supplied, \(x\) units, and the yield, \(y\) units, was investigated. Six values of \(x\) were chosen and for each value of \(x\) the corresponding value of \(y\) was measured. The results are shown in the table. These results, together with the regression line of \(y\) on \(x\), are plotted on the graph. \includegraphics{figure_1}

1. Give a reason why the regression line of \(x\) on \(y\) is not suitable in this context. [1]
2. Explain the significance, for the regression line of \(y\) on \(x\), of the distances shown by the vertical dotted lines in the diagram. [2]
3. Calculate the value of the product moment correlation coefficient, \(r\). [3]
4. Comment on your value of \(r\) in relation to the diagram. [2]