Least squares regression and residualsEdexcel A-Level Further Maths: Mind map

Exam questions on Least squares regression and residuals

1. A researcher models the relationship between a variable $x$ and a variable $y$ using the least squares regression line of $y$ on $x$, $y=a+bx$. Summary statistics from the data give $\bar x=5$, $\bar y=14$, $S_{xx}=40$ and $S_{xy}=88$.
   One observation is $x=6$, $y=15$. Calculate the residual for this observation.2 marks
2. A student records the number of hours, $x$, spent revising and the score, $y$, out of 40, on a test for 5 students: $(1,12)$, $(2,15)$, $(3,21)$, $(4,22)$, $(5,30)$. The summary statistics are $n=5$, $\sum x=15$, $\sum y=100$, $\sum x^2=55$, $\sum xy=343$ and $\sum y^2=2194$.
   The regression line is $y=7.1+4.3x$. Interpret the value $4.3$ in context, and explain why the line should not be used to predict the score of a student who revises for 15 hours.2 marks
3. A café owner records the midday temperature, $x\,^\circ$C, and the number of cold drinks sold, $y$, on six days: $(14,40)$, $(16,45)$, $(18,55)$, $(20,58)$, $(22,68)$, $(24,76)$. The summary statistics are $n=6$, $\sum x=114$, $\sum y=342$, $\sum x^2=2236$, $\sum xy=6750$ and $\sum y^2=20414$.
   Find the equation of the regression line of $y$ on $x$.3 marks