Least squares regression and residualsEdexcel A-Level Further Maths: Subtopic test

1. 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 $S_{xy}=88$.

   (a)

   Find the value of $b$.

   [1 mark]

   • A$0.45$
   • B$2.2$
   • C$8.8$
   • D$3$

   (b)

   Find the value of $a$.

   [1 mark]

   • A$25$
   • B$11$
   • C$-3$
   • D$3$

   (c)

   One observation is $x=6$, $y=15$. Calculate the residual for this observation.

   [2 marks]

   Total for question 1: 4 marks

2. 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 $\sum y^2=2194$.

   (a)

   Find the value of $S_{xy}$.

   [1 mark]

   • A$43$
   • B$343$
   • C$300$
   • D$86$

   (b)

   Find the residual sum of squares for the regression line of $y$ on $x$.

   [1 mark]

   • A$184.9$
   • B$194$
   • C$9.1$
   • D$4.3$

   (c)

   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]

   Total for question 2: 4 marks

3. 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 $\sum y^2=20414$.

   (a)

   Find the equation of the regression line of $y$ on $x$.

   [3 marks]

   (b)

   Using $y=-11.4+3.6x$, find the residual for the day on which the temperature was $20^\circ$C. Hence find the residual sum of squares for the six days.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A student measures the height, $y$ cm, of a seedling at the end of each of weeks $x=1,2,\ldots,7$ and records $3, 7, 8, 17, 12, 15, 15$. The summary statistics are $n=7$, $\sum x=28$, $\sum y=77$, $\sum x^2=140$, $\sum xy=364$ and $\sum y^2=1005$.

   (a)

   Find the equation of the regression line of $y$ on $x$, and calculate the residual sum of squares.

   [6 marks]

   (b)

   The week 4 height is suspected to be a misreading. Find the residual for week 4. With week 4 removed, the summary statistics are $n=6$, $\sum x=24$, $\sum y=60$, $\sum x^2=124$, $\sum xy=296$ and $\sum y^2=716$. Find the new regression line and its residual sum of squares, and comment on whether removing the week 4 value is justified.

   [6 marks]

   Total for question 4: 12 marks

End of questions