Further Statistics 2: Linear regressionEdexcel A-Level Further Maths: Topic test

1. 1
   A dealer records the age $x$ years and the price $y$, in hundreds of pounds, of $10$ second-hand cars of the same model. The summary statistics are $\bar x=5$, $\bar y=40$, $S_{xx}=60$ and $S_{xy}=-270$.

   (a)

   What is the gradient $b$ of the regression line of $y$ on $x$?

   [1 mark]

   • A$-0.22$
   • B$-4.5$
   • C$4.5$
   • D$-45$

   (b)

   What is the value of $a$ in the regression line $y=a+bx$?

   [1 mark]

   • A$40$
   • B$17.5$
   • C$62.5$
   • D$-22.5$

   (c)

   Interpret the values of $a$ and $b$ in context.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The regression line of $y$ on $x$ for a set of five observations is $\hat y=3.2+1.5x$.

   (a)

   What is the residual for the observation $(4,10.5)$?

   [1 mark]

   • A$1.3$
   • B$-1.3$
   • C$9.2$
   • D$0.7$

   (b)

   Which quantity does the least squares regression line of $y$ on $x$ make as small as possible?

   [1 mark]

   • AThe sum of the residuals
   • BThe sum of the perpendicular distances from the points to the line
   • CThe sum of the squares of the horizontal distances from the points to the line
   • DThe sum of the squares of the residuals

   (c)

   Four of the five residuals are $0.4$, $-0.9$, $0.5$ and $0.3$. Find the fifth residual.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A firm records the number of advertisements $x$ broadcast in a week and the sales $y$, in thousands of pounds, for six weeks. The data give $\sum x=36$, $\sum y=152$, $\sum x^2=258$ and $\sum xy=1037$.

   (a)

   Find $S_{xx}$ and $S_{xy}$, and hence find the gradient $b$ of the regression line of $y$ on $x$.

   [3 marks]

   (b)

   Find the equation of the regression line of $y$ on $x$, and find the residual for the week in which $7$ advertisements were broadcast and sales were $29$ thousand pounds.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   An online shop records, for $9$ months, the amount $x$, in hundreds of pounds, spent on advertising (with $4\leq x\leq16$) and the number $y$ of orders received. The summary statistics are $\bar x=10$, $\bar y=30$, $S_{xx}=240$, $S_{xy}=360$ and $S_{yy}=588$.

   (a)

   Find the equation of the regression line of $y$ on $x$. Interpret the gradient in context and comment on using the line to predict the number of orders when $x=40$.

   [6 marks]

   (b)

   Find the residual sum of squares for this model. A quadratic model fitted to the same data has a residual sum of squares of $30$. Comment on whether the quadratic model is an improvement.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   For $12$ observations of $(x,y)$, the means are $\bar x=6$ and $\bar y=20$, and $S_{xx}=40$, $S_{xy}=-100$ and $S_{yy}=330$.

   (a)

   What is the residual sum of squares, $\mathrm{RSS}=S_{yy}-\dfrac{S_{xy}^2}{S_{xx}}$?

   [1 mark]

   • A$250$
   • B$230$
   • C$80$
   • D$580$

   (b)

   What is the gradient of the regression line of $y$ on $x$?

   [1 mark]

   • A$2.5$
   • B$-2.5$
   • C$-0.4$
   • D$-0.3$

   (c)

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

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A model $y=a+bx$ is fitted to $8$ observations. Taken in order of increasing $x$, the residuals are $2.0$, $0.9$, $-0.6$, $-1.8$, $-2.0$, $-0.9$, $0.8$ and $1.6$.

   (a)

   What is the residual sum of squares?

   [1 mark]

   • A$10.6$
   • B$0$
   • C$2.05$
   • D$16.42$

   (b)

   What do the residuals suggest about the model?

   [1 mark]

   • AThe signs form a systematic pattern, so a curve may fit better than a straight line
   • BThe residuals sum to zero, so the straight line is a good fit
   • CThe largest residual is $2.0$, so that observation must be an error
   • DThe residuals are small, so the straight line must be appropriate

   (c)

   Describe what the residuals would look like if a straight line were an appropriate model.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A technician measures the extension $y$ mm of a spring when a load of $x$ newtons is applied, for $x=1,2,\dots,6$. The extensions are $4.2$, $6.1$, $7.9$, $10.2$, $12.1$ and $19.0$. The summary statistics are $\sum x=21$, $\sum y=59.5$, $\sum x^2=91$ and $\sum xy=255.4$.

   (a)

   Find the equation of the regression line of $y$ on $x$ for all six measurements.

   [3 marks]

   (b)

   Find the residual for the measurement at $x=6$. This measurement is thought to be faulty and is removed. Find the equation of the regression line for the other five measurements.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A delivery van's fuel use is recorded on $10$ journeys. For each journey, $x$ is the distance in km and $y$ is the fuel used in litres. The summary statistics are $\bar x=12$, $\bar y=39.6$, $S_{xx}=250$, $S_{xy}=700$ and $S_{yy}=2050$.

   (a)

   Find the equation of the regression line of $y$ on $x$ and interpret both coefficients in context.

   [6 marks]

   (b)

   Find the residual sum of squares. One journey has $x=20$ and $y=69.5$. Find its residual and comment on its effect on the model.

   [6 marks]

   Total for question 8: 12 marks

End of questions