Scatter diagrams and linear regressionEdexcel International A Level Maths: Subtopic test

1. 1
   A teacher records the number of hours of revision per week, $x$, and the mark out of 40, $y$, in a test for 8 students. The summary statistics are $\bar{x}=5$, $\bar{y}=22$, $S_{xx}=40$ and $S_{xy}=62$. The regression line of $y$ on $x$ is $y=a+bx$.

   (a)

   Find the value of $b$.

   [1 mark]

   • A$1.55$
   • B$0.645$
   • C$7.75$
   • D$4.4$

   (b)

   Find the value of $a$.

   [1 mark]

   • A$29.75$
   • B$20.45$
   • C$7.75$
   • D$14.25$

   (c)

   Estimate the mark of a student who revises for 6 hours per week.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A cafe owner models the daily sales, $s$ pounds, against the midday temperature, $t$ °C, using the regression line $s=120+8.5t$. The model was fitted to data collected on days when the temperature was between 12 °C and 28 °C.

   (a)

   Which statement correctly interprets the value $8.5$ in the model?

   [1 mark]

   • AWhen the temperature is 0 °C, the predicted daily sales are 8.50 pounds
   • BFor each 1 °C rise in temperature, the predicted daily sales increase by 8.50 pounds
   • CFor each 1 pound increase in sales, the temperature rises by 8.5 °C
   • DFor each 1 °C rise in temperature, the predicted daily sales increase by 8.5%

   (b)

   Which of these predictions from the model is likely to be the most reliable?

   [1 mark]

   • AThe sales when $t=5$
   • BThe sales when $t=35$
   • CThe sales when $t=20$
   • DThe sales when $t=0$

   (c)

   A student uses the model to predict the sales on a day when the midday temperature is 5 °C. Explain why this prediction may be unreliable.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A gardener applies $x$ grams of fertiliser to each of five plants of the same type and measures the height, $y$ cm, of each plant after four weeks. The values of $x$ are 2, 4, 6, 8, 10 and the heights are 5, 9, 10, 15, 16. For these data $\sum x=30$, $\sum y=55$, $\sum x^2=220$ and $\sum xy=386$.

   (a)

   Find the values of $S_{xx}$ and $S_{xy}$.

   [3 marks]

   (b)

   Find the equation of the regression line of $y$ on $x$ in the form $y=a+bx$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A shop records the price, $x$ pounds, of a product and the number sold per week, $y$, at five different prices. The prices are 40, 45, 50, 55, 60 and the numbers sold are 90, 82, 75, 66, 62. To simplify the working, the coded variable $u=\frac{x-40}{5}$ is used. For the five data points, $\sum u=10$, $\sum u^2=30$, $\sum y=375$ and $\sum uy=678$.

   (a)

   (i) Find the regression line of $y$ on $u$ in the form $y=a+bu$.
   (ii) Hence find the regression line of $y$ on $x$.
   (iii) Interpret the gradient of the line in part (ii).

   [6 marks]

   (b)

   (i) State which is the explanatory variable and which is the response variable.
   (ii) Use the regression line of $y$ on $x$ to estimate the number sold per week when the price is 52 pounds.
   (iii) The owner wants to use the line to predict sales at a price of 80 pounds. Comment on the reliability of this prediction.

   [6 marks]

   Total for question 4: 12 marks

End of questions