S1: Correlation and regressionEdexcel International A Level Maths: Topic test

1. 1
   A forester measures the age, $x$ years, and the trunk diameter, $y$ cm, of 12 trees. The data are summarised by $\bar x=6$, $\bar y=40$, $S_{xx}=42$, $S_{xy}=105$ and $S_{yy}=290$. The ages of the trees ranged from 2 to 11 years.

   (a)

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

   [1 mark]

   • A$0.4$
   • B$25$
   • C$2.5$
   • D$105$

   (b)

   Find the product moment correlation coefficient between $x$ and $y$.

   [1 mark]

   • A$0.951$
   • B$0.362$
   • C$0.4$
   • D$0.905$

   (c)

   Interpret the gradient of the regression line in context, and explain why the line should not be used to estimate the diameter of a tree aged 40 years.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A teacher uses the regression line $y=12.4+3.1x$ to model the mark $y$, out of 60, in a test for students who spent $x$ hours practising in a week. The values of $x$ in the data ranged from 2 to 10.

   (a)

   Which is the response variable?

   [1 mark]

   • AThe number of hours of practice.
   • BThe gradient 3.1.
   • CThe intercept 12.4.
   • DThe mark in the test.

   (b)

   Use the model to predict the mark of a student who practises for 8 hours.

   [1 mark]

   • A$24.8$
   • B$37.2$
   • C$15.5$
   • D$40.3$

   (c)

   A student suggests using the model to predict the mark for a student who practises for 20 hours. Calculate the predicted mark and explain why the prediction is not sensible.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A coach records, for 10 club runners, the distance $x$ km run in training in a week and the time $y$ minutes taken in a 5 km race. The data are summarised by $\sum x=300$, $\sum y=240$, $\sum x^2=9640$ and $\sum xy=6976$.

   (a)

   Find $S_{xx}$ and $S_{xy}$.

   [3 marks]

   (b)

   Find the equation of the regression line of $y$ on $x$, giving the constants to 3 significant figures.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A call centre records, for 10 trainee operators, the number of weeks $x$ of experience and the number of calls $y$ handled per day. The data, with $x$ ranging from 1 to 15 weeks, are summarised by $\sum x=90$, $\sum y=330$, $\sum x^2=1072$, $\sum y^2=12\,680$ and $\sum xy=3535$.

   (a)

   (i) Find $S_{xx}$, $S_{xy}$ and $S_{yy}$. [3]
   (ii) Find the product moment correlation coefficient. [2]
   (iii) Interpret your value in context. [1]

   [6 marks]

   (b)

   (i) Find the equation of the regression line of $y$ on $x$ in the form $y=a+bx$. [3]
   (ii) Interpret the value of $b$ in context. [1]
   (iii) Estimate the number of calls handled per day by an operator with 10 weeks of experience, and explain why the line should not be used for an operator with 40 weeks of experience. [2]

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A technician measures the operating temperature $x$ ($^\circ$C) of a machine and the time $y$ (hours) it runs before it needs servicing, for a sample of machines. The data are summarised by $S_{xx}=90$, $S_{yy}=40$ and $S_{xy}=-54$.

   (a)

   Find the product moment correlation coefficient.

   [1 mark]

   • A$-0.6$
   • B$-0.9$
   • C$-1.35$
   • D$0.9$

   (b)

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

   [1 mark]

   • A$-1.35$
   • B$-0.9$
   • C$0.6$
   • D$-0.6$

   (c)

   Describe the correlation between $x$ and $y$. State, with a reason, the effect on $r$ of measuring $y$ in minutes instead of hours.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A city records, for 40 fires, the number of fire engines $x$ sent to each fire and the cost $y$ of the damage caused. The product moment correlation coefficient is $r=0.82$.

   (a)

   What does the value of $r$ show?

   [1 mark]

   • AThere is strong positive linear correlation between $x$ and $y$.
   • BThere is weak positive linear correlation between $x$ and $y$.
   • CSending more engines causes more damage.
   • D82% of fires cause damage.

   (b)

   The regression line of $y$ on $x$ is used to estimate the number of engines that would be sent to a fire causing damage of £50 000. Which statement is correct?

   [1 mark]

   • AThis is appropriate, because the correlation is strong.
   • BThis is appropriate, provided that 50 000 lies within the range of the damage costs.
   • CThis is not appropriate, because the line of $y$ on $x$ is for estimating $y$ from $x$, and a different line is needed to estimate $x$ from $y$.
   • DThis is not appropriate, because $r$ must be at least 0.9 to use a regression line.

   (c)

   Explain why it would be wrong to conclude that sending fewer fire engines to a fire would reduce the cost of the damage.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A vineyard records the rainfall $x$ cm in the growing season and the grape yield $y$ tonnes per hectare for 12 seasons. The data are summarised by $\sum x=126$, $\sum y=174$, $S_{xx}=64$, $S_{yy}=100$ and $S_{xy}=48$.

   (a)

   Calculate the product moment correlation coefficient.

   [3 marks]

   (b)

   Find the equation of the regression line of $y$ on $x$, giving the constants to 3 significant figures.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A typing course records the number of weeks of training $x$ and the typing speed $y$ (words per minute) of six trainees. The values of $x$ are 2, 4, 6, 8, 10 and 12, and the values of $y$ are 24, 29, 38, 37, 46 and 42 respectively, so that $\sum x=42$, $\sum y=216$, $\sum x^2=364$, $\sum y^2=8110$ and $\sum xy=1652$.

   (a)

   (i) Find $S_{xx}$ and $S_{xy}$. [2]
   (ii) Find the equation of the regression line of $y$ on $x$. [2]
   (iii) Use the line to estimate the typing speed after 9 weeks of training, and comment on the reliability of your estimate. [2]

   [6 marks]

   (b)

   (i) Calculate the product moment correlation coefficient, given that $S_{yy}=334$. [2]
   (ii) Interpret your value in context. [1]
   (iii) The course manager says that the value of $r$ proves that the training causes the speed to increase. Comment on this claim. [2]
   (iv) The speeds are converted to words per second by dividing by 60. State the new value of the correlation coefficient. [1]

   [6 marks]

   Total for question 8: 12 marks

End of questions