4.4 Correlation and linear regressionIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   Over eight days a café records the maximum daily temperature, $T\,^\circ$C, and the number of cold drinks sold, $n$. $T$: 18, 21, 24, 26, 29, 31, 34, 36. $n$: 42, 55, 61, 70, 82, 85, 96, 104. The regression line of $n$ on $T$ has equation $n=aT+b$. Use your GDC.

   (a)

   Which best describes the correlation between $T$ and $n$?

   [1 mark]

   • AStrong positive linear correlation
   • BWeak positive linear correlation
   • CStrong negative linear correlation
   • DNo correlation

   (b)

   For the regression line, $a=3.36$ (3 s.f.). Which is the correct interpretation of $a$?

   [1 mark]

   • AThe café sells 3.36 drinks when the temperature is $0\,^\circ$C
   • BOn average, each $1\,^\circ$C rise in temperature is associated with about 3.36 more drinks sold
   • CThe temperature rises by $3.36\,^\circ$C for each extra drink sold
   • DEach $1\,^\circ$C rise in temperature increases drink sales by 3.36%

   (c)

   Use your regression line to estimate the number of cold drinks sold on a day when the maximum temperature is $28\,^\circ$C.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Over 12 months, a coastal town records its monthly ice-cream sales and the number of sunburn cases treated at its clinic. Pearson's product-moment correlation coefficient for the data is $r=0.91$.

   (a)

   Which conclusion is justified by the data?

   [1 mark]

   • AEating ice cream causes sunburn
   • BThe correlation between sales and sunburn cases is weak
   • CSunburn cases cause people to buy ice cream
   • DSales and sunburn cases tend to increase together, but this does not show that one causes the other

   (b)

   Which statement about $r=0.91$ is correct?

   [1 mark]

   • AThere is a strong negative linear correlation
   • B91% of the points lie exactly on the line of best fit
   • CThere is a strong positive linear correlation
   • DEach extra ice cream sold produces 0.91 extra sunburn cases

   (c)

   Suggest a third variable that could explain the correlation between ice-cream sales and sunburn cases.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A teacher records the number of hours, $x$, that eight students revised for a test and their test score, $y$ (out of 100). $x$: 2, 4, 5, 6, 8, 9, 10, 12. $y$: 45, 52, 50, 61, 63, 70, 66, 78. Use your GDC.

   (a)

   Find the value of Pearson's product-moment correlation coefficient $r$, and the equation of the regression line of $y$ on $x$.

   [3 marks]

   (b)

   (i) Interpret the gradient of the regression line in this context.
   (ii) Interpret the $y$-intercept in this context.
   (iii) Estimate the score of a student who revised for 7 hours.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A dealer records the age, $x$ years, and the value, $V$ thousand AED, of eight used cars of the same model. $x$: 1, 2, 3, 4, 5, 6, 7, 8. $V$: 82, 71, 66, 55, 52, 41, 38, 30. Use your GDC.

   (a)

   (i) Find the value of $r$.
   (ii) Describe the correlation between $x$ and $V$.
   (iii) Find the equation of the regression line of $V$ on $x$.
   (iv) Show that the regression line passes through the mean point $(\bar{x},\bar{V})$.

   [6 marks]

   (b)

   (i) Interpret the gradient of the regression line in this context.
   (ii) Estimate the value of a car that is $5.5$ years old.
   (iii) The dealer uses the line to estimate the value of a $15$-year-old car. Find this estimate and comment on its reliability.
   (iv) The dealer solves $45=-7.23x+86.9$ to estimate the age of a car worth $45$ thousand AED. Explain why this method is not reliable.

   [6 marks]

   Total for question 4: 12 marks

End of questions