Bivariate data, correlation and regressionEdexcel A-Level Maths: Subtopic test

1. 1
   A study of children aged from 5 to 12 years finds that the regression line of height $h$ cm on age $a$ years is $h=75+5.2a$.

   (a)

   What height does the model predict for a child aged 8 years?

   [1 mark]

   • A41.6 cm
   • B80.2 cm
   • C116.6 cm
   • D83 cm

   (b)

   Which statement correctly interprets the gradient 5.2?

   [1 mark]

   • AOn average, height increases by 5.2 cm for each extra year of age
   • BEach extra year of age causes every child to grow exactly 5.2 cm
   • CAge increases by 5.2 years for each extra centimetre of height
   • DHeight increases by 5.2% each year

   (c)

   Use the model to predict the height of a 30-year-old, and comment on the reliability of this prediction.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A scientist believes that two variables are related by $y=ax^n$. She plots $\log_{10}y$ against $\log_{10}x$ and finds that the points lie close to a straight line with gradient 1.5 that crosses the vertical axis at 0.477.

   (a)

   What is the value of $n$?

   [1 mark]

   • A0.477
   • B3
   • C31.6
   • D1.5

   (b)

   What is the value of $a$, to 2 significant figures?

   [1 mark]

   • A0.48
   • B3.0
   • C1.5
   • D0.72

   (c)

   Use the model to estimate $y$ when $x=100$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The number $N$ of bacteria, in thousands, in a culture $t$ hours after the start of an experiment is modelled by $N=kb^t$. A graph of $\log_{10}N$ against $t$ is a straight line passing through the points $(0,\,1.20)$ and $(4,\,2.00)$.

   (a)

   Show that $\log_{10}N=\log_{10}k+t\log_{10}b$.

   [3 marks]

   (b)

   Find the values of $k$ and $b$, correct to 3 significant figures.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A researcher records the hours of exercise per week, $x$, and the resting heart rate, $y$ beats per minute, of 40 adults who exercise between 1 and 10 hours per week. The regression line of $y$ on $x$ is $y=78-2.5x$.

   (a)

   (i) State which is the explanatory variable and which is the response variable.
   (ii) Interpret the gradient of the line.
   (iii) Predict the resting heart rate of an adult who exercises for 6 hours per week.
   (iv) A student uses the line to predict the heart rate for 20 hours of exercise per week. Comment on this prediction.

   [6 marks]

   (b)

   A newspaper claims that "exercising more reduces your resting heart rate". The data show a strong negative correlation, and the points for adults under 35 and adults aged 35 and over form two distinct groups, with the older group having higher heart rates for the same hours of exercise. Evaluate the claim.

   [6 marks]

   Total for question 4: 12 marks

End of questions