2016 H2 Mathematics Paper 2 Question 8

Linear Correlation and Regression

Answers

(i)

(ii)

The scatter diagram shows that, as x{x} increases, y{y} increases at a decreasing rate, which is not consistent with a linear model.

(iii)

c{c} is negative because y{y} increases as x{x} increases.
As x→∞,{x \to \infty, } y{y} approaches a positive value so d{d} is positive.

(iv)

r=−0.980{r=-0.980}
c=−17.5{c=-17.5}
d=91.8{d=91.8}

(v)

y=85.9{y=85.9}
The estimate is reliable because
  • x=3{x=3} lies within the given data range 1≤x≤50.{1\leq x \leq 50.}
  • ∣r∣≈1{|r|\approx 1} which suggest a strong linear correlation between y{y} and 1x.{\frac{1}{x}.}

Full solutions

(i)

(ii)

The scatter diagram shows that, as x{x} increases, y{y} increases at a decreasing rate, which is not consistent with a linear model. ■{\blacksquare}

(iii)

c{c} is negative because y{y} increases as x{x} increases. ■{\blacksquare}
As x→∞,{x \to \infty, } y{y} approaches a positive value so d{d} is positive. ■{\blacksquare}

(iv)

Using a GC,
r=−0.980 (3 sf)  ■c=−17.484=−17.5 (3 sf)  ■d=91.750=91.8 (3 sf)  ■\begin{align*} r &= -0.980 \textrm{ (3 sf)} \; \blacksquare \\ c &= -17.484 \\ &= -17.5 \textrm{ (3 sf)} \; \blacksquare \\ d &= 91.750 \\ &= 91.8 \textrm{ (3 sf)} \; \blacksquare \\ \end{align*}

(v)

When x=3,{x=3,}
y=−17.4843+91.750=85.9 (3 sf)  ■\begin{align*} y &= \frac{-17.484}{3} + 91.750 \\ &= 85.9 \textrm{ (3 sf)} \; \blacksquare \end{align*}
The estimate is reliable because
  • x=3{x=3} lies within the given data range 1≤x≤50.  ■{1\leq x \leq 50. \; \blacksquare}
  • ∣r∣≈1{|r|\approx 1} which suggest a strong linear correlation between y{y} and 1x.  ■{\frac{1}{x}. \; \blacksquare}