A random sample of eight pairs of values of $x$ and $y$ is used to obtain the following equations of the regression lines of $y$ on $x$ and of $x$ on $y$ respectively.
$y=-\frac{7}{10}x+\frac{151}{10}$, $x=-\frac{7}{6}y+20$
Seven of the pairs of data are given in the table.
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Find the eighth pair of values of $x$ and $y$. [5]
Determine the value of the product moment correlation coefficient, and say what it leads you to expect about the scatter diagram for this sample. [4]
Let $Y$ be the value obtained by substituting a sample value of $x$ into the equation of the regression line of $y$ on $x$. Evaluate $Y$ for each of the eight values of $x$ and verify that ${\sum (y - Y)^2 = 8.8}$.[4]
For each of the eight sample values of $x$, ${Y}'$ is given by ${Y}'=a+bx$, where $a$ and $b$ are any constants. What can you say about the value of $\sum{{{\left( y-{Y}' \right)}^{2}}}$?[2]

A random sample of eight pairs of values of $x$ and $y$ is used to obtain the following equations of the regression lines of $y$ on $x$ and of $x$ on $y$ respectively.

$y=-\frac{7}{10}x+\frac{151}{10}$, $x=-\frac{7}{6}y+20$

Seven of the pairs of data are given in the table.

Find the eighth pair of values of $x$ and $y$. 

Determine the value of the product moment correlation coefficient, and say what it leads you to expect about the scatter diagram for this sample. 

Let $Y$ be the value obtained by substituting a sample value of $x$ into the equation of the regression line of $y$ on $x$. Evaluate $Y$ for each of the eight values of $x$ and verify that ${\sum (y - Y)^2 = 8.8}$.

For each of the eight sample values of $x$, ${Y}'$ is given by ${Y}'=a+bx$, where $a$ and $b$ are any constants. What can you say about the value of $\sum{{{\left( y-{Y}' \right)}^{2}}}$?

Correlation & Regressions - General

Correlation & Regressions

International Baccalaureate Higher Level Mathematics

IB HL Math

N2003 P2 Q11

Let the eighth pair be $(a, b)$.

Regression lines: $y = -\dfrac{7}{10}x + \dfrac{151}{10}$ and $x = -\dfrac{7}{6}y + 20$

$\dfrac{7}{10}x + y = \dfrac{151}{10}$ — ①

$x + \dfrac{7}{6}y = 20$ — ②

Solving ① and ②, from GC: $x = 13$, $y = 6$.

$\Rightarrow \bar{x} = 13$, $\bar{y} = 6$

$\dfrac{10 + 11 + \cdots + a}{8} = 13 \Rightarrow \dfrac{94 + a}{8} = 13 \Rightarrow a = 10$

$\dfrac{9 + 8 + 7 + \cdots + b}{8} = 6 \Rightarrow \dfrac{40 + b}{8} = 6 \Rightarrow b = 8$

$\therefore$ Eighth pair of values are $(10, 8)$.

From GC (LinReg): $r = -0.9036$

$r = -0.904$ (3 s.f.)

This value is close to $-1$, indicating a strong negative linear correlation between $x$ and $y$. This suggests that the scatter diagram would show points lying close to the regression lines with a downward slope, i.e. as $x$ increases, $y$ decreases.

| $x$ | 10 | 11 | 12 | 11 | 17 | 14 | 19 | 10 |
|---|---|---|---|---|---|---|---|---|
| $y$ | 9 | 8 | 7 | 6 | 5 | 4 | 1 | 8 |
| $Y$ | 8.1 | 7.4 | 6.7 | 7.4 | 3.2 | 5.3 | 1.8 | 8.1 |

$\sum(y - Y)^2 = (9 - 8.1)^2 + (8 - 7.4)^2 + (7 - 6.7)^2 + \cdots + (8 - 8.1)^2 = 8.8$ (verified)

The regression line $y = -\dfrac{7}{10}x + \dfrac{151}{10}$ minimises $\sum(y - Y')^2$.

Since $\sum(y - Y)^2$ is the sum of least squares for the data set,

$\sum(y - Y')^2 > \sum(y - Y)^2$

$\Rightarrow \sum(y - Y')^2 > 8.8$

N2003 P2 Q11

N2003 P2 Q11

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