GATE 2023 EE Q.35: solution and answer

GATE 2023 · EEQ.35 · 1 mark · NAT

Three points in the $x$ - $y$ plane are $(- 1, 0.8)$ , $(0, 2.2)$ and $(1, 2.8)$ . The value of the slope of the best fit straight line in the least square sense is ________ (Round off to 2 decimal places).

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Answer and solution

ANSWER

The least-squares line is $y \approx 1.933 + 1.00 x$ , so the slope is $1.00$ .

Official answer: 0.9 to 1.1Source: Official answer key, GATE 2023 EE (IIT Kanpur)

Fit $y = a + b x$ by least squares. Minimising $\sum (y_{i} - a - b x_{i})^{2}$ with respect to $a$ and $b$ gives the normal equations $\sum y = na + b \sum x$ and $\sum x y = a \sum x + b \sum x^{2}$ , with $n = 3$ .
Tabulate the sums: $\sum x = - 1 + 0 + 1 = 0$ , $\sum y = 0.8 + 2.2 + 2.8 = 5.8$ , $\sum x^{2} = 1 + 0 + 1 = 2$ , $\sum x y = (- 1) (0.8) + (0) (2.2) + (1) (2.8) = - 0.8 + 2.8 = 2.0$ .
Solving the normal equations for the slope gives $b = \frac{n \sum x y - \sum x \sum y}{n \sum x ^{2} - ( \sum x ) ^{2}} = \frac{3 ( 2.0 ) - ( 0 ) ( 5.8 )}{3 ( 2 ) - 0 ^{2}} = \frac{6}{6} = 1$
Check: because $\sum x = 0$ , the two equations separate. The second gives $b = \frac{\sum x y}{\sum x ^{2}} = \frac{2.0}{2} = 1$ and the first gives $a = \frac{\sum y}{n} = \frac{5.8}{3} \approx 1.933$ .
So the best-fit line is $y \approx 1.933 + 1.00 x$ . Its slope, to 2 decimal places, is $1.00$ .

KEY IDEA

For a least-squares line $y = a + b x$ the slope is $b = \frac{n \sum x y - \sum x \sum y}{n \sum x ^{2} - ( \sum x ) ^{2}}$ . When the $x$ -values are symmetric about zero ( $\sum x = 0$ ), this reduces to $b = \sum x y / \sum x^{2}$ , and the intercept is just the mean of $y$ .

WATCH OUT FOR

Dropping the minus sign in $(- 1) (0.8)$ , which gives $\sum x y = 3.6$ and a wrong slope of $1.8$ . A second mistake is to regress $x$ on $y$ , which minimises horizontal distances instead of vertical ones and generally gives a different line. The question asks for $y$ fitted against $x$ .

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GATE 2023 EE Q.35: solution and answer

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