GATE 2024 IN Q.50: solution and answer

GATE 2024 · INQ.50 · 2 marks · NAT

The readings recorded from a 20-psig pressure gauge are given in the Table. The regression line obtained for the data is $y = 0.04 x + 10.32$ . The regression coefficient of determination, $R^{2} =$ ________ (rounded off to three decimal places).

| $x$ | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| $y$ (psig) | 10.3 | 10.5 | 10.4 | 10.5 | 10.5 |

GATE 2024 IN question 50: A two-row data table. — Figure from the GATE 2024 IN paper. It is part of the question.

Answer and solution

ANSWER

$R^{2} = 0.500$ .

Official answer: 0.450 to 0.550Source: Official answer key, GATE 2024 IN (IISc Bengaluru)

Means: $\overset{x}{ˉ} = \frac{15}{5} = 3$ and $\overset{y}{ˉ} = \frac{52.2}{5} = 10.44$ .
Deviations from the mean: $x_{i} - \overset{x}{ˉ} = - 2, - 1, 0, 1, 2$ and $y_{i} - \overset{y}{ˉ} = - 0.14, 0.06, - 0.04, 0.06, 0.06$ .
Sums of squares and cross-products: $S_{xx} = 4 + 1 + 0 + 1 + 4 = 10$ , $S_{x y} = 0.28 - 0.06 + 0 + 0.06 + 0.12 = 0.40$ , $S_{y y} = 0.0196 + 0.0036 + 0.0016 + 0.0036 + 0.0036 = 0.032$ .
Check against the given line: slope $b = S_{x y} / S_{xx} = 0.40/10 = 0.04$ and intercept $\overset{y}{ˉ} - b \overset{x}{ˉ} = 10.44 - 0.12 = 10.32$ . The given line is the least-squares fit, so the usual formula for $R^{2}$ applies.
Coefficient of determination for simple linear regression: $R^{2} = \frac{S _{x y}^{2}}{S _{xx} S _{y y}} = \frac{( 0.40 ) ^{2}}{10 \times 0.032} = \frac{0.16}{0.32} = 0.5$
Cross-check with residuals: the fitted values $\overset{y}{^} = 10.36, 10.40, 10.44, 10.48, 10.52$ give residuals $- 0.06, 0.10, - 0.04, 0.02, - 0.02$ , so $S S_{E} = 0.0036 + 0.0100 + 0.0016 + 0.0004 + 0.0004 = 0.016$ . With $S S_{T} = S_{y y} = 0.032$ , $R^{2} = 1 - \frac{S S _{E}}{S S _{T}} = 1 - \frac{0.016}{0.032} = 0.500$

KEY IDEA

$R^{2}$ is the fraction of the spread in $y$ that the line accounts for. For a least-squares line it equals $1 - S S_{E} / S S_{T}$ , which for a straight line simplifies to $S_{x y}^{2} / (S_{xx} S_{y y})$ (the square of the correlation coefficient). First confirm that the given line really is the least-squares fit; here $b = S_{x y} / S_{xx} = 0.04$ and $a = \overset{y}{ˉ} - b \overset{x}{ˉ} = 10.32$ .

WATCH OUT FOR

Common mistakes: reporting the correlation coefficient $r = 0.5 \approx 0.707$ in place of $R^{2}$ ; using raw sums of $y^{2}$ instead of deviations from the mean, which loses the small differences (of order $0.01$ ) to rounding; and dividing by $n - 1$ in only one of $S_{x y}$ , $S_{xx}$ or $S_{y y}$ . The ratio stays correct only when all three are treated the same way.

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GATE 2024 IN Q.50: solution and answer

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