F-Test Statistic in Simple Linear Regression | GATE ST 2025 | Problem-46 | RitwikMath

Описание: Crack the *F-test* for simple linear regression! Given 24 observations with summary statistics:
\(S_{XX} = 22.82\), \(S_{XY} = 15.48\), \(S_{YY} = 43.62\)

*Step-by-step calculation* for testing \(H_0: \beta = 0\) vs \(H_1: \beta \neq 0\):

Least squares estimator: \(\hat{\beta} = \frac{S_{XY}}{S_{XX}} = \frac{15.48}{22.82}\)
Regression Sum of Squares: \(SSR = \hat{\beta}^2 S_{XX} = \frac{15.48^2}{22.82} \approx 10.50\)
Error Sum of Squares: \(SSE = S_{YY} - SSR \approx 43.62 - 10.50 = 33.12\)
**F-statistic**:
\[
F = \frac{SSR/1}{SSE/22} = \frac{10.50}{33.12/22} \approx 6.98 \sim F_{1,22}
\]

Perfect for mastering *ANOVA table construction* and *regression significance testing* in GATE Statistics!

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