Section 28 Estimates: Hypothesis Testing

28.1 Estimates: Effects

\[ \large fm \leftarrow lm(SBP \sim BMI, \space data=BP) \]

\[ \large summary(fm) \]

	Estimate	Std. Error	t value	Pr(>|t|)
(Intercept)	36.2037	1.4801	24.4601	0
BMI	2.6323	0.0590	44.5940	0

28.2 Explanation

Statistical Model

\[ \large y_{i} = \beta_0 + \beta_1 x_{i} + \epsilon_{i} \]

Hypothesis

Intercept

\[ \large H_O: \beta_0 = 0 \] \[ \large H_A: \beta_0 \ne 0 \]

Regression coefficient

\[ \large H_O: \beta_1 = 0 \]

\[ \large H_A: \beta_1 \ne 0 \]

Test Statistic under the Null Hypothesis

Intercept

\[ \large \hat\beta_0 / SE(\hat\beta_0) \sim t_{df_{residual}} \]

Regression coefficient

\[ \large \hat\beta_1 / SE(\hat\beta_1) \sim t_{df_{residual}} \]