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H_0: $$[ H_0: \beta_2 = 0 ]$$

H_alternative: $$[ H_a: \beta_2 \neq 0 ]$$

t-score: $$[ t = \frac{\hat{\beta}_2}{\text{SE}(\hat{\beta}_2)} ]$$

$$
(\hat{\beta}_2)
$$

 coefficient for (X_2),

$$
(\text{SE}(\hat{\beta}_2))
$$

standard error

Question 2.b

$$
E(Y | X_1, X_2) = \alpha + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_1 X_2
$$

H_0 $$
H_0: \beta_3 = 0
$$

H_a $$
H_a: \beta_3 \neq 0
$$

t-score $$
t = \frac{\hat{\beta}_3}{\text{SE}(\hat{\beta}_3)}
$$

Question 2.c

$$
[SE(\beta_!) = \sqrt{\text{Var}(\hat{\beta}_!)}$$

stander error $$ (SE\beta_1)) $$ for the effect in $(X_1)$.  

$(\text{Var}(\hat{\beta}_1))$ is the coefficient for $(X_1)$

And the  $(\text{Var}(\hat{\beta}_1))$ is the coeficient variance for $(X_1)$.