Hypothesis Testing in Multiple Regression

Hard Multiple Regression Analysis Hypothesis Testing

A research analyst is conducting a multiple regression analysis to examine the effect of independent variables on the dependent variable, Y. The regression model is specified as follows:

$$ Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \epsilon $$

Where:

$$ Y $$ = the dependent variable
$$ X_1 $$ = an independent variable
$$ X_2 $$ = another independent variable
$$ \beta_0 $$ = the y-intercept
$$ \beta_1 $$ and $$ \beta_2 $$ = the coefficients of the independent variables
$$ \epsilon $$ = the error term

The analyst performs a hypothesis test to determine whether $\beta_1$ is significantly different from zero, stating the null hypothesis as $H_0: \beta_1 = 0$ and the alternative hypothesis as $H_a: \beta_1 \neq 0$. From the regression output, the t-statistic for $\beta_1$ is found to be 2.56, with a critical value of 2.45 (two-tailed) at a 5% significance level.

Based on this information, what should the analyst conclude about the significance of $\beta_1$?

The analyst should reject the null hypothesis, concluding that $\beta_1$ is statistically significant.

The analyst should fail to reject the null hypothesis, concluding that $\beta_1$ is not statistically significant.

The analyst cannot make a conclusion as the t-statistic value is greater than the critical value only in one tail.

Hint

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