In hypothesis testing, the Bonferroni correction is a method to address the problem of multiple comparisons. When testing multiple hypotheses simultaneously, the Bonferroni correction adjusts the significance level for each individual test in order to maintain the overall significance level.
Given you're testing three hypotheses \( H_0: (\beta_1, \beta_2, \beta_3) = (0, 0, 0) \) against \( H_1: (\beta_1, \beta_2, \beta_3) \neq (0, 0, 0) \), and your overall significance level \( \alpha = 0.01 \), the Bonferroni correction would lead to the following calculation:
- The number of hypotheses being tested \( m = 3 \).
- To preserve the overall significance level \( \alpha \), the adjusted significance level for each individual hypothesis test, \( \alpha_i \), would be:
\[ \alpha_i = \frac{\alpha}{m} = \frac{0.01}{3} \approx 0.00333333... \]
In the context of the Bonferroni test, \( \gamma \) would represent this adjusted significance level. Therefore, you can express \( \gamma \) as:
\[ \gamma \approx 0.00333. \]
To input this rounding to the nearest \( 10^{-5} \):
\[ \gamma \approx \frac{1}{300} = 0.00333. \]
So, the numerical value of the significance level \( \gamma \) adjusted for the Bonferroni correction for this test is:
\[ \boxed{0.00333}. \]