Suppose we instead wish to test the hypotheses H

Question

0

:(β
1

,β
2

,β
3

)=(0,0,0),H
1

:(β
1

,β
2

,β
3

)

=(0,0,0). Let γ be some appropriate value corresponding to the significance level, to be determined later. Choose all ψ that correctly represents the Bonferroni Test of H
0

against H
1

.
ψ=1{
3
∣
ϕ
^


1

+
β
^


2

+
β
^


3

∣

>q
γ

}
ψ=1
⎩
⎨
⎧


σ
2
/τ


max(
∣
∣


β
^


^


1


∣
∣

,
β
^


2

∣,∣
β
^


3

)

>q
γ


⎭
⎬
⎫


ψ=∏
i=1
3

1{
σ
2
/τ


∣
β
^


i

∣

>q
γ

}
ψ=1{
∣
∣


β
^


1

−
β
^


2

−
β
^


3


∣
∣

>q
γ

}
ψ=1{(
∣
∣


β
^


1

/
σ
2
/τ


∣
∣

>q
γ

) and (
∣
∣


β
^


2

/
σ
2
/τ


∣
∣

>q
γ

) and (
∣
∣


β
^


3

/
σ
2
/τ


∣
∣

>q
γ

)}
ψ=1{(
∣
∣


β
^


1

/
σ
2
/τ


∣
∣

>q
γ

) or (
∣
∣


β
^


2

/
σ
2
/τ


∣
∣

>q
γ

) or (
∣
∣


β
^


3

/
σ
2
/τ


∣
∣

>q
γ

)}

In the Bonferroni test of significance level α=0.01 for testing this particular H
0

against H
1

, what is the numerical value of γ ? Input a fraction or round to the nearest 10
−5
, if necessary.

Answer 1

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:

1. The number of hypotheses being tested \( m = 3 \).
2. 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:

\[ \boxed{0.00333}. \]