(c) Prove the Bonferroni inequality P(A1^c cap A2^c) ge 1 -...

(c) Prove the Bonferroni inequality P(A_1^c cap A_2^c) ge 1 - P(A_1) - P(A_2) and hence justify the following 100(1 - alpha)% joint confidence region for (eta_0, eta_1) eta_i in b_i pm B s{b_i}; i = 0,1. where B = t(1 - alpha/4; n - 2).

Transcript

00:01 Hello students, here we are asked to prove the bufferoni inequality which is probability of a1 complement intersection a2 complement is greater than or equal to 1 minus probability of a1 minus probability of a2 and we are also asked to justify 100 into 1 minus alpha percent joint confidence region for beta 1 and beta 2.

00:28 So, first of all probability of a1 union a2 is equal to probability of a1 plus probability of a2 minus probability of a1 intersection intersection a, so probability of a1 union a2 is less than or equal to probability of a1 plus probability of a2 this is because of a1 intersection a2 is greater than or equal to 0.

01:12 Then probability of a1 complement intersection a2 complement is equal to probability of a1 union a2 complement which is equal to 1 minus probability of a1 union a2, so this is equal to 1 minus probability of a1 minus a2 that is probability of a2, so we proved probability of a1 complement intersection a2 complement is greater than or equal to 1 minus probability a1 minus probability of a2 this minus not is equal to.

02:01 So, then answer for the second next question minus beta i plus b i by s of b i that is standard error which is equal to b i minus beta i by s of b i that follows t distribution with n minus 2 degrees of freedom where i is equal to 0 comma 1 and we know that t distribution formula is x bar minus mu by standard error.

02:33 So, probability of b i minus beta i by standard error of b i is greater than b which is equal to 2 into alpha by 4 which is equal to alpha by 2 this is 2 and this is alpha.

02:57 So, this we can represent the diagram this is minus b this is b this is alpha by 4 this is also alpha by 4 and this is 1 minus alpha by 2.

03:18 So, now probability of confidence intervals we are writing that is probability of b i belongs to b plus or minus b into standard error of b i where i from 0 to 1...

Question

(c) Prove the Bonferroni inequality P(A_1^c cap A_2^c) ge 1 - P(A_1) - P(A_2) and hence justify the following 100(1 - alpha)% joint confidence region for (eta_0, eta_1) eta_i in b_i pm B s{b_i}; i = 0,1. where B = t(1 - alpha/4; n - 2).

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(c) Prove the Bonferroni inequality P(A_1^c cap A_2^c) ge 1 - P(A_1) - P(A_2) and hence justify the following 100(1 - alpha)% joint confidence region for (eta_0, eta_1) eta_i in b_i pm B s{b_i}; i = 0,1. where B = t(1 - alpha/4; n - 2).