1.2 Type I and Type II Errors The next question is"how do we choose E " If e is to small we run the risk of rejecting Ho when it is true; we call this type i error. On the other hand if e is too large we run the risk of accepting Ho when it is false; we call this type ii error. That is, if we were to choose e too small we run a higher risk of committing a type i error than of committing a type il error and vice versa That is, there is a trade off between the probability of type i error, i.e Pr(x∈C1;b∈60)=a, and the probability B of type Ii error, i.e Pr(x∈Co;6∈1)=. Ideally we would like a= 6=0 for all oEe which is not possible for a fixed n. Moreover we cannot control both simultaneously because of the trade-off between them. The strategy adopted in hypothesis testing where a small value of a is chosen and for a given a, B is minimized. Formally, this amounts to choose a* such that Pr(x∈C1;b∈60)=a(6)≤afo6∈6 and Pr(x∈Co;b∈O1)=B(0), is minimized for 8∈61 y choosing C1 or Co appropriately. In the case of the above example if we were to choose a, say a*=0.05, then Pr(Xn-60|>;6=60)=0.05 How do we determine E, then? " The only random variable involved in the tatement X and hence it has to be its sampling distribution. For the above1.2 Type I and Type II Errors The next question is ”how do we choose ε ?” If ε is to small we run the risk of rejecting H0 when it is true; we call this type I error. On the other hand, if ε is too large we run the risk of accepting H0 when it is false; we call this type II error. That is, if we were to choose ε too small we run a higher risk of committing a type I error than of committing a type II error and vice versa. That is, there is a trade off between the probability of type I error, i.e. Pr(x ∈ C1; θ ∈ Θ0) = α, and the probability β of type II error, i.e. Pr(x ∈ C0; θ ∈ Θ1) = β. Ideally we would like α = β = 0 for all θ ∈ Θ which is not possible for a fixed n. Moreover we cannot control both simultaneously because of the trade-off between them. The strategy adopted in hypothesis testing where a small value of α is chosen and for a given α, β is minimized. Formally, this amounts to choose α ∗ such that Pr(x ∈ C1; θ ∈ Θ0) = α(θ) ≤ α ∗ for θ ∈ Θ0, and Pr(x ∈ C0; θ ∈ Θ1) = β(θ), is minimized for θ ∈ Θ1 by choosing C1 or C0 appropriately. In the case of the above example if we were to choose α, say α ∗ = 0.05, then Pr(|X¯ n − 60| > ε; θ = 60) = 0.05. ”How do we determine ε, then ?” The only random variable involved in the statement is X¯ and hence it has to be its sampling distribution. For the above 4