Let X1, X2, ..., X25 be independent and identically distributed (i.i.d.) Bernoulli(p) random variables, with 0 < p < 1. Let \(\bar{X}=\frac{1}{25} \sum_{i=1}^{25} X_i,\)
\(T_1=\left\{\begin{array}{cc} \frac{5(\bar{X}-0.5)}{\sqrt{\bar{X}(1-\bar{X})}}, & \text { if } 0<\bar{X}<1 \\ -5, & \text { if } \bar{X}=0 \\ 5, & \text { if } \bar{X}=1 \end{array}\right. \)
and T2 = 10 (X̅ - 0.5).
For testing H0 : p = 0.5 against H1 : p > 0.5, consider two tests ψ1 and ψ2 such that ψi rejects H0 if and only if Ti > 2, i = 1 and 2. If observed X̅ ∈ (0.5, 0.75), then which of the following statements are true?
1
If ψ1 rejects H0, then ψ2 also rejects H0
2
If ψ1 does not reject H0, then ψ2 also does not reject H0
3
If ψ2 rejects H0, then ψ1 also rejects H0
4
If ψ2 does not reject H0, then ψ1 also does not reject H0