Which of the following is not a rejection rule for one sample t-test: (Where \( \alpha \) is the level of significance)
(A) Reject null hypothesis \( H_{0} \) : if \( t\le-t_{\alpha} \).
(B) Reject null hypothesis \( H_{0} \) : if \( t\ge-t_{\alpha} \).
(C) Reject null hypothesis \( H_{0} \) : if \( t\ge t_{\alpha} \).
(D) Reject null hypothesis \( H_{0} \) : if \( t\le-t_{\alpha/2} \) or \( t\ge t_{\alpha/2} \).
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (B) only
In a t-test the null hypothesis is rejected when t falls in a tail of the distribution whose total probability is \( \alpha \):
left-tailed test: \( t\le-t_{\alpha} \)
right-tailed test: \( t\ge t_{\alpha} \)
two-tailed test: \( t\le-t_{\alpha/2} \) or \( t\ge t_{\alpha/2} \)
So the valid rejection rules are \( t\le-t_{\alpha} \), \( t\ge t_{\alpha} \), and \( t\le-t_{\alpha/2} \) or \( t\ge t_{\alpha/2} \); only (B) is not a rejection rule.
Checking the statements
(A) Reject null hypothesis \( H_{0} \) : if \( t\le-t_{\alpha} \). — Correct. This is the rule for a left-tailed test.
(B) Reject null hypothesis \( H_{0} \) : if \( t\ge-t_{\alpha} \). — Incorrect. \( t\ge-t_{\alpha} \) has probability \( 1-\alpha \); it is the acceptance region of the left-tailed test, not a tail.
(C) Reject null hypothesis \( H_{0} \) : if \( t\ge t_{\alpha} \). — Correct. This is the rule for a right-tailed test.
(D) Reject null hypothesis \( H_{0} \) : if \( t\le-t_{\alpha/2} \) or \( t\ge t_{\alpha/2} \). — Correct. This is the rule for a two-tailed test, with \( \frac{\alpha}{2} \) in each tail.