In a one-sample t-test of \(H_0\) about the population mean \(\mu\), with test value \(\mu_0\) and table value \(t_\alpha\) (\(t_\alpha \gt 0\)) for the chosen level of significance, which of the following statements are correct?
(A) The equality sign (=, ≤ or ≥) is placed in the alternative hypothesis \(H_1\).
(B) For \(H_1: \mu \ne \mu_0\), the test is two-tailed.
(C) For \(H_1: \mu \gt \mu_0\), \(H_0\) is rejected when the calculated t is less than \(t_\alpha\).
(D) For \(H_1: \mu \lt \mu_0\), \(H_0\) is rejected when the calculated t is less than \(-t_\alpha\).
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (B) and (D) only
The alternative hypothesis decides the tail: \(\mu \ne \mu_0\) gives a two-tailed test, \(\mu \gt \mu_0\) a right-tailed test and \(\mu \lt \mu_0\) a left-tailed test. \(H_0\) is rejected only when t falls in that tail beyond the table value.
So the equality sign is in \(H_0\); reject for \(t \gt t_\alpha\) in a right-tailed test and for \(t \lt -t_\alpha\) in a left-tailed test.
Checking the statements
(A) The equality sign (=, ≤ or ≥) is placed in the alternative hypothesis \(H_1\). — Incorrect. The equality sign is always in \(H_0\); \(H_1\) uses \(\ne\), \(\gt\) or \(\lt\).
(B) For \(H_1: \mu \ne \mu_0\), the test is two-tailed. — Correct. A difference in either direction is looked for.
(C) For \(H_1: \mu \gt \mu_0\), \(H_0\) is rejected when the calculated t is less than \(t_\alpha\). — Incorrect. In a right-tailed test \(H_0\) is rejected when t is greater than \(t_\alpha\).
(D) For \(H_1: \mu \lt \mu_0\), \(H_0\) is rejected when the calculated t is less than \(-t_\alpha\). — Correct. A left-tailed test rejects for t below \(-t_\alpha\).