Which of the following statements are correct?
(A) The null hypothesis states that there is no difference between a population parameter and a specific value.
(B) If the null hypothesis is rejected, the alternative hypothesis is accepted.
(C) In a one-sample t-test with a sample of size n, the number of degrees of freedom is n.
(D) A statistical hypothesis is a statement about a sample statistic such as the sample mean \(\bar{x}\).
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A) and (B) only
A hypothesis is an assumption about the population. The null hypothesis \(H_0\) says there is no difference; the alternative hypothesis \(H_1\) says there is one. For a one-sample t-test the degrees of freedom are n − 1.
So \(H_0\) is the “no difference” statement, rejecting \(H_0\) means accepting \(H_1\), df = n − 1, and hypotheses are about population parameters.
Checking the statements
(A) The null hypothesis states that there is no difference between a population parameter and a specific value. — Correct. \(H_0\) is the statement of no difference, for example \(\mu = \mu_0\).
(B) If the null hypothesis is rejected, the alternative hypothesis is accepted. — Correct. The two hypotheses are opposite views, so rejecting one means accepting the other.
(C) In a one-sample t-test with a sample of size n, the number of degrees of freedom is n. — Incorrect. The degrees of freedom are n − 1.
(D) A statistical hypothesis is a statement about a sample statistic such as the sample mean \(\bar{x}\). — Incorrect. A hypothesis is about a population parameter such as \(\mu\); \(\bar{x}\) is known once the sample is drawn.