Some table values of t at 5% level of significance are given below.
| Degrees of freedom | 10 | 19 | 20 |
| One-tailed | 1.812 | 1.729 | 1.725 |
| Two-tailed | 2.228 | 2.093 | 2.086 |
For one-sample t-tests at 5% level of significance, which of the following statements are correct?
(A) For a sample of 11 observations, the table value for a two-tailed test is 2.228.
(B) For the same degrees of freedom, the two-tailed table value is greater than the one-tailed table value.
(C) For a right-tailed test with a sample of 21 observations, \(H_0\) is rejected if t > 1.725.
(D) For a sample of 20 observations, the table value for a two-tailed test is 2.086.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A), (B) and (C) only
For a sample of n observations the degrees of freedom are n − 1, so samples of 11, 20 and 21 observations use the rows for 10, 19 and 20 degrees of freedom.
In every column of the table, the two-tailed value is larger than the one-tailed value (2.228 > 1.812, 2.093 > 1.729, 2.086 > 1.725), because the 5% is split between the two tails.
So n = 11 uses 2.228 (two-tailed), n = 21 uses 1.725 (one-tailed), n = 20 uses 2.093 (two-tailed), and two-tailed values are the larger ones.
Checking the statements
(A) For a sample of 11 observations, the table value for a two-tailed test is 2.228. — Correct. df = 11 − 1 = 10, two-tailed value 2.228.
(B) For the same degrees of freedom, the two-tailed table value is greater than the one-tailed table value. — Correct. For example 2.228 > 1.812 for 10 degrees of freedom.
(C) For a right-tailed test with a sample of 21 observations, \(H_0\) is rejected if t > 1.725. — Correct. df = 20, one-tailed value 1.725, rejection in the right tail.
(D) For a sample of 20 observations, the table value for a two-tailed test is 2.086. — Incorrect. df = 19, so the value is 2.093; 2.086 is for 20 degrees of freedom.