In a one-sample t-test, a random sample of 9 observations gives t = 1.2. If a random sample of 36 observations had given the same sample mean and the same sample standard deviation, the value of t would have been
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 2.4
The t-statistic can be written as
\( \displaystyle t = \frac{(\bar{x} - \mu_0)\sqrt{n}}{s} \)
With \(\bar{x}\), \(\mu_0\) and s unchanged, t is proportional to \(\sqrt{n}\).
When n changes from 9 to 36, \(\sqrt{n}\) changes from 3 to 6, that is, it doubles. So
\(t = 1.2 \times 2 = 2.4\)
Why the other options are incorrect
Option (1) 4.8 — Incorrect. This multiplies t by 4, the factor for n, but t changes with \(\sqrt{n}\).
Option (2) 1.2 — Incorrect. The standard error \(s/\sqrt{n}\) becomes smaller for the larger sample, so t does change.
Option (3) 0.6 — Incorrect. A larger sample makes the standard error smaller and t larger, not smaller.