In a one-sample t-test, a random sample of 16 observations has mean 54 and sample standard deviation 8 (calculated with divisor n − 1). If the value of the t-statistic is 2.5, the test value \(\mu_0\) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 49
The one-sample t-statistic is
\( t = \displaystyle\frac{\bar{x} - \mu_0}{s/\sqrt{n}} \)
The standard error is
\( \displaystyle\frac{s}{\sqrt{n}} = \frac{8}{\sqrt{16}} = 2 \)
So
\( 2.5 = \displaystyle\frac{54 - \mu_0}{2} \)
\(54 - \mu_0 = 5\)
\(\mu_0 = 49\)
Why the other options are incorrect
Option (1) 59 — Incorrect. With \(\mu_0 = 59\), \(t = (54 - 59)/2 = -2.5\); the sign is wrong.
Option (2) 34 — Incorrect. This takes the standard error as 8, forgetting to divide s by \(\sqrt{n}\).
Option (4) 52.75 — Incorrect. This takes the standard error as 8 ÷ 16 = 0.5, dividing by n instead of \(\sqrt{n}\).