For a certain data test statistic ‘t’ is calculated as: \( |t| = \left| \displaystyle\frac{65-68}{\frac{4}{\sqrt{15}}} \right| = 2.90\), then select the correct option :
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\bar{x} = 65; \mu = 68, n = 16\)
For a small sample, the one-sample t statistic (with s the standard deviation of the sample, divisor n) is
\( \displaystyle t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n-1}}} \)
Comparing with the given expression \( \displaystyle \frac{65 - 68}{\frac{4}{\sqrt{15}}} \):
numerator \(\bar{x} - \mu = 65 - 68\), so \(\bar{x} = 65\) and \(\mu = 68\)
\(s = 4\)
\(n - 1 = 15\), so \(n = 16\)
Check: \( \displaystyle |t| = \frac{3\sqrt{15}}{4} \approx \frac{3 \times 3.873}{4} \approx 2.90 \).
Why the other options are incorrect
Option (1) \(\bar{x} = 68; \mu = 65, n = 16\) — Incorrect. It swaps the sample mean and the population mean; the numerator would then be written \(68 - 65\), not \(65 - 68\).
Option (3) \(\bar{x} = 15; n = 4, \mu = 3\) — Incorrect. These numbers do not even fit the formula; 15 is \(n - 1\) and 4 is s.
Option (4) \(\bar{x} = 65; \mu = 68, n = 14\) — Incorrect. With \(n = 14\) the denominator would contain \(\sqrt{13}\), not \(\sqrt{15}\).