Consider the following hypothesis test
\( H_0:\mu\le 3530 \)
\( H_a:\mu\gt 3530 \)
A sample of 100 provided a sample mean of \( \bar{x}=3740 \) and population standard deviation S = 700. What is the value of t-statistic?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 3
The test statistic for a mean is
\( \displaystyle t=\frac{\bar{x}-\mu_0}{S/\sqrt{n}} \)
Here \( \bar{x}=3740 \), \( \mu_0=3530 \), S = 700 and n = 100.
\( \displaystyle \bar{x}-\mu_0=3740-3530=210 \)
\( \displaystyle \frac{S}{\sqrt{n}}=\frac{700}{\sqrt{100}} \)
\( \displaystyle =\frac{700}{10}=70 \)
\( \displaystyle t=\frac{210}{70}=3 \)
Why the other options are incorrect
Option (2) \( -3 \) — Incorrect. The sign is wrong: \( \bar{x}=3740 \) is above \( \mu_0=3530 \), so t is positive.
Option (3) 7 — Incorrect. The value is 3, not 7.
Option (4) 21 — Incorrect. 21 = 210 ÷ 10 divides the difference by \( \sqrt{n} \) only; it must be divided by the standard error \( \frac{S}{\sqrt{n}}=70 \).