Consider the following hypothesis test :
\( \mathrm{H_0}:\mu\le 35 \)
\( \mathrm{H_a}:\mu\gt 35 \)
A sample of 49 provided a sample mean of 38.2. The population standard deviation is 8. The value of test-statistic is,
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 2.8
The population standard deviation is known, so the test statistic is z:
\( \displaystyle z=\frac{\bar{x}-\mu_0}{\sigma/\sqrt{n}} \)
\( \displaystyle =\frac{38.2-35}{8/\sqrt{49}} \)
\( \displaystyle =\frac{3.2}{8/7} \)
\( \displaystyle =\frac{3.2\times 7}{8}=2.8 \)
Why the other options are incorrect
Option (1) 1.4 — Incorrect. The value is 2.8, not 1.4.
Option (2) −2.8 — Incorrect. −2.8 has the wrong sign; the sample mean 38.2 is above 35, so z is positive.
Option (3) −1.6 — Incorrect. −1.6 is negative, although \( \bar{x}\gt\mu_0 \) makes z positive.