Consider the following hypothesis test:

$H_0: μ≤12$
$H_a:μ> 12$

A sample of 25 provided a sample mean $\bar x = 14$ and a sample standard deviation S = 4.32. If $t_{0.05} = 1.711$, then which of the following is correct?

(A) The test statistic is defined as $t=\frac{\bar x-μ}{\frac{S}{\sqrt{n}}}$
(B) The value of the test statistic is 1.31.
(C) At $α = 0.05$, the null hypothesis is rejected.
(D) If the value of the t-statistic is less than $t_α$, then null hypothesis is accepted.

Choose the correct answer from the options given below:

(A), (C) and (D) only

The correct answer is Option (3) → (A), (C) and (D) only

Given

$H_0:\mu\le12,\;H_a:\mu>12$

$n=25,\;\bar{x}=14,\;s=4.32$

Test statistic

$t=\frac{\bar{x}-\mu}{\frac{s}{\sqrt{n}}}$

$t=\frac{14-12}{\frac{4.32}{\sqrt{25}}}=\frac{2}{\frac{4.32}{5}}=\frac{2}{0.864}=2.315$

Given $t_{0.05}=1.711$

Since $2.315>1.711$, reject $H_0$ at $0.05$ level.

Check options

(A) Correct.

(B) Incorrect, $t=2.315$.

(C) Correct.

(D) Correct decision rule for right tailed test.

The correct options are (A), (C) and (D).