Consider the following hypothesis test:
\(H_0: \mu \le 12\)
\(H_a: \mu \gt 12\)
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 \(t=\displaystyle\frac{\bar x-\mu}{\frac{S}{\sqrt{n}}}\).
(B) The value of the test statistic is 1.31.
(C) At \(\alpha = 0.05\), the null hypothesis is rejected.
(D) If the value of the t-statistic is less than \(t_\alpha\), then null hypothesis is accepted.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A), (C) and (D) only
Since \(H_a: \mu \gt 12\), this is a right-tailed one-sample t-test with n = 25, \(\bar x = 14\), S = 4.32 and n − 1 = 24 degrees of freedom. H0 is rejected when \(t \gt t_\alpha\). The test statistic is
\( \displaystyle t = \frac{14 - 12}{\frac{4.32}{\sqrt{25}}} = \frac{2}{0.864} = 2.31 \)
So \(t = 2.31\), which is greater than \(t_{0.05} = 1.711\).
Checking the statements
(A) The test statistic is defined as \(t=\displaystyle\frac{\bar x-\mu}{\frac{S}{\sqrt{n}}}\). — Correct. Here S is the sample standard deviation computed with divisor n − 1, so the standard error of the mean is \(S/\sqrt{n}\).
(B) The value of the test statistic is 1.31. — Incorrect. The value is 2.31.
If the standard deviation is taken with divisor n instead, the same test uses \(s/\sqrt{n - 1}\) and gives \(t = 2/(4.32/\sqrt{24}) = 2.27\). Either way t is not 1.31.
(C) At \(\alpha = 0.05\), the null hypothesis is rejected. — Correct. \(2.31 \gt 1.711\), so t lies in the rejection region.
(D) If the value of the t-statistic is less than \(t_\alpha\), then null hypothesis is accepted. — Correct. In a right-tailed test, H0 is rejected only when \(t \gt t_\alpha\); a value below \(t_\alpha\) lies outside the rejection region, so H0 is accepted (not rejected).