Consider the following hypothesis test:

$H_0: \mu \leq 12$

$H_a: \mu > 12$

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

(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:

Given:

• Null hypothesis: $H_0: \mu \leq 12$
• Alternative hypothesis: $H_a: \mu > 12$ (right-tailed test)
• Sample size: $n = 25$
• Sample mean: $\bar{x} = 14$
• Sample standard deviation: $s = 4.32$
• Critical value: $t_{0.05} = 1.711$
• Significance level: $\alpha = 0.05$

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Statement (A): The test statistic is defined as $t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}$

This is the standard t-test statistic formula used when the population standard deviation is unknown. The denominator $\frac{s}{\sqrt{n}}$ is the standard error.

Statement (A) is correct.

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Statement (B): The value of the test statistic is 1.31

The standard error is:

$SE = \frac{s}{\sqrt{n}}$

$SE = \frac{4.32}{\sqrt{25}}$

$SE = \frac{4.32}{5}$

$SE = 0.864$

The t-statistic using $\mu_0 = 12$ from $H_0$ is:

$t = \frac{\bar{x} - \mu_0}{SE}$

$t = \frac{14 - 12}{0.864}$

$t = \frac{2}{0.864}$

$t = 2.31$

The actual test statistic is 2.31, not 1.31.

Statement (B) is incorrect.

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Statement (C): At $\alpha = 0.05$, the null hypothesis is rejected

For a right-tailed test, reject $H_0$ if $t > t_{\alpha}$.

Comparison:

• Calculated test statistic: $t = 2.31$
• Critical value: $t_{0.05} = 1.711$

Since $2.31 > 1.711$, the null hypothesis is rejected.

There is sufficient evidence at the 5% significance level to conclude that $\mu > 12$.

Statement (C) is correct.

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Statement (D): If the value of the t-statistic is less than $t_\alpha$, then null hypothesis is accepted

For a right-tailed test:

• If $t < t_{\alpha}$ → Fail to reject $H_0$
• If $t > t_{\alpha}$ → Reject $H_0$

This statement correctly describes the acceptance region for the test.

Statement (D) is correct.

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Statements (A), (C), and (D) are correct.

The answer is Option 3.