Let's solve the hypothesis test step by step.

Given:

• Sample size ($n$) = 18
• Sample mean ($\bar{x}$) = 12.4
• Sample standard deviation ($s$) = 3.86
• Null hypothesis ($H_0$): $\mu \geq 14.1$
• Alternative hypothesis ($H_1$): $\mu < 14.1$
• This is a left-tailed test because $H_1$ states that the mean is less than 14.1.

---

Test statistic formula for $t$-test:

$t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$ where:

• $\bar{x}$ = sample mean
• $\mu_0$ = population mean under $H_0$
• $s$ = sample standard deviation
• $n$ = sample size

---

Substituting the values:

1. $\bar{x} = 12.4$
2. $\mu_0 = 14.1$
3. $s = 3.86$
4. $n = 18$

$t = \frac{12.4 - 14.1}{3.86 / \sqrt{18}}$

Step 1: Compute the denominator ($s / \sqrt{n}$):

$s / \sqrt{n} = 3.86 / \sqrt{18} = 3.86 / 4.2426 \approx 0.910$

Step 2: Compute the numerator ($\bar{x} - \mu_0$):

$\bar{x} - \mu_0 = 12.4 - 14.1 = -1.7$

Step 3: Calculate $t$:

$t = \frac{-1.7}{0.910} \approx -1.868$

---

Final Answer:

$t = -1.868$

Let me know if you'd like further details on interpreting this test or need help with p-values!

---

Related Questions:

1. What is the critical value for this test at a 0.05 significance level?
2. How do we interpret the $t$-test result in the context of hypothesis testing?
3. What assumptions must be satisfied to use the $t$-test in this scenario?
4. How does the sample size affect the test statistic and conclusions?
5. Can this test be conducted using a $z$-test instead, and why or why not?

---

Tip:

Always check the direction of the alternative hypothesis ($H_1$) to correctly determine whether the test is one-tailed or two-tailed.