The problem requires calculating the value of the test statistic $t$ for a hypothesis test.

Given Information:

• Null Hypothesis: $H_0: \mu = 20$
• Alternative Hypothesis: $H_1: \mu > 20$
• Sample Size ($n$) = 39
• Sample Mean ($\bar{x}$) = 24
• Sample Standard Deviation ($s$) = 2

Formula for the t-test statistic:

The t-test statistic is calculated as: $t = \frac{\bar{x} - \mu}{s / \sqrt{n}}$ where:

• $\bar{x}$ is the sample mean,
• $\mu$ is the population mean under the null hypothesis,
• $s$ is the sample standard deviation, and
• $n$ is the sample size.

Substituting the values:

1. Difference between the sample mean and the hypothesized mean: $\bar{x} - \mu = 24 - 20 = 4$
2. Standard error of the mean: $\frac{s}{\sqrt{n}} = \frac{2}{\sqrt{39}} \approx \frac{2}{6.244} \approx 0.32$
3. Calculating $t$: $t = \frac{4}{0.32} \approx 12.50$

Answer:

The value of the test statistic $t$ is approximately $12.50$.

Would you like further details or explanations on hypothesis testing or related topics?

Related Questions:

1. How do we interpret the results of a t-test in hypothesis testing?
2. What is the significance of the degrees of freedom in a t-test?
3. How do we determine the critical t-value for this test?
4. What are Type I and Type II errors in hypothesis testing?
5. How does sample size impact the test statistic and the conclusion of a hypothesis test?

Tip:

Always ensure your sample size is large enough to approximate the normal distribution in t-tests, which helps in obtaining reliable results.