We are conducting a one-sample t-test to test the claim. Since the population standard deviation is unknown and the sample size is small ($n = 14$), we will use the t-distribution.

Step 1: Formulate the hypotheses

• Null hypothesis ($H_0$): $\mu = 79.8$
• Alternative hypothesis ($H_1$): $\mu > 79.8$ (right-tailed test)

Step 2: Calculate the test statistic

The formula for the test statistic in a t-test is:

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

Where:

• $\bar{x}$ is the sample mean (90.8)
• $\mu_0$ is the population mean under the null hypothesis (79.8)
• $s$ is the sample standard deviation (10.6)
• $n$ is the sample size (14)

Substitute the values:

$t = \frac{90.8 - 79.8}{\frac{10.6}{\sqrt{14}}}$

First, compute the denominator:

$\frac{10.6}{\sqrt{14}} = \frac{10.6}{3.7417} \approx 2.834$

Now compute the t-statistic:

$t = \frac{90.8 - 79.8}{2.834} = \frac{11}{2.834} \approx 3.882$

Thus, the test statistic is approximately:

$t \approx 3.882$

Step 3: Find the p-value

To find the p-value, we look up the value of $t = 3.882$ in a t-distribution table with $n - 1 = 14 - 1 = 13$ degrees of freedom.

Using statistical software or a t-distribution table, the one-tailed p-value for $t = 3.882$ with 13 degrees of freedom is approximately:

$p \approx 0.0012$

Final Answers:

• Test statistic: $t \approx 3.882$
• p-value: $p \approx 0.0012$

Would you like further clarification or details?

Here are some related questions:

1. How do you interpret the p-value in hypothesis testing?
2. What are the critical values for a t-test at the 0.02 significance level?
3. How does sample size affect the t-statistic?
4. What assumptions are necessary for conducting a t-test?
5. What is the difference between a one-tailed and two-tailed t-test?

Tip: Always check if the assumptions for a t-test are met (e.g., normality of the population distribution).