To calculate the degrees of freedom for a two-sample t-test (assuming unequal variances), you can use the following formula:

$\text{Degrees of Freedom (df)} = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{\left(\frac{s_1^2}{n_1}\right)^2}{n_1 - 1} + \frac{\left(\frac{s_2^2}{n_2}\right)^2}{n_2 - 1}}$

Where:

• $s_1$ and $s_2$ are the standard deviations of Class A and Class B, respectively.
• $n_1$ and $n_2$ are the number of students in Class A and Class B, respectively.

Given:

• $s_1 = 8.2$, $n_1 = 18$
• $s_2 = 5.2$, $n_2 = 21$

Let’s calculate the degrees of freedom.

First, we calculate the numerator:

$\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2} = \frac{8.2^2}{18} + \frac{5.2^2}{21}$

Then we calculate the denominator:

$\frac{\left(\frac{s_1^2}{n_1}\right)^2}{n_1 - 1} + \frac{\left(\frac{s_2^2}{n_2}\right)^2}{n_2 - 1}$

Finally, we divide the numerator by the denominator to find the degrees of freedom.

I'll compute this now.The degrees of freedom for this two-sample t-test is approximately 27.92.

Do you want more details or have any other questions?

Here are some related questions you might consider:

1. How do you perform a two-sample t-test using these degrees of freedom?
2. What is the significance of degrees of freedom in hypothesis testing?
3. How does the standard deviation affect the degrees of freedom?
4. What assumptions are made in the unequal variance t-test?
5. How would the degrees of freedom change if the sample sizes were different?
6. How do you interpret the degrees of freedom in the context of this problem?
7. What is the impact of using pooled variance versus unequal variances in a t-test?
8. Can you calculate the confidence interval for the difference in means between Class A and Class B?

Tip: When conducting statistical tests, always check if the assumptions of the test (such as normality or equality of variances) are met for your data.