https://file.aiquickdraw.com/mathbot/file/1725246568884ttnqsvyj.jpg

The image you've provided contains a formula related to the **two-sample t-test** for comparing the means of two independent groups.

### Formula Breakdown:
The formula given is for the t-statistic:

\[
t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2} - 2\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}}
\]

Where:
- \(\bar{x}_1\) and \(\bar{x}_2\) are the sample means of groups 1 and 2, respectively.
- \(s_1^2\) and \(s_2^2\) are the sample variances of groups 1 and 2, respectively.
- \(n_1\) and \(n_2\) are the sample sizes of groups 1 and 2, respectively.

### Data from the Image:
From the image, the following values are noted:
- \(\sum x_1 = 174\)
- \(\sum (x_1 - \bar{x}_1)^2 = 146.8\)
- \(\sum x_2 = 303\)
- \(\sum (x_2 - \bar{x}_2)^2 = 348.7\)
- \(\bar{x}_1 = 5.8\)
- \(\bar{x}_2 = 10.1\)

### Steps to Calculate t-value:
1. **Calculate sample means**:
   - The values of \(\bar{x}_1\) and \(\bar{x}_2\) are provided.

2. **Calculate sample variances \(s_1^2\) and \(s_2^2\)**:
   - Use the sum of squares \(\sum (x_i - \bar{x}_i)^2\) and the sample size to calculate the variance.

3. **Substitute into the t-formula**:
   - Substitute the means, variances, and sample sizes into the formula to compute the t-value.

### Next Steps:
Would you like to proceed with calculating the t-value using the given data? If so, please confirm the sample sizes \(n_1\) and \(n_2\) if they are available.

---

#### Relative Questions:
1. How do sample size variations affect the t-value?
2. What is the significance of the degrees of freedom in a t-test?
3. How is the critical value for the t-test determined?
4. What assumptions must be met for using a two-sample t-test?
5. How would the formula change for a one-sample t-test?

#### Tip:
Always ensure that the data follows a normal distribution or that the sample sizes are large enough for the Central Limit Theorem to apply before using a t-test.

Learn how to calculate the t-value for a two-sample t-test using provided data. Includes step-by-step instructions on finding sample means, sample variances, and applying the t-test formula. Suitable for advanced high school students and beyond.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation