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.

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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.