What factors influence the size of the t-statistics( mean difference, variance, sample size) and which way?

The t-statistic is a measurement that determines the statistical significance of a mean difference between two groups. It is influenced by several factors, including the mean difference itself, the variance, and the sample size. Let's break down how each of these factors influences the size of the t-statistic and in which direction:

1. Mean Difference:
The mean difference refers to the difference between the means of the two groups being compared. A larger mean difference indicates a more significant difference between the groups. As the mean difference increases, the t-statistic also tends to increase, making it more likely to reject the null hypothesis.

2. Variance:
Variance measures the spread or dispersion of data within each group. A lower variance means that the data points are less scattered, while a higher variance indicates greater dispersion. When the variance is smaller, the t-statistic tends to be larger, leading to increased statistical significance and a higher likelihood of rejecting the null hypothesis.

3. Sample Size:
The sample size refers to the number of observations or individuals in each group. A larger sample size provides more information and reduces the uncertainty associated with estimating population parameters. A larger sample size generally results in a larger t-statistic, making it easier to detect a significant mean difference between the groups.

In summary, a larger mean difference, a smaller variance, and a larger sample size all tend to increase the size of the t-statistic, making it more likely to reject the null hypothesis and indicate a significant difference between the groups. Conversely, smaller mean differences, larger variances, and smaller sample sizes reduce the t-statistic's size and decrease the likelihood of finding statistical significance.