T-Test Calculator

What is a T-Test?

A t-test is a statistical hypothesis test used to determine if there is a significant difference between the means of two groups. It is particularly useful when dealing with small sample sizes or when the population standard deviation is unknown.

Formula and Its Meaning

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

\[t = \frac{\bar{x_1} - \bar{x_2}}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}\]

Where:

• \(\bar{x_1}\) and \(\bar{x_2}\) are the means of the two groups
• \(s_1^2\) and \(s_2^2\) are the variances of the two groups
• \(n_1\) and \(n_2\) are the sample sizes of the two groups

Calculation Steps

1. Calculate the mean of each group
2. Calculate the variance of each group
3. Determine the sample sizes
4. Plug these values into the t-statistic formula
5. Calculate the result

Example Calculation

Let's calculate for two groups:

Group 1: 5, 20, 40, 80, 100

Group 2: 1, 29, 46, 78, 99

1. Means: \(\bar{x_1} = 49\), \(\bar{x_2} = 50.6\)
2. Variances: \(s_1^2 = 1530\), \(s_2^2 = 1334.8\)
3. Sample sizes: \(n_1 = n_2 = 5\)
4. T-statistic: \[t = \frac{49 - 50.6}{\sqrt{\frac{1530}{5} + \frac{1334.8}{5}}} \approx -0.0731\]

Visual Representation

This scatter plot represents both data sets. Red points and dashed line represent the first group, green represents the second group. The closeness of the mean lines illustrates the small t-value, indicating little difference between the groups.