How do you find the t statistic from t-value?

It appears there's a misunderstanding in the question. The terms "t-statistic" and "t-value" are often used interchangeably to refer to the same thing: the calculated value from a t-test. Therefore, you don't "find" the t-statistic from the t-value; they are the same thing.

To clarify, the t-statistic (or t-value) is calculated using a formula that depends on the specific type of t-test being performed. Here's a breakdown with examples:

One-Sample t-test

This test compares the mean of a single sample to a known population mean.

• Formula: t = (x̄ - μ) / (s / √n)

  • Where:
    • x̄ is the sample mean
    • μ is the population mean
    • s is the sample standard deviation
    • n is the sample size

Independent Two-Sample t-test (Unpooled or Pooled)

This test compares the means of two independent groups. There are two versions depending on whether you assume the variances of the two groups are equal.

• Unpooled (Welch's t-test - Variances NOT assumed equal):

  t = (x̄₁ - x̄₂) / √((s₁² / n₁) + (s₂² / n₂))

  • Where:
    • x̄₁ and x̄₂ are the sample means of the two groups
    • s₁² and s₂² are the sample variances of the two groups
    • n₁ and n₂ are the sample sizes of the two groups

• Pooled (Variances assumed equal):

  First, calculate the pooled variance: s_p² = ((n₁ - 1)s₁² + (n₂ - 1)s₂²) / (n₁ + n₂ - 2)

  Then, calculate the t-statistic: t = (x̄₁ - x̄₂) / (s_p√(1/n₁ + 1/n₂))

Paired Sample t-test

This test compares the means of two related groups (e.g., before and after measurements on the same subjects).

• Formula: t = d̄ / (s_d / √n)

  • Where:
    • d̄ is the average of the differences between the paired observations
    • s_d is the standard deviation of the differences
    • n is the number of pairs

Using the t-statistic

Once you calculate the t-statistic, you compare it to a critical t-value from a t-distribution table (or use statistical software to calculate a p-value). The critical t-value depends on the degrees of freedom and the chosen significance level (alpha). The degrees of freedom depend on the type of t-test:

• One-sample t-test: df = n-1
• Independent two-sample t-test (unpooled): df is calculated using a complex formula, often approximated by software.
• Independent two-sample t-test (pooled): df = n₁ + n₂ - 2
• Paired sample t-test: df = n-1

If the absolute value of the calculated t-statistic is greater than the critical t-value, you reject the null hypothesis.

In summary, the t-statistic is a calculated value used to perform hypothesis testing. It is not derived from a "t-value" – the t-statistic is the t-value in this context.