How to find a P-value from a t-Score?

Finding the p-value involves determining the probability of observing a test statistic as extreme as, or more extreme than, the one calculated from the sample data, assuming the null hypothesis is true. The steps for finding the p-value depend on the type of statistical test being performed.

Here, we’ll provide a general guide for finding p-values in hypothesis testing using common statistical test, t-test.

Steps for calculating P-value from a T-Score

The p-value in a t-test represents the probability of observing a T-score as extreme as, or more extreme than, the one calculated from your sample data, assuming the null hypothesis () is true.

Steps to finding p-values in t-tests:

1. Calculate the T-score:

T-score is a numerical value that measures the difference between the observed sample mean () and the hypothesized mean () relative to the standard error of the mean (SEM). It indicates how much the sample mean is likely to vary from the true population mean due to random sampling.

The specific formula for the t-statistic depends on the type of t-test being performed.

Two common t-tests are as follows:

1. One-sample t-test:

Used to compare the mean of a single sample to a hypothesized mean.

Where,

• is the sample mean.
• is the hypothesized population mean.
• s is the sample standard deviation.
• n is the sample size.

2. Two-sample t-test:

Used to compare the means of two independent samples. There are two types:

• Independent Samples T-Test: Used when you want to compare the means of two independent groups to determine if there is a significant difference between them.
• Paired Samples T-Test: Used when you want to compare the means of two related groups (paired observations) to determine if there is a significant difference between them.

In this article, we’ll further proceed with Independent samples two t-test.

where,

• are the mean of 1st and 2nd sample.
• are the standard deviation of 1st and 2nd sample.
• are total number of observations in each sample.

2. Determine the degrees of freedom (df):

Degrees of freedom (df) are the number of independent values in your sample that contribute to the variability of the data.

The specific formula for One-sample t-test:

Formula for Two-Sample t-test:

3. Identify the appropriate t-distribution:

• The t-distribution is a theoretical probability distribution that describes the behavior of the t-score under the null hypothesis.
• The shape of the t-distribution depends on the degrees of freedom (df).

4. Find the p-value using the t-distribution:

Use a T-distribution table that provides the probability of obtaining a specific T-score value given the corresponding degrees of freedom (df) and the type of test (one-tailed or two-tailed).

5. Interpret the p-value:

Compare the p-value to the chosen significance level (), typically 0.05.

• If the p-value is less than :
  This indicates strong evidence against the null hypothesis and suggests a statistically significant difference.
• If the p-value is greater than \alpha" title="Rendered by QuickLaTeX.com" height="20" width="166" style="vertical-align: -2px;">:
  This fails to reject the null hypothesis and suggests no statistically significant difference.

Both one-sample t-tests and two-sample t-tests can be either left-tailed, right-tailed, or two-tailed depending on the specific research question and the directionality of the hypothesis.

In the realm of statistical analysis, the p-value stands as a pivotal metric, guiding researchers in drawing meaningful conclusions from their data. This article delves into the significance and computation of p-values in Python, focusing on the t-test, a fundamental statistical tool.