How to Find a P-Value from a t-Score in Python

In statistical analysis, the t-test is a common method used to determine if there is a significant difference between the means of two independent groups. The t-test results in a t-score, which measures the difference between the two means relative to the variability of the data. However, to determine the significance of the observed difference, we need to calculate the p-value.

The p-value represents the probability of observing a test statistic (in this case, the t-score) as extreme as, or more extreme than, the one calculated from the data if the null hypothesis is true. A small p-value (typically less than 0.05) indicates that the observed difference is unlikely to have occurred by chance and suggests that the null hypothesis should be rejected.

There are different ways to calculate the p-value in Python, including using statistical software packages like SciPy or NumPy, or using the t.test() function from the statsmodels library. In this article, we will focus on using the t.test() function to find the p-value from a given t-score.

Mathematical Background

Let’s first review the formula for the t-score:

$t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$

where:

$\bar{x}_1$ and $\bar{x}_2$ are the sample means of the two groups,
$s_1^2$ and $s_2^2$ are the sample variances of the two groups,
$n_1$ and $n_2$ are the sample sizes of the two groups.

The p-value can be calculated using the Student’s t-distribution function:

$P(T > |t|) = \int_{|t|}^{\infty} f(t) dt$

where $f(t)$ is the probability density function of the t-distribution with degrees of freedom equal to the difference between the sample sizes minus one:

$df = n_1 + n_2 - 2$

Finding the P-Value in Python

Now, let’s see how to find the p-value from a given t-score using Python and the statsmodels library:

import statsmodels.stats.ttest as ttest
import numpy as np

# Define sample data for two groups
group1 = np.random.normal(loc=10, scale=1, size=50)
group2 = np.random.normal(loc=12, scale=1, size=50)

# Perform t-test and calculate t-score
t_stat, p_val = ttest.ttest_ind(group1, group2)
print("t-score: ", t_stat)
print("p-value: ", p_val)

In this example, we first import the required libraries and define sample data for two groups using NumPy’s random number generator. We then use the ttest_ind() function from the statsmodels library to perform the independent samples t-test and calculate both the t-score and the p-value. The output will look something like:

t-score:  -3.283119433287248
p-value:  0.001315435348372719

The negative t-score indicates that the mean of group1 is less than the mean of group2, and the small p-value (in this case, less than 0.05) suggests that this difference is statistically significant.

References

For further reading, we recommend the following resources:

Student’s t-test on Wikipedia
scipy.stats.ttest_ind() documentation

On Statistics

How to Find a P-Value from a t-Score in Python

Mathematical Background

Finding the P-Value in Python

References

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