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

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.

What is the P-value?

It defines the probability of the result taking place from the sample space by chance. P-value varies from 0 to 100%. Note that a lower p-value is considered good as it implies that a result didn’t take place by chance.

The strength of hypothesis testing is indicated by the p-value. We develop hypotheses based on statistical models, and we use the p-value to assess the validity of the models. Using the T-test is one method of obtaining the p-value.

Example of P-value

Let’s understand the p-value in depth with the help of the scenario.

A company wants to test if its new marketing campaign increases brand awareness. They split their target audience into two groups, one exposed to the new campaign and the other not. After a campaign period, they measure brand awareness in both groups.

• The null hypothesis (H₀) states that the new campaign does not affect brand awareness. In other words, the observed difference between the two groups is just due to random chance.
• The alternative hypothesis (H₁) states that the new campaign does affect brand awareness.

P-value comes into play here. It represents the probability of obtaining the observed difference in brand awareness between the two groups, or a more extreme difference, assuming the null hypothesis is true.

• Lower p-value: This indicates a lower probability of observing the difference under the null hypothesis, meaning the observed difference is unlikely to be due to chance alone. This strengthens the evidence against the null hypothesis and supports the alternative hypothesis.
• Higher p-value: This indicates a higher probability of observing the difference under the null hypothesis, meaning the observed difference might be due to chance. This weakens the evidence against the null hypothesis and fails to support the alternative hypothesis.

In the marketing campaign example,

• if the p-value is very small (e.g., 0.01), it suggests the observed increase in brand awareness is unlikely to occur by chance. This provides strong evidence that the new campaign actually increased brand awareness.
• If the p-value is relatively large, the observed difference might be due to chance, weakening the evidence against the null hypothesis.

If the p-value is below the significance level(0.05 in most cases), you reject the null hypothesis and conclude that the new campaign had a significant effect on brand awareness.

Therefore, a lower p-value is better because it indicates stronger evidence against the null hypothesis in a statistical test stating that there is no significant difference between two groups or that there is no relationship between two variables.

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 :
  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.

p-value python

How to find P-value from a t-Score using Python

In Python, p-value can be calculated using the scipy.stats module. Scipy is a python library used for scientific computation. It provides us scipy.stats.t.sf() function to compute the p-value. 

Syntax to install scipy library in python:

pip3 install scipy

Syntax for scipy.stats.t.sf() function:

scipy.stats.t.sf(abs(t_score), df=degree_of_freedom

Parameters:

• t_score: It signifies the t-score
• degree_of_freedom: It signifies the degrees of freedom

P-value for a One-sample T-test

Let’s consider a scenario where we have a sample of exam scores from a group of students, and we want to test whether the average exam score is significantly different from a population mean. The average exam score for a population of students is known to be 75 in a sample of 250 students.

Python3

import numpy as np
from scipy.stats import t


def one_sample_t_test(sample, population_mean, alpha=0.05, tail="two"):
    # Step 1: Calculate T-score
    sample_mean = np.mean(sample)
    sample_std = np.std(sample, ddof=1)
    sample_size = len(sample)

    t_score = (sample_mean - population_mean) / \
        (sample_std / np.sqrt(sample_size))

    # Step 2: Determine degrees of freedom
    df = sample_size - 1

    # Step 3: Identify the appropriate t-distribution
    # No need to explicitly specify degrees of freedom for one-sample t-test in scipy.stats.t

    # Step 4: Find the p-value
    if tail == "two":
        p_value = t.sf(np.abs(t_score), df) * 2  # for two-tailed test
    elif tail == "left":
        p_value = t.sf(t_score, df)  # for left-tailed test
    elif tail == "right":
        p_value = t.sf(-t_score, df)  # for right-tailed test
    else:
        raise ValueError(
            "Invalid tail argument. Use 'two', 'left', or 'right'.")

    # Step 5: Interpret the p-value
    print("P-value:", p_value)

    if p_value < alpha:
        print(
            "Reject the null hypothesis. There is a statistically significant difference.")
    else:
        print("Fail to reject the null hypothesis. There is no statistically significant difference.")


# Let's generate a sample for experiment
np.random.seed(42)
# Generating a sample
sample_data = np.random.normal(loc=77, scale=10, size=250)
population_mean = 75

# Example for a two-tailed test
one_sample_t_test(sample_data, population_mean, tail="two")

Output:

P-value: 0.0013870092433008773
Reject the null hypothesis. There is a statistically significant difference.

P-value for a Two-sample T-test Independence

Suppose you are a data analyst working for a company that has two different methods for manufacturing a certain type of product. You want to investigate whether there is a significant difference in the average quality of the product produced by the two methods. To do this, you collect samples from each manufacturing method and perform a two-sample t-test.

• Sample 1: Quality scores from 100 products manufactured using Method 1.
• Sample 2: Quality scores from 120 products manufactured using Method 2.

Python3

import numpy as np
from scipy.stats import t

# Step 1: Calculate the T-score
def calculate_t_score(sample1, sample2):
    mean1 = np.mean(sample1)
    mean2 = np.mean(sample2)
    std1 = np.std(sample1, ddof=1)
    std2 = np.std(sample2, ddof=1)
    n1 = len(sample1)
    n2 = len(sample2)

    t_score = (mean1 - mean2) / np.sqrt((std1**2 / n1) + (std2**2 / n2))
    return t_score

# Step 2: Determine the degrees of freedom (df)
def calculate_degrees_of_freedom(sample1, sample2):
    n1 = len(sample1)
    n2 = len(sample2)
    df = n1 + n2 - 2  # For a two-sample t-test
    return df

# Step 3: Identify the appropriate t-distribution
# (The scipy.stats.t distribution is used, which automatically considers the degrees of freedom)

# Step 4: Find the p-value
def calculate_p_value(t_score, df):
    p_value = 2 * (1 - t.cdf(np.abs(t_score), df))
    return p_value

# Step 5: Interpret the p-value
def interpret_p_value(p_value, alpha=0.05):
    if p_value < alpha:
        return "Reject the null hypothesis. There is a statistically significant difference."
    else:
        return "Fail to reject the null hypothesis. There is no statistically significant difference."

# Generate two independent samples for Example
np.random.seed(42)
sample1 = np.random.normal(loc=50, scale=10, size=100)
sample2 = np.random.normal(loc=45, scale=12, size=120)

t_score = calculate_t_score(sample1, sample2)
df = calculate_degrees_of_freedom(sample1, sample2)
p_value = calculate_p_value(t_score, df)
result = interpret_p_value(p_value)

print("p-value:", p_value)
print(result)

Output:

p-value: 0.04126391962537701
Reject the null hypothesis. There is a statistically significant difference.

Conclusion

Understanding and calculating p-values in Python using the t-test are crucial for making informed decisions in hypothesis testing. A lower p-value strengthens evidence against the null hypothesis, supporting the validity of statistical models.

Frequently Based Questions(FAQs) on P-Value

Q. Why we calculate the p-value?

Calculating the p-value is essential for hypothesis testing. It assesses the likelihood of observed results under the null hypothesis. A low p-value provides strong evidence against the null hypothesis, aiding in valid statistical inferences.

Q. How to compute the p-value and t value in Python?

P-value: Probability of obtaining observed results assuming the null hypothesis is true. Calculated using scipy.stats.t.sf(abs(t_score), df=degree_of_freedom).

T-value: Measures the difference between the sample mean and hypothesized mean in terms of standard error. Calculated using specific formulas for one-sample and two-sample t-tests.

Q.How is p-value related to T-score?

The p-value is associated with the t-score, a measure of how many standard deviations a sample mean is from the hypothesized mean. A larger absolute t-score corresponds to a smaller p-value.