F-distribution and its Application in Hypothesis Testing

Understanding the F-distribution

The F-distribution is a probability distribution that arises frequently as the null distribution of a test statistic, particularly in the analysis of variance (ANOVA), the F-test, and in comparing variances.

It is used when comparing two samples to find out if they come from populations with equal variances. The shape of the F-distribution is positively skewed and depends on two parameters: degrees of freedom for the numerator and degrees of freedom for the denominator.

Variance: Definition and Concept

Variance is a statistical measure that represents the degree of spread in a dataset or the amount of variation from the average (mean). In simpler terms, it measures how much the numbers in a data set differ from the mean of the data set. A high variance indicates that the data points are spread out over a wider range of values, while a low variance signifies that they are clustered closely around the mean.

Mathematical Expression for Variance

Sample Variance vs Population Variance

The formula above calculates the population variance, assuming that the data set represents the entire population. However, when working with samples (a subset of a population), we typically use sample variance. The sample variance adjusts the denominator to consider the fact that we're working with a sample rather than the entire population. This adjustment, known as Bessel's correction, reduces the denominator by 1, resulting in the sample variance formula:

Why Bessel's Correction?

The rationale behind Bessel's correction (using n-1 instead of n) for sample variance is to provide an unbiased estimator of the population variance. When estimating population parameters from a sample, there's an inherent bias because we're using the sample mean ￣x instead of the true population mean. By dividing by n?1 rather than n, we compensate for this bias, making the sample variance an unbiased estimator of the true population variance.

"ddof" in Variance Calculation

The term ddof stands for "delta degrees of freedom." In the variance calculation method .var(ddof=1) used in many statistical software packages like pandas in Python, the ddof parameter allows you to adjust the degrees of freedom. Setting ddof=1 applies Bessel’s correction, ensuring the calculation is for sample variance. If ddof=0, the calculation would return the population variance (assuming the data set represents the entire population).

In summary, variance is a fundamental statistical measure used to quantify the degree to which individual data points in a dataset deviate from the mean value, and understanding whether to use sample or population variance (reflected in the use of ddof) is crucial in statistical analyses and interpretation.

Hypothesis Testing Overview

Hypothesis testing is a statistical method used to make inferences or draw conclusions about a population based on sample data. It involves making an initial assumption (a hypothesis), and then testing whether this hypothesis holds true based on the sample data. The two key types of hypotheses are:

1. Null Hypothesis (H0): This is a statement of no effect or no difference and is the hypothesis that is initially assumed to be true.
2. Alternative Hypothesis (H1 or Ha): This is what you would believe if you find sufficient evidence against the null hypothesis.

The outcome of a hypothesis test is usually determined through a p-value, which measures the probability of observing the data, or something more extreme, under the assumption that the null hypothesis is true. If the p-value is less than a predefined significance level (commonly 0.05), the null hypothesis is rejected in favor of the alternative.

Applying Hypothesis Testing to F-distribution

The F-distribution often arises when comparing the variances of two different populations and is used in the analysis of variance (ANOVA) and the F-test.

Scenario: Comparing Variances with the F-test

Let's consider you want to test if the variances of two normal populations are equal. The F-test can be useful in this scenario.

Steps for the F-test:

• State the Hypotheses:H0: (σ1)^2=(σ2)^2 (The variances are equal). H1: (σ1)^2 ≠ (σ2)^2 (The variances are not equal)

# Calculate the variances
var_pre = df_pre['time'].var(ddof=1)
var_post = df_post['time'].var(ddof=1)        

• Test Statistic:The F-statistic is calculated as the ratio of the two sample variances ((s1)^2 and (s2)^2). Ensure that the larger variance is in the numerator to get an F-statistic greater than or equal to 1. F=(s2)^2/(s1)^2

# Calculate F statistic (test statistic)
F = max(var_post, var_pre) / min(var_post, var_pre)        

• Decision Rule: Determine the critical value of F from the F-distribution tables for a chosen significance level (α, often 0.05) and the degrees of freedom from both samples.If the calculated F-statistic is greater than the critical value from the F-distribution table, reject H0.
• Calculate P-value:The p-value can be found using statistical software or F-distribution tables. If the p-value is less than α, reject H0.
• Conclude:Based on the rejection (or non-rejection) of H0, conclude whether there is enough evidence to support that the variances are significantly different.

Example of Using the F-distribution

Scenario: You are a data scientist working for an e-commerce company. Recently, the user interface team redesigned the product page, and they want to know if the new design has made any difference in the average time users spend on that page. They provide you with two datasets: one containing the time (in seconds) users spent on the product page before the redesign (pre_redesign_times.csv), and the other containing the time users spent after the redesign (post_redesign_times.csv). Using the F-distribution, can you determine if there's a statistically significant difference in the variances of user engagement times between the two designs? Write a Python program to conduct this hypothesis test, compute the F-statistic, calculate the associated p-value, and draw a conclusion based on a significance level of 0.05. Display your findings visually to make it comprehensible for the UI team.

In the given scenario, you're tasked with determining if there's a statistically significant difference in the variances of user engagement times on a product page before and after a UI redesign. This is a perfect use case for an F-test since it compares the variances of two independent samples.

Steps for Hypothesis Testing:

1. Null Hypothesis (H0): The variances of user engagement times before and after the redesign are equal.
2. Alternative Hypothesis (H1): The variances are not equal.
3. Significance Level: Typically set at 0.05.

To generate the .csv files and conduct the F-test in Google Colab, you will first create the CSV files using the following code snippet, and then proceed with loading these files into your analysis. Since Google Colab doesn't persist files across sessions, if you want to work with these files in the future, you might consider saving them to Google Drive. Below, I outline the steps, including how to save to and load from Google Drive:

1. Setting Up and Saving Files in Google Colab

First, let's generate and save the CSV files in the Colab environment:

import pandas as pd
import numpy as np

# Number of samples
n_samples = 1000

# Simulating engagement times
mean_pre, std_dev_pre = 300, 50
mean_post, std_dev_post = 310, 60

pre_redesign_times = np.random.normal(mean_pre, std_dev_pre, n_samples)
post_redesign_times = np.random.normal(mean_post, std_dev_post, n_samples)

# Save to CSV
df_pre = pd.DataFrame({'time': pre_redesign_times})
df_post = pd.DataFrame({'time': post_redesign_times})

df_pre.to_csv('pre_redesign_times.csv', index=False)
df_post.to_csv('post_redesign_times.csv', index=False)

print("CSV files created successfully!")
        

2. Saving Files to Google Drive for Persistent Storage

To save the CSV files to Google Drive:

from google.colab import drive
drive.mount('/content/drive')

# Specify your own path in Google Drive
path = '/content/drive/MyDrive/'

# Save files to Google Drive
df_pre.to_csv(path + 'pre_redesign_times.csv', index=False)
df_post.to_csv(path + 'post_redesign_times.csv', index=False)

print("CSV files saved to Google Drive successfully!")
        

3. Loading Files from Google Drive in Future Sessions

In a new Colab session, you can load the files directly from Google Drive:

from google.colab import drive
drive.mount('/content/drive')

# Adjust the path according to where you saved the files in Google Drive
path = '/content/drive/MyDrive/'
df_pre = pd.read_csv(path + 'pre_redesign_times.csv')
df_post = pd.read_csv(path + 'post_redesign_times.csv')

# Now df_pre and df_post are loaded and can be used for further analysis.
        

Now, let's check df_pre

	time
0	317.087799
1	393.808542
2	347.521192
3	271.154817
4	255.079266
...	...
995	275.604430
996	407.865411
997	269.714254
998	337.104769
999	314.964629
1000 rows × 1 columns        

And df_post

	time
0	388.104477
1	403.690672
2	311.920249
3	264.794928
4	337.598329
...	...
995	252.570945
996	330.627273
997	307.080859
998	311.967820
999	264.490280
1000 rows × 1 columns        

4. Conducting the F-test

Step1: Calculate Variances

# Calculate the variances
var_pre = df_pre['time'].var(ddof=1)
var_post = df_post['time'].var(ddof=1)

print(f'var_pre = {var_pre}, var_post = {var_post}')        

var_pre = 2531.3317175906764, var_post = 3829.6227455764333

        

1. DataFrame Access: df_pre['time'] and df_post['time'] are accessing the column named 'time' in the DataFrames df_pre and df_post, respectively. These columns contain the engagement times (in seconds) of users on the product page before and after a website redesign.
2. Variance Calculation:.var() is a pandas DataFrame method used to calculate the variance of a given set of values.ddof=1 is an argument that defines the "Delta Degrees of Freedom." For the variance calculation, the standard formula is:Variance= ∑(xi?￣x)^2 / (n?ddof)Where:xi = each value in the dataset ￣x = mean of the values, n = number of values, and ddof = delta degrees of freedom
3. Degrees of Freedom (ddof):In statistics, when calculating the sample variance, we often divide by n?1 (where n is the sample size). This approach, where ddof=1, corrects the bias in the estimation of the population variance from a sample and provides an unbiased estimate. It's known as Bessel's correction. If ddof=0 were used, the formula would calculate the population variance of the dataset.
4. Interpretation:var_pre holds the variance in user engagement times on the product page before the redesign.var_post holds the variance in engagement times after the redesign.

Understanding how the variance is calculated and interpreted is crucial, especially when applying statistical tests like the F-test, which relies on the variance of datasets to determine the significance of differences between group variances. The correct application of degrees of freedom (ddof) in calculating sample variance ensures that statistical inferences, such as comparing pre- and post-redesign user engagement variances, are accurate and reliable.

Step 2: Calculate F statistic (test statistic)

# Calculate F statistic (test statistic)
F = max(var_post, var_pre) / min(var_post, var_pre)


print(F)        

1.5128885396424738
        

Test Statistic:The F-statistic is calculated as the ratio of the two sample variances ((s1)^2 and (s2)^2). Ensure that the larger variance is in the numerator to get an F-statistic greater than or equal to 1. F=(s2)^2/(s1)^2

Step3: Degree of Freedom (fd)

The degrees of freedom correspond to the number of independent observations in each sample minus one. In this scenario, both samples have 1000 observations, so their degrees of freedom are both 999.

dfn = df_pre['time'].size - 1  # Degrees of freedom for the numerator
dfd = df_post['time'].size - 1  # Degrees of freedom for the denominator

print(f'dfn = {dfn} \ndfd = {dfd}')        

dfn = 999 
dfd = 999        

Step 4: P-Value

# Calculate the p-value
p_value = 1 - stats.f.cdf(F, dfn, dfd)        

The Full Python Program to Conduct the F-test

import pandas as pd
from scipy import stats
import matplotlib.pyplot as plt
import numpy as np

from google.colab import drive
drive.mount('/content/drive')

# Adjust the path according to where you saved the files in Google Drive
path = '/content/drive/MyDrive/'

# Load the datasets
df_pre = pd.read_csv(path + 'pre_redesign_times.csv')
df_post = pd.read_csv(path + 'post_redesign_times.csv')




# Calculate the variances
var_pre = df_pre['time'].var(ddof=1)
var_post = df_post['time'].var(ddof=1)

# Compute the F-statistic
F = max(var_pre, var_post) / min(var_pre, var_post)

# Degrees of freedom
dfn = len(df_pre) - 1 if var_pre > var_post else len(df_post) - 1
dfd = len(df_post) - 1 if var_pre > var_post else len(df_pre) - 1

# Calculate the p-value
p_value = 1 - stats.f.cdf(F, dfn, dfd)

# Compute the critical value for alpha = 0.05
alpha = 0.05
F_critical = stats.f.ppf(1 - alpha, dfn, dfd)

# Visualization
x = np.linspace(0, 3, 1000)
y = stats.f.pdf(x, dfn, dfd)

plt.plot(x, y, label="F-distribution PDF")
plt.axvline(F, color="black", linestyle="--", label=f'F-statistic = {F:.2f}')
plt.axvline(F_critical, color="red", linestyle="--", label=f'Critical value = {F_critical:.2f}')
plt.fill_between(x, y, where=(x > F_critical), color='lightgrey', label="Rejection region")
plt.annotate(f'p-value = {p_value:.2e}', (2.1, 1), color="blue")
plt.title("F-distribution with F-statistic")
plt.xlabel("F value")
plt.ylabel("Probability density")
plt.legend()
plt.show()

# Conclusion
if p_value < alpha:
    print("Reject the null hypothesis: The variances of user engagement times are significantly different.")
else:
    print("Fail to reject the null hypothesis: No significant difference in variances detected.")
        

Let's break down this diagram's components:

1. F-distribution PDF (blue curve): This represents the probability density function (PDF) of the F-distribution. The F-distribution is used to compare the variances of two populations, in this case, the user engagement times before and after the page redesign. The shape of the F-distribution is determined by the degrees of freedom of the two datasets.
2. F-statistic (dashed black line): This is the computed value of the test statistic from your data. It's calculated as the ratio of the larger variance to the smaller variance. Here, its value is 1.51, which means the variance of one dataset is 1.51 times that of the other dataset.
3. Critical value (dashed red line): The critical value is the threshold beyond which we would reject the null hypothesis. This value is based on the significance level (α = 0.05) and the degrees of freedom of the two datasets. Here, its value is 1.11.
4. Rejection region (shaded area): This is the region beyond the critical value. If the F-statistic falls into this region, then the difference in variances is considered statistically significant, and we reject the null hypothesis. In your case, the F-statistic (1.51) falls to the right of the critical value (1.11), which places it within the rejection region.
5. p-value (annotated in blue): The p-value is the probability of obtaining test results at least as extreme as the results actually observed, assuming that the null hypothesis is true. It provides a measure of evidence against the null hypothesis. A small p-value (typically ≤ 0.05) indicates strong evidence against the null hypothesis, so you reject it in favor of the alternative hypothesis. In this case, the p-value is 3.56e-11, which is much smaller than 0.05. This means the difference in variances is highly statistically significant.

Interpretation of the diagram:

• The F-statistic (1.51) falls in the rejection region.
• The p-value (3.56e-11) is much less than 0.05.

Thus, you reject the null hypothesis, concluding that there's a statistically significant difference in the variances of user engagement times between the two designs.

Degree of Freedom (df):

Refers to the number of independent values or quantities which can be assigned to a statistical distribution. In the context of the F-distribution, there are typically two degrees of freedom involved: one for the numerator (df1) and one for the denominator (df2). The degrees of freedom are often related to the sample size. For example, in an ANOVA test, the degrees of freedom for the numerator are related to the number of groups being compared and the degrees of freedom for the denominator are related to the total sample size minus the number of groups. The shape of the F-distribution depends on these degrees of freedom.

import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import f

# Define a range for the x-axis (F values)
x = np.linspace(0, 3, 1000)

# Define the different degrees of freedom to be plotted
dfs = [(1,1), (2,1), (5,2), (10,1), (100,100)]

# Plot the F-distribution for each degree of freedom
for df in dfs:
    y = f.pdf(x, df[0], df[1])
    plt.plot(x, y, label=f'd1={df[0]}, d2={df[1]}')

plt.title('F-distribution with Different Degrees of Freedom')
plt.xlabel('F value')
plt.ylabel('Probability density')
plt.legend()
plt.grid(True)
plt.ylim(0, 2.5)
plt.xlim(0, 3)
plt.show()
        

This diagram visualizes the probability density function (PDF) of the F-distribution for various degrees of freedom. Let's break it down:

Degrees of Freedom (df):

The shape of the F-distribution is governed by two degrees of freedom parameters, often denoted d1 and d2.

• d1 (numerator degrees of freedom): This is typically associated with the variance of the first group or sample.
• d2 (denominator degrees of freedom): This is associated with the variance of the second group or sample.

Observations from the Diagram:

1. Peak and Skewness: The peak of the F-distribution moves to the right (towards higher F-values) as d1 increases while d2 remains fixed. The distribution also becomes more skewed to the right with increasing d1.
2. Spread and Height: As either d1 or d2 increases, the distribution becomes flatter (less peaky) and spreads out more. For instance, the curve for d1=100, d2=100 is broader and less peaky compared to d1=1, d2=1.
3. Approaching Normality: With very high degrees of freedom (e.g., d1=100, d2=100), the F-distribution starts to resemble a normal distribution, although it is still right-skewed.

Why does it vary by df?

The F-distribution is derived from the ratio of two chi-squared distributions (which are themselves governed by degrees of freedom). The variation in the shape of the F-distribution with changing degrees of freedom arises due to the underlying properties of these chi-squared distributions.

• When both d1 and d2 are small, the variance in each sample is based on fewer observations, leading to a sharper and more peaked distribution.
• As d1 and d2 increase, each sample's variance is based on more observations, leading to a smoother distribution. This is why the distribution becomes broader and less peaked.

In summary, the degrees of freedom effectively represent the amount of information or data underlying the variance estimates, and this, in turn, influences the shape and properties of the F-distribution.

Two-Tails vs One-Tail

The F-test is primarily used to compare the variances of two populations. It does this by taking the ratio of two sample variances, leading to an F-distribution under the null hypothesis that both populations have equal variances.

Let's first discuss the tails in the context of the F-test:

1. Right-tailed F-test:

- You use a right-tailed test when you want to determine if the variance of the first population is greater than the variance of the second population.

- Rejection region: The right (upper) tail of the F-distribution.

The plot shows:

• The F-distribution curve.
• The rejection region in the right tail of the F-distribution.
• A dashed blue line indicating the computed F-value.
• A dash-dot green line indicating the critical F-value.
• The p-value corresponding to the computed F-value.

The code to generate this graph at the end of the article

2. Left-tailed F-test:

- This is when you want to determine if the variance of the first population is less than the variance of the second population.

- Rejection region: The left (lower) tail of the F-distribution.

In this plot:

• The F-distribution curve is displayed.
• The rejection region is in the left tail of the F-distribution.
• A dashed blue line indicates the computed F-value.
• A dash-dot green line indicates the critical F-value.
• The p-value corresponding to the computed F-value is shown.

The code to generate this graph at the end of the article

3. Two-tailed F-test:

- Used when you're simply interested in determining if the two variances are unequal, without a specific direction in mind.

- Rejection regions: Both tails of the F-distribution.

In this plot:

• The F-distribution curve is shown.
• The rejection regions are in both tails of the F-distribution.
• A dashed blue line represents the computed F-value.
• Dash-dot green lines represent the critical F-values in both tails.
• The p-value corresponding to the computed F-value is displayed.

The code to generate this diagram at the end of the article

Coding Differences:

1. Right-tailed:

    p_value = 1 - f.cdf(f_stat, df1, df2)        

2. Left-tailed:

    p_value = f.cdf(f_stat, df1, df2)        

3. Two-tailed:

    p_value = 2 * min(f.cdf(f_stat, df1, df2), 1 - f.cdf(f_stat, df1, df2))        

### Diagrams:

Unfortunately, I can't create live plots directly, but I'll describe what the diagrams would look like:

1. Right-tailed:

- The F-distribution curve would be plotted, and the area to the right of the critical F-value would be shaded, representing the rejection region.

2. Left-tailed:

- The F-distribution curve would be plotted, but this time the area to the left of the critical F-value would be shaded, indicating the rejection region.

3. Two-tailed:

- The F-distribution curve would again be plotted, and both tails would have shaded areas, each representing the rejection region.

In all diagrams, the computed F-statistic would be marked on the curve, allowing you to visually compare it to the rejection region(s) and determine whether to reject the null hypothesis.

It's worth noting that the F-distribution is not symmetric. This means that for a given significance level, the critical values for the left and right tails won't be simple reciprocals. You'd need to look up or calculate them separately.

When is the Largest variance over the smaller variance when calculating F?

For the F-test, the general formula is:

Now, to ensure that the F-value is always greater than or equal to 1 (since F-distribution values less than 1 are symmetric around the value of 1), it's common to place the larger variance in the numerator and the smaller variance in the denominator. This ensures the test is always right-tailed.

When doing a two-tailed test, you compare the computed F-value to both the upper and lower critical values. However, because of the nature of the F-distribution, this isn't simply a matter of looking at both tails in the manner you might with a t-test. Instead:

• If you already placed the larger variance in the numerator, you only need to compare the F-value with the upper critical value from the F-distribution.
• If the F-value exceeds this critical value, you reject the null hypothesis at the chosen significance level, suggesting the variances are different.
• The symmetry of the F-distribution values less than 1 ensures that if the larger variance is in the numerator and the F-value is significant in a two-tailed test, the inverse of the F-value would also be significant if you were to switch the variances.

To sum it up: For a two-tailed F-test, you typically place the larger variance in the numerator to ensure the F-value is >= 1. You then compare this F-value against the upper critical value from the F-distribution. If it's greater, you conclude that the variances are significantly different at the given significance level. The nature of the F-distribution ensures this approach is valid for testing inequality in both directions.

Conclusion

By conducting the F-test and interpreting the F-statistic and p-value, we can determine whether the redesign had a significant impact on the variability of the time users spend on the product page. This information, combined with other metrics like mean engagement time, can offer a comprehensive view of the redesign's effectiveness.

Additional Readings:

Code for right-tail plot:

import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import f

# Given data (these are example values)
df1 = 10  # degrees of freedom for sample 1
df2 = 10  # degrees of freedom for sample 2
alpha = 0.05  # significance level

# Compute critical F-value for right-tailed test
f_critical = f.ppf(1-alpha, df1, df2)

# Example F-value (for illustration purposes, you'd compute this from your samples)
f_value = 2.5  

# Compute p-value
p_value = 1 - f.cdf(f_value, df1, df2)

# Plot
x = np.linspace(0, 5, 1000)
y = f.pdf(x, df1, df2)

plt.plot(x, y, label="F-distribution")
plt.fill_between(x, y, where=(x > f_critical), color='red', label="Rejection Region")
plt.axvline(f_value, color='blue', linestyle="--", label=f"F-value = {f_value:.2f}")
plt.axvline(f_critical, color='green', linestyle="-.", label=f"Critical Value = {f_critical:.2f}")
plt.legend()
plt.title("Right-tailed F-test")
plt.xlabel("F value")
plt.ylabel("Probability density")
plt.annotate(f"p-value={p_value:.4f}", xy=(f_value, 0), xytext=(f_value-0.5, 0.1),
             arrowprops=dict(facecolor='black', arrowstyle='->'))
plt.show()        

Code for left-tailed diagram

import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import f

# Given data (these are example values)
df1 = 10  # degrees of freedom for sample 1
df2 = 10  # degrees of freedom for sample 2
alpha = 0.05  # significance level

# Compute critical F-value for left-tailed test
f_critical = f.ppf(alpha, df1, df2)

# Example F-value (for illustration purposes, you'd compute this from your samples)
f_value = 0.5  

# Compute p-value
p_value = f.cdf(f_value, df1, df2)

# Plot
x = np.linspace(0.1, 5, 1000)
y = f.pdf(x, df1, df2)

plt.plot(x, y, label="F-distribution")
plt.fill_between(x, y, where=(x < f_critical), color='red', label="Rejection Region")
plt.axvline(f_value, color='blue', linestyle="--", label=f"F-value = {f_value:.2f}")
plt.axvline(f_critical, color='green', linestyle="-.", label=f"Critical Value = {f_critical:.2f}")
plt.legend()
plt.title("Left-tailed F-test")
plt.xlabel("F value")
plt.ylabel("Probability density")
plt.annotate(f"p-value={p_value:.4f}", xy=(f_value, 0.1), xytext=(f_value+0.5, 0.2),
             arrowprops=dict(facecolor='black', arrowstyle='->'))
plt.show()        

The code to generate the two-sided

import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import f

# Given data (these are example values)
df1 = 10  # degrees of freedom for sample 1
df2 = 10  # degrees of freedom for sample 2
alpha = 0.05  # significance level

# Compute critical F-values for both tails
f_critical_left = f.ppf(alpha/2, df1, df2)
f_critical_right = f.ppf(1 - alpha/2, df1, df2)

# Example F-value (for illustration purposes, you'd compute this from your samples)
f_value = 1.5  

# Compute p-value
if f_value < f_critical_left:
    p_value = f.cdf(f_value, df1, df2)
elif f_value > f_critical_right:
    p_value = 1 - f.cdf(f_value, df1, df2)
else:
    p_value = 1

# Plot
x = np.linspace(0.1, 5, 1000)
y = f.pdf(x, df1, df2)

plt.plot(x, y, label="F-distribution")
plt.fill_between(x, y, where=(x < f_critical_left) | (x > f_critical_right), color='red', label="Rejection Regions")
plt.axvline(f_value, color='blue', linestyle="--", label=f"F-value = {f_value:.2f}")
plt.axvline(f_critical_left, color='green', linestyle="-.", label=f"Critical Value Left = {f_critical_left:.2f}")
plt.axvline(f_critical_right, color='green', linestyle="-.")
plt.legend()
plt.title("Two-tailed F-test")
plt.xlabel("F value")
plt.ylabel("Probability density")
plt.annotate(f"p-value={p_value:.4f}", xy=(f_value, 0.1), xytext=(f_value+0.5, 0.2),
             arrowprops=dict(facecolor='black', arrowstyle='->'))
plt.show()