Normal Distribution Random Normal T-test Type I Error

 

 

This test is based on the standard probability density function (PDF) following a Gaussian distribution.

The PDF is configured with a mean of 0 and a standard deviation of 1 to generate the random values.

A series of these randomly generated values is then tested using a T-test to determine whether there is sufficient evidence to reject the null hypothesis (mean = 0).

In most cases, with a 95% confidence level (the standard probability level used in economics and finance), there is typically insufficient evidence to reject the null hypothesis, i.e., the mean is 0.

However, with a significantly high number of trials (e.g., 1000), under the same confidence level, there are instances where there is sufficient evidence to reject the null hypothesis, i.e., the mean is not 0.

# For statistics

import random

import numpy as np

from scipy import stats

from scipy.stats import norm

# For plotting

import matplotlib.pyplot as plt

from matplotlib.patches import Polygon

# For table output

import pandas as pd

# Setting Plot Size

plt.rcParams["figure.figsize"] = (10,7)

# Reference list

LinkList=["List of colours: https://matplotlib.org/stable/gallery/color/named_colors.html",

          "Random Normal Distribution (Gausian PDF): https://numpy.org/doc/stable/reference/random/generated/numpy.random.normal.html",

          "Normal Distribution Functions: https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.t.html",

          "T-test: ",

          "   - One sample: https://builtin.com/data-science/t-test-python",

          "   - Independent: https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.ttest_ind.html",

          "   - Confidence Interval:",

          "      https://stackoverflow.com/questions/15033511/compute-a-confidence-interval-from-sample-data",

          "      https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.t.html",

          "Plot Filling https://matplotlib.org/stable/gallery/lines_bars_and_markers/fill_between_demo.html"]

for k in range(len(LinkList)):

    print(LinkList[k])

# Setting parameters

n=1000; # Number of elements

mu=0 # mean of the distribution

sigma=1 # Standard deviation

# Random normally distributed elements with n data points

x=np.random.normal(mu, sigma, n) # A variable of elements

# Gausian Probability Density Function (PDF)

def Gausian_PDF(bins,mu,sigma):

    g=1/(sigma*np.sqrt(2*np.pi))*np.exp(-(bins-mu)**2/(2*sigma**2)) # PDF formula

    return g

# Plotting function

def Disn_Hist(x,Confidence_Level):

    # Histogram and Probability-Density setting

    No_Bins=100 #Setting number of bins i.e. how many slices of bars

    Probability_Density, bins, ignored = plt.hist(x, No_Bins, density=True, color = "lightgreen",  label='Probability Density')

    # Mean and Standard Deviation of the sample data x

    x_bar=np.mean(x); s=np.std(x)

    # Normal distribution of the sample

    g_x=Gausian_PDF(bins,x_bar,s) # Calling PDF

    plt.plot(bins,g_x,linewidth=2, color='Green', label='Sample Distribution') # Plotting line graph of PDF

    plt.axvline(x=x_bar, label=f'x_bar is {round(x_bar,3)}', color='Green') # Sample mean

    # Normal distribution of the null hypothesis

    mu=0; sigma=1

    g_0=Gausian_PDF(bins,mu,sigma) # Calling PDF

    plt.plot(bins,g_0,linewidth=2, color='Grey', label='Null Hypothesis', linestyle='dashed') # Plotting line graph of PDF

    plt.axvline(x=x_bar, label=f'mu is {round(mu,3)}', color='Grey') # Sample mean

    # Confidence Interval: qnorm

    # Graoh Output

    plt.legend(loc="upper left")

    plt.title("Histogram & Mean Comparison")

    plt.show()

Confidence_Level=95

Disn_Hist(x,Confidence_Level)

# T-test testing the null-hypothesis

def Ttest(x, mu, Confidence_Level, Print):

    x_bar=np.mean(x)

    t,p=stats.ttest_1samp(x, mu, alternative='two-sided')

    z=stats.t.ppf((1+Confidence_Level/100)/2.,n-1) # Z-value

    CI=[x_bar-stats.sem(x)*z, x_bar+stats.sem(x)*z] # Confidence Interval

    #Plotting a Gausian PDF

    points=np.linspace(-3,3,100)

    SEM=1 #stats.sem(x)

    g_0=Gausian_PDF(points,mu,SEM)

    plt.plot(points,g_0,linewidth=2, color='Grey', label='Sample Distribution') # Plotting line graph of PDF

    plt.axvline(x=0, color='Grey') # Null-Hypothesis

    plt.title('t-test')

    plt.xlabel('t-value')

    # T-test critical point

    SigLevel=1-Confidence_Level/100

    # Lower Fill

    LowerT_Pnts=np.linspace(min(points),stats.norm.ppf(SigLevel/2))

    LowerT_Dsts=Gausian_PDF(LowerT_Pnts,0,1)

    #LowerT=[(bins[k],LowerT_Dsts[k]) for k in range(len(bins))]

    plt.fill_between(LowerT_Pnts,LowerT_Dsts, color='darkseagreen')

    # Upper Fill

    UpperT_Pnts=np.linspace(stats.norm.ppf(1-SigLevel/2),max(points))

    UpperT_Dsts=Gausian_PDF(UpperT_Pnts,0,1)

    #UpperT=[(bins[k],UpperT_Dsts[k]) for k in range(len(bins))]

    plt.fill_between(UpperT_Pnts,UpperT_Dsts, color='darkseagreen')

   

    plt.axvline(x=t, label=f't is {round(t,3)}', color='Magenta') # Null-Hypothesis

    h=['H0: mu = x_bar ∴ No sufficient evidence to reject the null hypothesis (H0).',

       'H1: mu ≠ x_bar ∴ Sufficient evidence to reject the null hypothesis (H0).']

    if Print==1:

        print(f'Confidence Interval: {CI}')

        print(f'T-value: {t},  p-value: {p}')

        print(h[int(np.floor((1-p)-0.05))])

    return t,p

print('\n')

print('T-test testing the null-hypothesis')

Confidence_Level=95 #%

ShowingTstatDetail=1

x=np.random.normal(mu, sigma, n) # A variable of elements

plt.figure(); Ttest(x, 0, Confidence_Level, ShowingTstatDetail)

# Why there is sometimes no significant evidence proving it is zero.

def Type_I_Error_test(Confidence_Level,Number_of_datapoints, Number_of_trial,Print):

    X_bar=[];T=[];P=[];SigLevel=1-Confidence_Level/100

    for l in range(Number_of_trial):

        mu=0; sigma=1; n=Number_of_datapoints

        x=np.random.normal(mu, sigma, n)

        t,p=stats.ttest_1samp(x, mu)

        if p<SigLevel:

            #plt.figure(); Disn_Hist(x,Confidence_Level)

            plt.figure(); Ttest(x, 0, Confidence_Level, Print)

            x_bar=np.mean(x)

            X_bar.append(x_bar)

            T.append(t)

            P.append(p)

    return X_bar,T,P

print('\n')

print('Why there is sometimes no significant evidence proving it is zero.')

Confidence_Level=95 #%

Number_of_datapoints=1000

Number_of_trial=100

ShowingTstatDetail=0

X_bar,T,P=Type_I_Error_test(Confidence_Level, Number_of_datapoints, Number_of_trial, ShowingTstatDetail)

print(f"There are {len(P)} times out of {Number_of_trial} times of the trial not be able to reject the null hypothesis.")

print(f"The percentage of no sufficient evidence to reject the null hypothesis is {round(len(P)/Number_of_trial,2)}%")

d = {'x_bar':X_bar,'t':T, 'p':P}

df = pd.DataFrame(data=d)

display(df)