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Hypothesis Testing with Python

Exercise 1: t-test degrees of freedom

Using the formula below (which applies to t-tests with unequal degrees of freedom), create a python function that performs this calculation

png

  • s1 = standard deviation of sample 1

  • n1 = sample 1 size

Calculate the degrees of freedom for:

  • s1 = 5.32, n1 = 10
  • s2 = 7.03, n2 = 12
def degrees_of_freedom_t_test(s1,n1,s2,n2):
    s1n1 = (s1**2)/n1
    s2n2 = (s2**2)/n2
    top = (s1n1+s2n2)**2
    bottom = ((s1n1)**2)/(n1-1) + ((s2n2)**2)/(n2-1)
    return top/bottom

degrees_of_freedom_t_test(5.32,10,7.03,12)

19.853795244472487

Exercise 2: Perform a two sample t-test

Create a python function performs a two sample t-test of unequal variances. Use your result from problem 1 to calculate the degrees of freedom. The function arguments should be;

  • sample_mean1
  • sample_mean2
  • sample_std1 (standard deviation)
  • sample_std2
  • n1 (sample 1 size)
  • n2
  • tails (the number of tails for the test)
  • alpha (the significance level)

Make the function print the;

  • p_value
  • test statistic
  • critical value
  • alpha

Make the function return one of two conclusions;

  • Reject null hypothesis (if p_value < alpha and |test_statistic|>|critical_value|)
  • Otherwise, Fail to reject null hypothesis

RECALL

We calculate the test_statistic by; png

HINTS

  • p_value can be calculated using (1 - t.cdf(abs(t_statistic), df)) FOR EACH TAIL
  • critical_value can be calculate using t.ppf(1 - alpha / tails, df)

Run this function for the following values (from lecture notes):

  • sample_mean1 = 22.29
  • sample_mean2 = 16.3
  • sample_std1 = 5.32
  • sample_std2 = 7.03
  • n1 = 10
  • n2 = 12
  • tails = 2
  • alpha = 0.05
from scipy.stats import t

def calculate_t_test(sample_mean1, sample_mean2, sample_std1, sample_std2, n1, n2, tails,alpha):
    # Calculate the t-statistic
    df = round(degrees_of_freedom_t_test(sample_std1,n1,sample_std2,n2))
    standard_error = ((sample_std1 ** 2) / n1 + (sample_std2 ** 2) / n2) ** 0.5
    test_statistic = (sample_mean1 - sample_mean2) / (standard_error)

    # Calculate the p-value
    p_value = tails*(1 - t.cdf(abs(test_statistic), df))

    # Calculate the critical value
    critical_value = t.ppf(1 - alpha / tails, df)

    print('p_value: {}'.format(p_value))
    print('critical_value: {}'.format(critical_value))
    print('test_statistic: {}'.format(test_statistic))
    print('Significance level alpha: {}'.format(alpha))

    if p_value<=alpha:
        if critical_value<0 and test_statistic<critical_value:
            return "Reject Null Hypothesis"
        elif critical_value>0 and test_statistic>critical_value:
            return "Reject Null Hypothesis"
    else:
        return "Fail to Reject Null Hypothesis"

calculate_t_test(22.29, 16.3, 5.32, 7.03, 10, 12, tails=2,alpha=0.05)

p_value: 0.03425038740869879
critical_value: 2.0859634472658364
test_statistic: 2.2723574890239813
Significance level alpha: 0.05





'Reject Null Hypothesis'

Exercise 3: Plotting a t-distribution

Using matplotlib make a plot of the t-distribution for 20 degrees of freedom.

  • Define the horizontal values using x = np.linspace(-4,4,500)
  • Define the vertical values using t.pdf(horizontal_values, degrees of freedom)
  • Use the plt.plot(horizontal_values, vertical_values, label = ....) to plot the graph
  • Give the horizontal axis a label t-score
  • Give the vertical axis a label Probability density
  • Make a vertical line for the test statistic using plt.axviline(test_statistic,linestyle=':',label='test_statistic')
  • Give it a legend using the plt.legend() function
  • Shade the tails using the following;
    • alpha: plt.fill_between(x, pdf, where=(x < -critical_value) | (x > critical_value),...) give it the color red and an alpha (transparency parameter) value of 0.2 and label of alpha = 0.05
    • p-value: plt.fill_between(x, pdf, where=abs(x) > test_statistic, color='darkred',...) transparency parameter= 0.4 and label p_value
  • Show the plot
import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import t

# Set the degrees of freedom
df = 20

# Set the range of x-values
x = np.linspace(-4, 4, 500)

# Compute the probability density function (PDF) for the t-distribution
pdf = t.pdf(x, df)

# Create the plot
plt.plot(x, pdf, 'b-', label='t-distribution (df=20)')

# Compute the critical values for alpha = 0.05 (two-tailed test)
alpha = 0.05
critical_value = t.ppf(1 - alpha/2, df)

test_statistic = 2.272

# Shade the critical regions
plt.fill_between(x, pdf, where=(x < -critical_value) | (x > critical_value), color='red', alpha=0.2, label='Alpha = 0.05')
plt.fill_between(x, pdf, where=abs(x) > test_statistic, color='darkred', alpha=0.4,label='p-value')
# Add labels and a legend
plt.xlabel('t score')
plt.ylabel('Probability Density')
plt.title('t-Distribution with 20 Degrees of Freedom')
plt.axvline(test_statistic,linestyle=':',label='test statistic')
plt.legend()
# Display the plot
plt.show()

png

Exercise 4: Chi2 distribution

Using the observations from the lecture slides;

                      [[60, 40, 30]
                       [80, 70, 50]
                       [70, 90, 40]]
  • Perform a chi2 test using the chi2_contingency() function from the scipy.stats package
  • Let alpha=0.05

  • Find the critical_value using chi2.ppf(1 - alpha, df) from function from scipy.stats

  • Print Reject the null hypothesis if p_value<=alpha & test_statistic>critical_value

  • Otherwise, Fail to reject null hypothesis
from scipy.stats import chi2_contingency,chi2

# Observed frequencies (contingency table)
alpha = 0.05
observed_freq = np.array([[60, 40, 30],
                          [80, 70, 50],
                          [70, 90, 40]])

# Perform chi-square test of independence
test_statistic, p_value, df, expected_freq = chi2_contingency(observed_freq)
critical_value = chi2.ppf(1 - alpha, df)

# Print the chi-square test statistic, degrees of freedom, and p-value
print("Chi-square statistic:", test_statistic)
print("Degrees of freedom:", dof)
print("p-value:", p_value)
print('alpha:',alpha)
print('critical_value:',critical_value)

if p_value<=alpha and test_statistic>critical_value:
    print("Reject Null hypothesis")
else:
    print("Fail to reject null hypothesis")

Chi-square statistic: 8.469322344322341
Degrees of freedom: 4
p-value: 0.07582258667376501
alpha: 0.05
critical_value: 9.487729036781154
Fail to reject null hypothesis