Z - Test

Z Test for Testing Means

One Sample Z Test

Two Sample Z Test

import numpy as np

def TwoSampZTest(samp_mean_1, samp_mean_2, samp_std_1, samp_std_2, n1, n2):
    denominator = np.sqrt((pow(samp_std_1, 2) / n1) + (pow(samp_std_2, 2) / n2))
    z_score = (samp_mean_1 - samp_mean_2) / denominator
    return z_score

When series data is given use following code

from statsmodels.stats import weightstats as stests
z_score, p_value = stests.ztest(list_1, list_2, alternative ='smaller')

Proportion test

2 sample z proportion test

The Quidditch teams at Hogwarts conducted tryouts for two positions: Chasers and Seekers.

In Group Chasers, out of 90 students who tried out, 57 were selected. In Group Seekers, out of 120 students who tried out, 98 were selected.

Is there a significant difference in the proportion of students selected for Chasers and Seekers positions?

Conduct a test at 90% confidence level.

Using Formula

n1 = 90
x1 = 57
n2 = 120
x2 = 98

# Calculate sample proportions
p1 = x1 / n1
p2 = x2 / n2

# Calculate pooled sample proportion
pooled_p = (x1 + x2) / (n1 + n2)

# Calculate standard error
se = np.sqrt(pooled_p * (1 - pooled_p) * (1/n1 + 1/n2))

# Calculate Z-test statistic
z_statistic = (p1 - p2) / se

# Two-tailed p-value
p_value = 2 * (1 - norm.cdf(abs(z_statistic)))

# Print the results
print("Z-statistic:", z_statistic)
print("p-value:", p_value)

# Check if the p-value is less than the significance level
alpha = 0.10
if p_value < alpha:
    print("Reject the null hypothesis: There is a significant difference in proportions.")
else:
    print("Fail to reject the null hypothesis: No significant difference in proportions.")

"""
Z-statistic: -2.990306921349541
p-value: 0.002786972588957992
Reject the null hypothesis: There is a significant difference in proportions.
"""

Using Scipy package

from statsmodels.stats.proportion import proportions_ztest

successes = [57, 98]
trial = [90, 120]
z_statistic, p_value = proportions_ztest(successes, trial, alternative='two-sided')
z_statistic, p_value # (-2.990306921349541, 0.002786972588958094)

As a product manager, you want to evaluate the user satisfaction for two different seasons of Naruto Shippuden (Season 1 and Season 2).

You collected feedback from 250 viewers who watched Season 1 of Naruto Shippuden, and 120 expressed satisfaction. Similarly, for Season 2, you gathered data from 300 viewers, and 150 of them expressed satisfaction.

Conduct an appropriate test at a 95% confidence interval to determine if there's a higher user satisfaction for Season 2 than for Season 1.

import numpy as np
# Data
n1 = 250
x1 = 120
n2 = 300
x2 = 150

# Calculate sample proportions
p1 = x1 / n1
p2 = x2 / n2

# Calculate pooled sample proportion
pooled_p = (x1 + x2) / (n1 + n2)

# Calculate standard error
se = np.sqrt(pooled_p * (1 - pooled_p) * (1/n1 + 1/n2))

# Calculate Z-test statistic
z_statistic = (p2 - p1) / se

# One-tailed p-value (since the alternative is p2 < p1)
p_value = 1-norm.cdf(z_statistic)

# Print the results
print("Z-statistic:", z_statistic)
print("p-value:", p_value)

# Check if the p-value is less than the significance level
alpha = 0.05
if p_value < alpha:
    print("Reject the null hypothesis: There is higher user satisfaction for Season 2.")
else:
    print("Fail to reject the null hypothesis: No evidence of higher user satisfaction for Season 2.")
"""
Z-statistic: 0.46717659215115714
p-value: 0.32018676972652416
Fail to reject the null hypothesis: No evidence of higher user satisfaction for Season 2.
"""

successes = np.array([120, 150])
trials = np.array([250, 300])

# Perform two-sample Z-test for proportions
z_statistic, p_value = proportions_ztest(successes, trials, alternative='smaller')  # 'smaller' for p2 < p1

# Print the results
print("Z-statistic:", z_statistic)
print("p-value:", p_value)

# Check if the p-value is less than the significance level
alpha = 0.05
if p_value < alpha:
    print("Reject the null hypothesis: There is higher user satisfaction for Season 2.")
else:
    print("Fail to reject the null hypothesis: No evidence of higher user satisfaction for Season 2.")

"""
Z-statistic: -0.46717659215115714
p-value: 0.3201867697265242
Fail to reject the null hypothesis: No evidence of higher user satisfaction for Season 2.
"""