1.4. One sample z-test and their relation with two-sample z-test

Z-test is a statistical test that is used to determine whether the mean of a sample is significantly different from a known population mean when the population standard deviation is known. It is particularly useful when the sample size is large (>30) [1].

This kind of statistical test had and the proper situation to apply had been described at Track 08, section 1.3.

A numerical example of a Z-test

The average service time of a company in 2018 was 12.44 minutes. Management wants to know whether the arithmetic mean current is different from 12.44 minutes. A sample with 150 values ​​had an arithmetic mean of 13.71 minutes and a standard deviation of 2.65 minutes. Using α = 5%, can you conclude whether the time is currently different?

• Null hypothesis (Ho): The mean had been not affected, i.e., μ = 12.44.

• Alternate hypothesis (Ha): Then the mean had been affected, i.e., μ ≠  12.44.

• From the table described in the step to choose a statistical test, the sign of the hypotheses Ho and Ha point that a two-tailed test should be carried.

• Since the sample is larger than 30, then a normal distribution could be employed.

• The next table helps to understand the relation between confidence level, alpha (α), and the critical value z α/2.

• To compute a statistical test is necessary to convert the observed value in the mean of the sample (x̄) to the scale of a standard normal distribution (Zobs). These could be done using the following equation:

zobs = (x̄ - μ)/(s/(n^0.5))

• This equation will result in the following numbers:

zobs = (13.71 - 12.44)/(2.65/(150^0.5)) = (13.71-12.44)/0.2164 = 5.87 

Since Zobs = 5.87 is higher than upper critical value z α/2 = 1.96, then we can reject the Null hypothesis.

When to use a Z-test

The following conditions must hold to apply a Z-test:

• The sample size should be equal or greater than 30. Otherwise, we should use the t-test.

• Samples should be drawn at random from the population.

• The standard deviation of the population should be known.

• Samples that are drawn from the population should be independent of each other.

• The data should be normally distributed, however, for a large sample size, it is assumed to have a normal distribution because central limit theorem.

Types of Z-test

A company wanted to compare the performance of its factory employees in two different factories located in two different parts of a country – Factory 1, and Factory 2, in terms of the number of manufactured products in a day. The company randomly selected 30 employees from the Factory 1 and 30 employees from the Factory 2. The following data was collected [2]:

• Step 1: Pose the research question and determine the proper statistical test.

The company wants to determine if the performance of the employees in Factory 1 is different from the performance of the employees in the Factory 2. To do this, we will use a two-sample z-test for means.

• Step 2: Obtain the samples statistics from the two factories (populations).

• Factory 1: x̄1 = 750, σ1 = 20.

• Factory 2: x̄2 = 780, σ2 = 25.

• Step 3: Formulate the null and alternate hypotheses and set the level of significance for the test.

  • Null hypothesis (Ho): There is no difference between the performance of employees at different Factories. There is no difference between the means, i.e., μ1 - μ2 = 0.

  • Alternate hypothesis (Ha): There is a difference in the performance of the employees. Then the mean had been affected, i.e., μ1 - μ2 ≠  0.

  • We will perform two-tailed test using confidence level (alpha),  α = 5%.

• Step 4: Use the formula for two-sample z-test for means to calculate the z-test statistic zobs.

  • zobs = (x̄1 – x̄2 ) / √((σ1 )²/n1 + (σ2)²/n2)

  • zobs = (-30) / √((20)²/30 + (25)²/30))

  • zobs = -5.13

• Step 5: Compare zobs with the critical value z α/2 from the following table: 

The next Python code is useful to automate the manual computation made previously [3, 4].

import numpy as np

import scipy.stats as stats

import matplotlib.pyplot as plt

# Step 2: Sample statistics

x1 = 750  # Sample mean of Factory 1

sigma1 = 20  # Population standard deviation of Factory 1

n1 = 30  # Sample size of Factory 1

x2 = 780  # Sample mean of Factory 2

sigma2 = 25  # Population standard deviation of Factory 2

n2 = 30  # Sample size of Factory 2

# Step 4: Calculate the z-test statistic

z_obs = (x1 - x2) / np.sqrt((sigma1 ** 2) / n1 + (sigma2 ** 2) / n2)

print(f"Zobs: {z_obs:.3f}")

# Step 5: Critical values for a two-tailed test at alpha = 0.05

alpha = 0.05

z_critical = stats.norm.ppf(1 - alpha/2)

print(f"Critical value (z_alpha/2): ±{z_critical:.3f}")

# Step 6: Conclusion

if abs(z_obs) > z_critical:

    conclusion = "Reject the null hypothesis"

else:

    conclusion = "Fail to reject the null hypothesis"

print(conclusion)

# Plotting the results

x = np.linspace(-4, 4, 1000)

y = stats.norm.pdf(x)

plt.figure(figsize=(10, 6))

plt.plot(x, y, label='Standard Normal Distribution')

# Critical regions

plt.fill_between(x, y, where=(x < -z_critical) | (x > z_critical), color='red', alpha=0.3, label='Critical regions')

# Zobs line

plt.axvline(z_obs, color='blue', linestyle='--', label=f'Zobs = {z_obs:.3f}')

plt.text(z_obs + 0.1, max(y)*0.5, f'Zobs = {z_obs:.3f}', color='blue', ha='left')

# Critical region text

plt.text(-z_critical, max(y)*0.1, f'Critical region: {z_critical:.3f}', color='red', ha='center')

plt.text(z_critical, max(y)*0.1, f'Critical region: {z_critical:.3f}', color='red', ha='center')

# Formatting the plot

plt.title('Two-Tailed Z-Test')

plt.xlabel('Z-value')

plt.ylabel('Probability Density')

plt.legend()

plt.grid(True)

plt.show()

Zobs: -5.132

Critical value (z_alpha/2): ±1.960

Reject the null hypothesis