T-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 1.6.

A numerical example of a T-test

Numerical example and its solution summary 

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 25 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?

Let's recall the development to solve the previously numerical example presented.

• 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 smaller than 30, then a student's T distribution could be employed.

• The next table helps to understand the relation between confidence level, alpha (α), and the critical value t α/2 for the degrees of freedom (sample size - 1).

Let’s consider that the first factory shares 21 samples of ball bearings where the mean diameter of the sample comes out to be 10.5 cm. On the other hand, the second factory shares 25 samples with a mean diameter of 9.5 cm. Both have a standard deviation of 1 cm [3]:

• 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 t-test for means.

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

• Factory 1: x̄1 = 10.5, σ1 = 1.

• Factory 2: x̄2 = 9.5, σ2 = 1.

• 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%.

  • Df (degree of freedom) = min(n1, n2) - 1 = 21 - 1 = 20

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

  • tobs = (x̄1 – x̄2 ) / √((s1 )²/n1 + (s2)²/n2)

  • tobs = (1) / √((1)²/21 + (1)²/25))

  • tobs = 3.378

• Step 5: Compare tobs with the critical value t α/2 from the previous table. 

The next Python code is useful to automate the manual computation made previously [2, 3].

import numpy as np

import scipy.stats as stats

import matplotlib.pyplot as plt

# Step 2: Sample statistics

x1 = 10.5  # Sample mean of Factory 1

s1 = 1  # Sample standard deviation of Factory 1

n1 = 21  # Sample size of Factory 1

x2 = 9.5  # Sample mean of Factory 2

s2 = 1  # Sample standard deviation of Factory 2

n2 = 25  # Sample size of Factory 2

# Degrees of freedom

df = min(n1, n2) - 1

# Step 4: Calculate the t-test statistic

t_obs = (x1 - x2) / np.sqrt((s1 ** 2) / n1 + (s2 ** 2) / n2)

print(f"Tobs: {t_obs:.3f}")

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

alpha = 0.05

t_critical = stats.t.ppf(1 - alpha/2, df)

print(f"Critical value (t_alpha/2): ±{t_critical:.3f}")

# Step 6: Conclusion

if abs(t_obs) > t_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 < -t_critical) | (x > t_critical), color='red', alpha=0.3, label='Critical regions')

# Tobs line

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

plt.text(t_obs - 0.1, max(y)*0.5, f'Tobs = {t_obs:.3f}', color='blue', ha='right')

# Critical region text

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

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

# Formatting the plot

plt.title('Two-Tailed t-Test')

plt.xlabel('t-value')

plt.ylabel('Probability Density')

plt.legend()

plt.grid(True)

plt.show()

Tobs: 3.378

Critical value (t_alpha/2): ±2.086

Reject the null hypothesis