In the realm of Lean Six Sigma, the Two-Sample t-Test, also known as the Independent Samples t-Test, is a statistical procedure used to determine whether the mean differences between two independent samples are significantly different. This test is particularly important in the Analyze phase of the DMAIC (Define, Measure, Analyze, Improve, Control) methodology, where identifying and validating the root causes of problems is crucial.

Understanding the Two-Sample t-Test

The Two-Sample t-Test compares the means from two independent groups under the assumption that both samples are random, independent, and come from normally distributed populations with unknown but equal variances. This test is useful in various situations, such as comparing the performance of two processes, the output of two shifts, or the quality metrics of products from two different suppliers.

Application in Lean Six Sigma

In Lean Six Sigma projects, the Two-Sample t-Test enables practitioners to make data-driven decisions by testing hypotheses about process improvements. For example, if a team wants to determine whether a new training program has improved the productivity of assembly line workers, they could use this test to compare the mean outputs of a group that received training to those who did not.

Steps to Perform a Two-Sample t-Test

1. Define the Hypotheses:

   1. Null Hypothesis (H0​): There is no difference in the population means (μ1​=μ2​).

   2. Alternative Hypothesis (Ha​): There is a significant difference in the population means (1≠2μ​).

      
      

2. Collect Data: Gather sample data from two independent groups. Ensure that the data collection methods do not introduce bias.

   
   

3. Check Assumptions: Verify that the data meets the test assumptions – independence, normality, and equal variances. Use statistical tests like the Shapiro-Wilk test for normality and Levene's test for equality of variances if necessary.

   
   

4. Conduct the Test: Calculate the t-statistic using the formula:

where xˉ1​ and xˉ2​ are the sample means, n is the sample size (assuming equal sizes for simplicity), and sp​ is the pooled standard deviation.

5. Determine Significance: Compare the calculated t-value with the critical t-value from the t-distribution table at the desired significance level (α). If the calculated t-value exceeds the critical value, reject the null hypothesis.

6. Draw Conclusions: Based on the test result, conclude whether there is sufficient evidence to support that there is a significant difference between the group means.

Practical Considerations

• Sample Size: Larger sample sizes can provide more reliable results.

  
  

• Effect Size: Consider the practical significance of the findings, not just statistical significance. A statistically significant result may have limited practical value.

  
  

• Software Tools: Utilize statistical software packages for accurate calculations and to conduct additional checks for assumptions.

  
  

Conclusion

The Two-Sample t-Test is a powerful tool in the Lean Six Sigma toolkit for comparing the means of two independent samples. By rigorously testing hypotheses about process improvements, practitioners can ensure that their decisions are based on solid statistical evidence, thereby enhancing the effectiveness of their Lean Six Sigma projects.

Real-Life Scenario Example

Imagine a company, "EcoPaper", that manufactures biodegradable paper cups. They've introduced a new material to make their cups more durable. Before fully transitioning to this new material, EcoPaper wants to ensure that the new cups' durability is significantly better than that of the old material. They conduct a test with two groups:

• Group A (Old Material): 30 cups are tested for durability under standardized conditions.

  
  

• Group B (New Material): 30 different cups made from the new material are tested similarly.

  
  

The durability is measured by the amount of liquid the cups can hold until they start leaking, recorded in milliliters (ml).

Step-by-Step Calculation

To perform the Two-Sample t-Test, follow these steps:

1. Collect Data: Assume the following results (simplified for the example):

   • Group A (Old Material): Mean (μ₁) = 350ml, Standard Deviation (σ₁) = 15ml

   • Group B (New Material): Mean (μ₂) = 365ml, Standard Deviation (σ₂) = 20ml

   • Sample Sizes: n₁ = n₂ = 30

     
     

2. State the Hypotheses:

   • Null Hypothesis (H₀): μ₁ = μ₂ (No difference in means)

   • Alternative Hypothesis (H₁): μ₁ ≠ μ₂ (A significant difference exists)
     

3. Select Significance Level:

   • Typically, a 5% level (α = 0.05) is chosen.
     

4. Calculate the t-Statistic: The formula for the t-statistic with equal variances is:

Where:

• xˉ1​ and xˉ2​ are the sample means,

• sp​ is the pooled standard deviation, calculated as:

• n is the sample size for each group (assuming they're equal).

5. Determine the Degrees of Freedom (df):

df=n1​+n2​−2=30+30−2=58 6. Find the Critical t-Value:

Using a t-distribution table or software, find the critical t-value for α = 0.05 and df = 58. This value is approximately ±2.001, see below:

7. Make the Decision: Since the calculated t-statistic (-4.47) is less than the critical value (-2.001), we reject the null hypothesis.

Conclusion

By rejecting the null hypothesis, we conclude that there is a significant difference between the durabilities of cups made from the old and new materials. EcoPaper can confidently switch to the new material, expecting an improvement in cup durability.

Reproducing the Exercise

To apply this test to new data, follow the steps above, replacing the sample means, standard deviations, and sample sizes with your data. Ensure your data meets the assumptions (normality and equal variances) for accurate results.