Introduction to Hypothesis Testing in Lean Six Sigma

Hypothesis testing is a statistical method used in Lean Six Sigma to make decisions using data. It involves making an assumption (the hypothesis) about a population parameter and then using sample data to test whether this assumption is likely to be true or not. Hypothesis testing is pivotal in the Analyze phase of the DMAIC (Define, Measure, Analyze, Improve, Control) methodology, as it helps identify significant factors and relationships within processes that could be the root cause of problems.

Understanding the Paired t-Test

The Paired t-Test, also known as the Dependent Sample t-Test, is a statistical procedure used to determine whether the mean difference between two sets of observations is zero. In Lean Six Sigma projects, it is commonly applied to compare the means of a continuous variable collected from the same subjects at two different times or under two different conditions. This test is essential when dealing with 'before and after' scenarios, such as assessing the effect of a process improvement or any intervention on the same group of subjects.

When to Use a Paired t-Test

1. Before and After Studies: To evaluate the impact of a change in the process on the same group of items or people.

2. Matched Pairs: To compare two treatments or conditions on the same subjects where the subjects have been matched in pairs, based on certain criteria.

3. Repeated Measures: When measurements are taken from the same subjects at different times or under different conditions.
   

Assumptions of the Paired t-Test

• Dependence: The samples are dependent, meaning that there is a meaningful pairing between the two sets of data.

• Scale of Measurement: The data should be continuous or ordinal.

• Distribution: The differences between pairs are approximately normally distributed.
  

Steps in Conducting a Paired t-Test

1. Formulate the Hypotheses:

   • Null Hypothesis (H0): There is no difference in the mean outcomes of the two paired groups.

   • Alternative Hypothesis (H1): There is a significant difference in the mean outcomes of the two paired groups.

2. Calculate the Mean Difference: Subtract one set of observations from the other for each subject to get the difference, then compute the mean of these differences.

3. Determine the Test Statistic: Use the mean difference, the standard deviation of the differences, and the number of pairs to calculate the t-statistic.

4. Compare to the Critical Value: Determine the critical t-value from the t-distribution table based on the desired level of significance (alpha) and the degrees of freedom. The degrees of freedom for a paired t-test is the number of pairs minus one.

5. Make a Decision: If the absolute value of the calculated t-statistic is greater than the critical t-value, reject the null hypothesis; otherwise, do not reject it.

6. Interpret the Results: Conclude whether there is evidence to support a significant difference between the paired groups after the intervention or change was applied.
   

Application in Lean Six Sigma Projects

In Lean Six Sigma projects, the Paired t-Test is invaluable for quantifying the effect of process improvements. For example, if a manufacturing company wants to assess the impact of a new machine lubricant on equipment performance, it could measure the machine's output speed before and after the lubricant application using the same set of machines. A Paired t-Test would help determine if the lubricant significantly improved the machines' performance by comparing the means of the two sets of speed measurements.

Conclusion

The Paired t-Test is a powerful tool in the Lean Six Sigma toolkit, allowing practitioners to make informed decisions based on statistical evidence. By understanding when and how to use this test, Lean Six Sigma professionals can effectively measure the impact of their improvement efforts, ensuring that changes lead to meaningful process enhancements.

Example: Employee Training Program Effectiveness

Scenario: A company wants to evaluate the effectiveness of a new training program designed to improve sales skills among its employees. The sales figures (in thousands of dollars) for ten employees are recorded one month before and one month after the training. The goal is to determine if the training led to a significant increase in sales.

Data:

• Before Training: [45, 50, 55, 48, 51, 54, 58, 47, 52, 49]

• After Training: [49, 53, 60, 52, 56, 58, 62, 51, 57, 54]
  

Step 1: Calculate the Differences

The first step in a paired t-test is to calculate the difference between the paired observations.

Step 2: Summarize the Differences Calculate the mean (Dˉ) and the standard deviation (SD) of the differences.

Using our data:

• SD of D = Calculate the square root of the average of squared deviations from the mean Dˉ.

  
  

Step 3: Conduct the t-Test

The t-value is calculated using the formula:

Where μ0​ is the hypothesized difference between the pairs (in this case, μ0​=0 if we assume there is no effect from the training).

Let's say the calculated SD of D is 0.82 (for illustrative purposes; normally, you would calculate this using the formula given above). Substituting the values:

Step 4: Determine Significance

Compare the calculated t-value against the critical t-value from the t-distribution table at your chosen significance level (α), typically 0.05 for a two-tailed test, with degrees of freedom df=n−1=9.

In this hypothetical example, assuming the critical t-value is around 2.262 (from t-tables for df=9 and α=0.05 for a two-tailed test), our calculated t-value of 16.47 is much higher, indicating a statistically significant increase in sales after the training.

(If the calculated t-value is greater than the critical value, the difference is statistically significant.)

Reproducing with New Data:

To apply this procedure with new data, follow these steps:

1. Collect paired observations for your specific scenario.

2. Calculate the difference between each pair of observations.

3. Compute the mean and standard deviation of these differences.

4. Perform the t-test calculation with the formula provided.

5. Compare the calculated t-value against the critical t-value to determine significance.

This approach allows any practitioner to assess the impact of interventions in paired observational settings, providing a powerful tool for decision-making based on empirical evidence.