In the realm of Lean Six Sigma, the One-Sample t-Test is a powerful statistical tool used to make informed decisions about process improvements. This test is particularly useful when comparing the mean of a single sample to a known or hypothesized population mean, especially when the sample size is small and the population standard deviation is unknown. The One-Sample t-Test helps in identifying whether significant differences exist between the sample mean and the population mean, thereby facilitating data-driven improvements in quality management and process optimization.

The Purpose of the One-Sample t-Test

The primary purpose of the One-Sample t-Test within Lean Six Sigma projects is to verify or challenge assumptions about a process performance. For instance, a company may want to determine if the average cycle time of a process is significantly different from the target or benchmark cycle time. By applying the One-Sample t-Test, teams can ascertain if observed deviations are due to random variation or indicate a true shift in the process performance, necessitating corrective actions.

How the One-Sample t-Test Works

The One-Sample t-Test compares the sample mean (xˉ) to the population mean (μ) using the t-statistic, which is calculated as follows:

Where:

• xˉ is the sample mean,

• μ is the population mean,

• s is the sample standard deviation, and

• n is the sample size.

  
  

The calculated t-statistic is then compared against the critical t-value from the t-distribution table at a specified confidence level (usually 95% or 99%). If the absolute value of the t-statistic is greater than the critical value, the null hypothesis (H0​:μ=μ0​), which states there is no significant difference between the sample and population means, is rejected in favor of the alternative hypothesis (H1​:μ=μ0​).

Applications in Lean Six Sigma Projects

The One-Sample t-Test is versatile and can be applied in various phases of a Lean Six Sigma project, including:

• Measure Phase: To validate measurement systems by comparing the measured value of a standard or calibration part against its known value.

  
  

• Analyze Phase: To identify significant factors that may be causing a deviation from the desired process performance.

  
  

• Improve Phase: To evaluate the effectiveness of pilot tests or simulations by comparing the before and after performance of a process.

  
  

Considerations and Best Practices

• Sample Size: The One-Sample t-Test is particularly useful for small sample sizes (typically less than 30). For larger samples, the z-test might be more appropriate.

  
  

• Normality Assumption: The test assumes that the data is approximately normally distributed. It's advisable to perform a normality test or use graphical methods like Q-Q plots to verify this assumption.

  
  

• Outliers: Be mindful of outliers, as they can significantly impact the test results. Investigate and address outliers appropriately before conducting the test.

  
  

Conclusion

The One-Sample t-Test is a fundamental tool in Lean Six Sigma methodology, offering a rigorous approach to hypothesis testing when comparing sample data to a known or hypothesized population mean. By leveraging this test, Lean Six Sigma practitioners can make evidence-based decisions, driving improvements in quality and efficiency in their processes. As with any statistical tool, understanding its assumptions, limitations, and proper application is key to unlocking its full potential in enhancing process performance.

Real-Life Based Scenario

Imagine a company, "Green Widget Manufacturing," is implementing Lean Six Sigma practices to improve its production process. A critical aspect of their production line is the thickness of the paint applied to their widgets. The company has set a standard that the average thickness should be 2.0 mm. To ensure this standard is consistently met, the quality control department samples 25 widgets from a day's production to measure the paint thickness.

The measurements (in mm) are as follows:

2.1,1.9,2.0,2.1,1.8,2.2,1.9,2.0,2.1,1.8,2.0,2.2,2.1,1.9,2.0,2.3,1.8,2.0,2.1,1.9,2.0,2.2,1.9,2.0,2.1

The quality control department wants to use a One-Sample t-Test to determine if there is a statistically significant difference between the sample mean and the known standard thickness of 2.0 mm. Hypotheses

• Null Hypothesis (H0​): The mean thickness of the paint on the widgets is equal to 2.0 mm μ=2.0
  

• Alternative Hypothesis (Ha): The mean thickness of the paint on the widgets is not equal to 2.0 mm μ = 2.0

Calculations

1.Calculate the sample mean (xˉ):

2.Calculate the sample standard deviation (s):

3. Calculate the t-statistic:

Where:

• xˉ is the sample mean

• μ0​ is the population mean (2.0 mm in this case)

• s is the sample standard deviation

• n is the sample size
  

4.Determine the critical t-value from the t-distribution table at the desired level of significance (α, typically 0.05 for a 95% confidence level) and degrees of freedom (df=n−1).Compare the calculated t-statistic to the critical t-value to decide whether to reject or fail to reject the null hypothesis. Example Calculation

Assuming the calculations from the sample data provided: (Maths for mean and standard deviation calculation are not detailed here)

• Sample mean (xˉ) = 2.02 mm

• Sample standard deviation (s) = 0.15 mm

• Sample size (n) = 25

• Degrees of freedom (df) = 24

• Population mean (μ0​) = 2.0 mm

Assuming a 95% confidence level, the critical t-value for df=24 is approximately ±2.064 (from t-distribution tables):

Conclusion

Since the calculated t-statistic (0.67) is less than the critical t-value (±2.064), we fail to reject the null hypothesis. This result suggests that there is no statistically significant difference between the sample mean and the population mean of 2.0 mm, implying that the paint thickness is on target according to the company standards.

This example illustrates how a One-Sample t-Test can be applied in a real-life scenario within the Lean Six Sigma framework to make data-driven decisions about process improvements and quality control.