This question below has been answered once but its based on ANOVA. For my question the answer needs to be centered around "fundamentals of hypothesis testing - one sample test"

Now, imagine you are a statistician called to do some analysis to show a quality improvement in a manufacturing plant. The plant manager wants to use the statistical analysis in a presentation to the company executives to show that quality improvements were made in the plant. The data is constructed with a before the improvement sample of size n equal to 300 items, and the after improvement data is composed of a sample size n equal to 300 items. The plant manager intents to show that quality of the products after the improved is significantly improved as compared to quality of the product before the improvement.

When ensuring the integrity of the data, you discover the following information:

The samples were drawn from 3 different crews who had historically different workmanship as follows:

• Crew A was known to generally produce products with poor quality,
• Crew B was known to generally produce products with average quality, and
• Crew C was known to generally produce products with excellent quality

Thebeforedata was composed as follows:

• 180 items (60% of the before sample) were selected from crew A,
• 60 items (20% of the before sample) were selected from crew B,
• 60 items (20% of the before sample) were selected from crew C.

Theafterdata was composed as follows:

• 60 items (20% of the after sample) were selected from crew A,
• 60 items (20% of the after sample) were selected from crew B,
• 180 items (60% of the after sample) were selected from crew C.

The data composition presents you with a concern about the sampling and analysis. Since the number of items coming from each crew is different in the before and after data, the analysis could possibly indicate that the after data (60% is from Crew C) is better than the before data (60% is from Crew A) regardless of any quality improvements.You feel obligated to discuss your concerns with the plant manager. From your previous discussions with the plant manager, you have the impression that the plant manager needs to present the analysis to upper management and will want you to perform the analysis with the present data. So, you want to have in your mind the reasons why the analysis should not proceed with all data as it is in the current form. From your past experience, you believe that your concern will become accepted by the plant manager if you can propose a solution to the potential problem.

You need to propose a solution of how to proceed with the analysis to test if an improvement was made, preferably without collecting any further data.