In your role as an engineering manager at a manufacturing company, you've been tasked with improving the quality of a critical product component. The manufacturing process involves several variables, including temperature, pressure, and production speed. Your team has collected data from the last six months and noticed some interesting patterns. This case study presents an opportunity to delve into the concepts of correlation and causation within the context of engineering management.

Month Temperature $(\backslash$deg C$)$ Pressure $($psi$)$ Production Speed Product Quality $(1-10)$

Jan $120501006$

Feb $11548955$

Mar $11851984$

Apr $122531027$

May $11950995$

Jun $123551058$

Based on the data collected, analyze the correlation between different process variables $($temperature$,$ pressure, production speed$)$ and the quality of the product component. What kind of correlation measures $($Pearson$,$ Spearman$)$ would you use and why? Interpret the correlation coefficients you calculate and discuss their implications for quality improvement.

Discuss the distinction between correlation and causation. Are there any process variables that show a strong correlation with quality? How would you approach investigating whether any of these correlations imply a causal relationship? What challenges might you encounter in establishing causation?