There is a significant variation in the extent to which different factors influence the development of the sporting goods manufacturing industry. According to data from October 28, 2014, at 09:10, empirical analysis shows that the impact of residents' cultural education levels and per capita income on this industry differs greatly across regions. For example, provinces like Zhejiang, Fujian, Jiangsu, Guangxi, and Tianjin, as well as areas in North and South China, may have similar levels of education and income, yet their contributions to the growth of the sporting goods sector vary considerably. Looking at the influence coefficients among these factors, and without considering other external variables, the main reason for such differences lies in the fact that the development of China's sporting goods manufacturing is not solely shaped by education and income levels. Other external factors, such as national or local political directions, industrial policies, market conditions, and regional resource advantages, also play critical roles, leading to distinct influence coefficients across regions. Empirical results indicate a long-term cointegration relationship between the level of cultural education, national income, and the time series of sporting goods manufacturing development. In most regions of China, both the cultural education level and per capita income of residents have a long-term positive impact on the industry's growth. However, the effect of national income levels on the development of the sporting goods manufacturing industry tends to be stronger than that of cultural education. This study focuses on the internal factors of residents—such as their education and income levels—to analyze their impact on the sporting goods manufacturing industry. Nevertheless, it's important to note that many other factors also influence this sector, including technological research and development, industrial structure, human resources, foreign investment, trade, brand strength, and even financial crises. These elements can significantly shape the development trajectory of the industry. The author plans to further explore these aspects in future research, aiming to provide a more comprehensive understanding of the factors driving the growth of the sporting goods manufacturing industry in China.

4.Mathmatical set

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