Now even united state of America as an example of a capitalist economy has the government regulating and also taking part in the decision making of the economy.does that makes it a mixed economic also?
Yes, the US is an example of a mixed economy. Some examples of how the government impacts the economy include laws and licensing requirements that control the ability to buy or sell certain goods/services (liquor laws, gambling laws, excise taxes, licensing for certain professions, etc.), and the government's involvement in building infrastructure such as educational institutions, roads, hospitals, etc. I don't know of any economy that is truly a free market economy.