How do I analyze data to find points where there is a "significant" increase in the "y" variable.

I am looking at the relationship between two variables 'x' and 'y'. What analysis do I perform if I had the following question. "At what point in the 'x' variable is there a significant change in the 'y' variable". Say for example, say my 'x' variable ranges from 1 - 10, and I want to be able to say "at point 4, there is a significant change in 'y', compared with points 1 - 3.

What do I look for in my dataset to know if I can answer such a question, and if so, how do I go about analyzing the data to answer such a question?

Thanks in advance.


No cake for spunky
Significant in this sense is not a statistical concept. You have to decide for yourself what significant change in Y is. Also in linear regression it is assumed that Y changes the same at different levels of X. So if you are finding that the impact of x on Y changes at various levels of X you are violating an important concept of linear regression and have to find a way to address this.
I would define significance in this case as the x where the second derivative y = f(x) exceeds zero.
x2 y136
x3 y140
x4 y144
x5 y173
I'd pick x = 5
A string regression helps?


No cake for spunky
I think the key here is if you are talking about statistical or substantive significance. Probably one of the most misunderstood concepts ever.