# How to compute minimum sample size of a simple linear regression model with given statistics values

Suppose the statistics values are given as follows: $$\sum_{i=1}^{n}x_i, \sum_{i=1}^{n}y_i, \sum_{i=1}^{n}x_iy_i, \sum_{i=1}^{n}x_i^2,\sum_{i=1}^{n}y_i^2$$

Firstly, we can compute the regression coefficient $$\beta_0, \beta_1$$ such that: $$Y =\beta_0 + \beta_1 X$$

An ANOVA (F-test) can also be carried out to see whether X and Y are dependent. Suppose $$F , p-value $$> 0.05$$ such that there is no strong evidence that $$\beta_1 \ne 0$$. i.e. X and Y are independent.

However, what if we were told that $$\sum_{i=1}^{n}(x_i-\bar x)^2, \sum_{i=1}^{n}(y_i-\bar y)^2, \sum_{i=1}^{n}(x_i-\bar x)(y_i-\bar y)$$ were doubled, X and Y are dependent by ANOVA, and find the minimum sample size $$n$$.

My approach: I am new to ANOVA and statistics test, the only I can think of is to start it with F-test formula , and if X and Y are dependent, it should fulfill $$F>F_{critical}$$ but $$F_{critical}$$ is unknown due to unknown sample size $$n$$.

Besides, the new model has the same $$\beta_1$$ as before. from doubling up $$\sum_{i=1}^{n}(y_i-\bar y)^2$$. We can also find out the new $$SS_{TOTAL}$$

Apologies for the unorganised ideas above. I have no idea of the right direction to solve this problem.

I would like to know how to solve this please. Thank you very much.