As you can see from the provided picture given Heart attack given rate per 100,000 population. I was able to successfully ran my regression; but now I am trying to figure out how to alter my coefficients if heart attack rates were reported per 1000 populations. (I was given a hint saying that demonstrate this algebraically using the formulas for the OLS coefficients: (i) 0 ,(ii) 1 but couldn't figure out) any directives would greatly appreciated.
You have $y_i = \beta_0 + x_i \beta_1 + \epsilon_i$ and we are replacing $x_i$ by $1000x_i$ to get $y_i = \beta_0' + (1000x_i) \beta_1' + \epsilon_i$. What would be nice is if the constant terms, $\beta_0$ and $\beta_0'$ are equal and also if the slopes $\beta_1$ and $1000\beta_1'$ are equal.
One way to show that this is what happens is to take the formulas:
$$\hat\beta_1 = \frac{\sum (x_i - \bar x)(y_i - \bar y)}{\sum (x_i - \bar x)^2}$$
What happens if you replace
$$x_i \gets 1000x_i \text{ and } \bar x \gets \frac1n\sum_{i=1}^n(1000x_i) = \frac{1000}{n}\sum_{i=1}^nx_i = 1000\bar x?$$
Then do the same for $\hat\beta_0 = \bar y - \bar x \hat \beta_1$.