Your hypothesis seems to be the correct way around. I would suggest that you calculate the p-values of all the different variables in your model. This will help you to see how significant the variables are relative to one another. Since you didn't specify, I am not sure if you are calculating this by hand of if you are using a program.
If you are doing it by hand, I suggest that you also try and do it in R using the anova() or summary() function. In this way you can check whether you calculations were correct. Remember if you at testing at 5% significance level, the p-values need to be smaller than 0.05 for you to reject the null hypothesis.
To answer our other question, if you are calculating by hand, you can use the F-test for testing the overall adequacy of the model. For each beta coefficient you will have to use a t-test. From those you can calculate the corresponding p-values.
When modelling the response with only one predictor (simple linear regression), the predictor might be statistically significant. However when you use it together with other predictors in the model (multiple regression) that predictor might not be significant at all. This is often due to multicollinearity.
Hope this answers your question!