Understanding T-Test and Statistical Significance in Regression Analysis
Let me give you a very special case which is just a simple linear regression with one variable. If the quiz gives you Y = aX + b, and they ask you what is the relationship between X and Y, and is the relationship significant at maybe 5% or 1%, we'll encounter questions like this.
There are two different scenarios: one is a simple linear regression, and the other is multiple regression, which means Y = a₁X₁ + a₂X₂ + ... + aₙXₙ + b + ε. You might get multiple regression, and they'll ask you about the relationship between X₁ and Y, or all the X₁, X₂, up to Xₙ and their relation to Y.
My suggestion is that if the quiz has a P-value, we are going to use the P-value, especially when analyzing simple regression like Y = aX + b. If we get the P-value of a, and the P-value is, say, 0.04, which is less than 5%, once you get the P-value less than the critical value, it is significant.
If you don't have a P-value, you can also use the T-statistics. T-statistics is also a good way, and for simple linear regression, you can calculate the T-statistics of the slope coefficient. You can calculate that and compare whether it's bigger or smaller than the critical value.
For multiple regression, the first method that should come to mind is to use F-statistics or calculate the F-value to evaluate whether X₁, X₂, all the way to Xₙ coefficients - whether all these variables can explain the variance of Y. F-value is more like taking all variables as a whole to determine whether they explain the variance of Y or not.
So we have three methods - P-value, T-statistics, and F-value - to make our judgment for whether something is significant at a certain level. Remember that the F-value is a one-tailed test, and we check the F-statistics table to find the critical value. We calculate the F-value, and if the F-value is bigger than the critical value, it is significant.