# Linear models and hypothesis testing

Studying for a final and I really can't figure out how to proceed beyond the first step of forming the linear model. Finding the matrix l for the null hypothesis lβ=0 is especially proving to be harder than I'd like to admit. Any help would be greatly appreciated.

• Add the self-study tag. – Michael R. Chernick Dec 13 '17 at 2:25
• Well, have you learned anything in the subject that you doing the final for? Did you derive a formula for $\hat\beta$ in the lectures? Why don't you write down $X$ in this case and then try forming $\hat\beta$ explicitly? The formula is very easy in this case. – Gordon Smyth Dec 13 '17 at 2:32

The null hypothesis says all beta are zero except intercept : $$\beta_1=\beta_2=\beta_3 = ... =\beta_n = 0$$ The alternate hypothesis says : At least one of them is not zero $$\beta_i \neq0$$

Essentially the hypothesis is telling you to compare the 2 models

Model A : $\quad\quad y = \beta_0$

Model B : $\quad\quad y = \beta_0 + \beta_1 * x_1+ \beta_2 * x_2 + \beta_3 * x_3 + .... + \beta_n * x_n$

To compare 2 nested models, ( nested means model A is completely inside model B) we will use an F test.

$F_{calc} = \frac{\frac{SSE(A) - SSE(B)}{Difference \ in\ parameters\ of\ model\ A \ and\ model\ B }}{\frac{SSE(B)}{Num\ of\ data\ points-number\ of\ param\ in\ Model\ B}}$ Here $Difference \ in\ parameters\ of\ model\ A \ and\ model\ B = n$

Here $Number\ of\ data\ points = 4\ (i=1 .. 4)$

Here $Number\ of\ param\ in\ Model\ B = N$

Calculate these values and get the value of $F_{calc}$ and compare it to the value of $F_{table}$.

If the $F_{calc} > F_{table}$, then you reject null hypothesis, else accept it.

• I think you have misunderstood the question. The OP does not have many predictor variables but only one taking on the values 1, 2, 3, 4. – mdewey Dec 27 '17 at 15:19
• Yes, you are correct. I just gave a general understanding of the model with multiple predictors, to give him a bigger picture. – user3808268 Dec 27 '17 at 15:22