More Regression

We set variables appropriately for the multiple linear regression model

$\displaystyle Y_j = \beta_0 + \beta_1 x_{1j} + \beta_2 x_{2j} + \cdots + \beta_k x_{kj} + \epsilon_j,
\quad j=1,\ldots,n,

starting from the Response $ Y_j$ to predictor X1 up to X9.

The summary of multiple linear regression is obtained in the table below.

The standard error $ S_i$ for the estimate $ \beta_i$ gives rise to the null hypothesis

$\displaystyle H_0:\: \beta_i = 0

for each $ i = 0,\ldots,k$. It can be constructed to find whether the response is dependent of the i-th predictor. Under the null hypothesis the test statistic $ T_i = \displaystyle\frac{\hat{\beta}_i}{S_i}$ is distributed as the t-distribution with $ (n-k-1)$ degrees of freedom. Thus, we reject $ H_0$ at significance level $ \alpha$ if $ \vert T_i\vert > t_{\alpha/2,n-k-1}$. By computing the p-value $ p_i^*$ we can equivalently reject $ H_0$ if $ p_i^* < \alpha$.