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Inference on Parameters

Hypothesis tests and confidence intervals are considered for linear regression models. Inferences on the slope parameter $ \beta_1$ is of particular interest since it determines the nature of relationship (positive or negative trend, or no relation) between the explanatory and the dependent variable.

The data set consists of

  1. explanatory variable for $ x_i$'s;
  2. dependent variable for $ Y_i$'s.
The analysis for simple linear regression are summarized in the following table.

It is important to know that the standard errors (SE's) can be calculated as follows:

  1. $ \displaystyle S_0 = \hat{\sigma} \sqrt{\frac{1}{n} + \frac{\bar{x}^2}{S_{xx}}}$ is the SE for the estimate $ \beta_0$ of intercept.
  2. $ \displaystyle S_1 = \frac{\hat{\sigma}}{\sqrt{S_{xx}}}$ is the SE for the estimate $ \beta_1$ of slope.
Once the SE's are obtained, we can proceed to construct the confidence interval of level $ (1-\alpha)$ = for the coefficients $ \beta_0$ and $ \beta_1$ as follows:

$\hat{\beta}_0 \pm t_{\alpha/2,n-2} S_0$ = ( , )

$\hat{\beta}_1 \pm t_{\alpha/2,n-2} S_1$ = ( , )

Under the standard assumption of regression model we can make hypotheses for the coefficients $ \beta_0$ and $ \beta_1$, and test them via p-value.

  1. The null hypothesis

    $\displaystyle H_0:\: \beta_0 = 0 $

    for the intercept parameter $ \beta_0$ may not be of particular interest, but the hypothesis test can be performed to see whether the intercept is significant or not. Under the null hypothesis the test statistic $ T_0 = \displaystyle\frac{\hat{\beta}_0}{S_0}$ has a t-distribution with df = (n-2) = degrees of freedom, and $ H_0$ is reject at significance level $ \alpha$ if $ \vert T_0\vert > t_{\alpha/2,n-2}$, or equivalently if p-value < $ \alpha$.

  2. The null hypothesis

    $\displaystyle H_0:\: \beta_1 = 0 $

    for the slope parameter $ \beta_1$ can be constructed in order to find whether the response variable $ Y_i$ is linearly dependent on the explanatory variable $ x_i$ (in favor of the alternative hypothesis $H_A:\:\beta_1\neq 0$) or not. Under the null hypothesis the test statistic $ T_1 = \displaystyle\frac{\hat{\beta}_1}{S_1}$ is distributed as t-distribution with $ (n-2)$ degrees of freedom. Thus, we reject $ H_0$ at significance level $ \alpha$ if $ \vert T_1\vert > t_{\alpha/2,n-2}$, or equivalently if p-value < $ \alpha$.

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