e-Statistics

## Inference on Parameters

The concepts of hypothesis tests and confidence intervals can be applied to linear regression models. Inferences on the slope parameter is of particular interest since it determines the nature of relationship between the explanatory and the dependent variable.

The data set consists of

1. explanatory variable for 's;
2. dependent variable for 's.
The analysis for simple linear regression are summarized in the following table. Under the standard assumption of regression model we can make hypotheses for the coefficients and , and test them via p-value.

1. is the standard error for the estimate of slope. The null hypothesis

can be constructed to find whether the response variable is dependent of the explanatory variable. Under the null hypothesis the test statistic is distributed as the -distribution with degrees of freedom. Thus, we reject at significance level if . By computing the p-value we can equivalently reject if . When is rejected, we can proceed to construct the confidence interval of level for the coefficient as

( , )

2. is the standard error for the estimate of intercept. The null hypothesis

may not be of particular interest, but the procedure for hypothesis testing can be similarly proceeded. Under the null hypothesis the test statistic has the -distribution with degrees of freedom, and is reject at significance level if , or, equivalently when the -value satisfies . Then the confidence interval of level for the coefficient as
( , )