Question

What are the typical null and alternate hypotheses for a significance test on slope?

Answer 1

SOLUTION :

The simple linear regression model is

Where is constant,   is the slope, x is the value of independent variable , y is the value of dependent variable and is the random error.

Hypothesis is given by

If there is a significant relationship between the independent variable x and dependent variable y, the slope will not equal to zero.

The null hypothesis is states that the slope is equal to zero and the alternative hypothesis States that the slope is not equal to zero.

Where we have specified a two-sided alternative. Since the errors are NID( 0, ) the observation

Now is normally distributed with mean and variance . Thus

........................(. )

If the null hypothesis is true. If were know, we could use Z to test the hypothesis. Typically, is unknown. We have already know MSres is an unbiased estimator of.

Estiblised that distribution. That MSres and are independent. By the definition of a t statistic is

. ..........................( . )

Follow a distribution if the null hypothesis. is true. The degree of freedom with are number of degree of freedom associated with MSres. Thus the ratio is the test statistic used to test . The test procedure computes And compares the observed value of from with the upper Percentage point of the distribution ( ). This procedure reject the null hypothesis if.

Alternatively, a P - value approach could also be used for decision making.

P- value : the P- value is probability of observing a sample statistic as extreme as the test statistics. Since the test statistics is a t statistic, use the t distribution calcutor to assess the probability associated with the test statistics.

Interpretation : if the sample finding are unlikely given the null hypothesis, the researcher reject the null hypothesis. Typically this involve comparing the P value to the significance level, the rejecting the null hypothesis when the P value is less than the significance level.