Question

In a simple regression analysis for a given data set, if the null hypothesis β = 0 is rejected, then the null hypothesis ρ = 0 is also rejected. This statement is ___________ true. always

Answer 1

Null hypothesis:
`\rho =0`

Alternative hypothesis:
`\rho \\eq 0`

The statistic to check the hypothesis is given by:

`t=(r √(n-2))/(√(1-r^2))`

And is distributed with n-2 degrees of freedom

And the statistic to check the significance of a coeffcient in a regression is given by:

`t_1 = \frac{\hat{\beta_1} -0}{S.E (\hat{\beta_1})}`

For this case is importantto remember that t1 and p value for test of slope coefficient is the same test statistic and p value for the correlation test so then the answer would be:

Always

Explanation:

In order to test the hypothesis if the correlation coefficient it's significant we have the following hypothesis:

Null hypothesis:
`\rho =0`

Alternative hypothesis:
`\rho \\eq 0`

The statistic to check the hypothesis is given by:

`t=(r √(n-2))/(√(1-r^2))`

And is distributed with n-2 degrees of freedom

And the statistic to check the significance of a coeffcient in a regression is given by:

`t_1 = \frac{\hat{\beta_1} -0}{S.E (\hat{\beta_1})}`

For this case is importantto remember that t1 and p value for test of slope coefficient is the same test statistic and p value for the correlation test so then the answer would be:

Always