Question

The accompanying table shows the regression results when estimating y = β₀ + β₁x + ε.

	Coefficients	Standard Error	t-stat	p-value
Intercept	0.083	3.56		0.9822
x	1.417	0.63		0.0745

When testing whether the slope coefficient differs from 1, the value of the test statistic is ____.

0.66

1.42

1.96

2.25

Answer 1

According to the table the final simple linear regression equation is :

Where is the response variable , is the only independent variable and is the error part in the model.

Now to test    vs    we need to follow the following procedure.

Now as the errors   , the observations .

Now   is a linear combination of the observations , so   is normally distributed with mean and variance . Therefore the statistic

is distributed if the null hypothesis   is true. if   is known , we could use to test the hypotheses . Typically is unknown , we know that is an unbiased estimator of .

Now and that and are independent .

therefore follows distribution if the null hypothesis is true.

Now

therefore

which is the desired value of the test statistic.