Test the hypothesis that the slope is positive against the alternative that it is negative at the 5% level of significance. What is the p-value?

Single Linear Regression Theory Review

1. Recall that we define our parameters of interest β0 and β1 as the parameters governing the “line of
   best fit” between Y and X:
   β0, β1 = arg min
   b0,b1
   E[(Y − b0 − b1X)
   2
   ]. (1)
   Once we define these parameters we define the regression error term  = Y − β0 − β1X which then
   generates the linear model
   Y = β0 + β1X + .
   (a) Using the first order conditions for β0 and β1 (set the derivatives of the right hand side of (1)
   with respect to b0 and b1 equal to zero at) show why E[] = E[X] = 0.
   (b) Using the definition of β0 and β1 as line of best fit parameters, give an intuitive explanation for
   why E[] = 0.
   Hypothesis Testing and Confidence Intervals
   In the following questions, whenver running a hypothesis test, please state the null and alternative hypotheses,
   show some work, and state the conclusion of the test.
2. In an estimated simple regression model based on n = 64, the estimated slope parameter, βˆ
   1, is 0.310
   and the standard error of βˆ
   1 is 0.082.
   (a) What is ˆσ
   2
   β1
   ? Recall σβ1
   is the terms such that, approximately for large n,
   √
   n(βˆ
   1 − β1) ∼ N(0, σβ1
   ).
   (b) Test the hypothesis that the slope is zero against the alternative that it is not at the 1% level of
   significance (α = 0.01).
   (c) Test the hypothesis that the slope is negative against the alternative that it is positive at the 1%
   level of significance (α = 0.01).
   (d) Test the hypothesis that the slope is positive against the alternative that it is negative at the 5%
   level of significance. What is the p-value?
   1