In: Statistics and Probability
The following is a regression summary from R for a linear regression model between an explanatory variable x and a response variable y. The data contain n = 50 points. Assume that all the conditions for SLR are satisfied.
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.1016 0.4082 -2.699 -------**
x 2.2606 0.0981 ---- < 2e-16 ***
(a) Write the equation for the least squares regression line.
(b) R performs a t-test to test whether the slope is significantly different than 0. State the null and alternative hypothesis for this test. Based on the p-value what is the conclusion of the test (i.e., reject or do not reject the null hypothesis)?
(c) Calculate the missing p-value for the intercept.
(d) Calculate the missing t-statistic for the slope.
(e) Calculate a 95% confidence interval for the slope of the regression line. Does this interval agree with the results of the hypothesis test?
(a)
Equation for the least squares regression line :
(b)
Null and alternative hypothesis for this test :
Let level of significance = 0.01
We have the P-value = 2 e-16 < 0.01 ( Level of significance) , we reject Ho and conclude that Slope is significantly different than 0.
(c)
We can either use T-tables or use excel function "TDIST() " to find the P-value
TDIST( | t value | ,df , 1 = one tailed or 2 = two-tailed)
Here, we have a two-tailed test.
df = Degrees of freedom = n - 2 = 50 - 2 =48
P-value for the intercept = TDIST(|-2.699|,48,2) = 0.00957
(d)
t-statistic for the slope can be calculated using the excel function " TINV() "
t-statistic for the slope = TINV(2e-16,48) = 12.282
(e)
100(1-)% confidence interval for the slope of the regression line :
95% confidence interval for the slope of the regression line :