Question

In: Statistics and Probability

The following is a regression summary from R for a linear regression model between an explanatory...

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?

Solutions

Expert Solution

(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 :


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