Question

1. Consider simple regression model (one independent variable)

a) How do you find a confidence interval for the coefficient of X?

b) How do you conduct a test if the coefficient of X is 10 or not?

c) How do you find a CI for the mean of the dependent variable if x=2?

Answer 1

(a)

Let suppose for simple regression model x(independent variable) and y(dependent variable).

The equation for the simple regression model is:

,  

Now we have to construct a confidence interval for , which is the slope of the population regression line.

Confidence interval for

Since the slope of the least square regression line is a point estimate of .

,

where, mean square error

(b) To conduct a test for the coefficient of X(or we say for ) whether it is 10 or not, we use a t-test.

The null and alternative hypothesis are as follows:

       

The formula to calculate the value of test statistics is:   

This test statistics is use to calculate the P-value. Ans the P-value is used to answer the question , that how likely we get a test statistic as extreme as we did, if the Null hypothesis, is TRUE, and then using a P-value and by comparing it with the significance level , we make a decision Whether to REJECT null hypothesis or not.

(c) The confidence interval for the mean of the dependent variable/response when the independent/predictor variable will take some value.

The general form for confidence interval in words is:

where,

fitted/ predicted value of the dependent variable when the independent/predictor takes the value .

independent variable value, in this case

mean square error,