) How many degrees of freedom are available for hypothesis tests and confidence intervals for single regression coefficient

We use students' t distribution therefore degrees of freedom is v= n-2

Step-by-step explanation:

Confidence Interval Estimate of Population Regression Co efficient β.

To construct the confidence interval for β, the population regression co efficient , we use b, the sample estimate of β. The sampling distribution of b is normally distributed with mean β and a standard deviation σ.y.x / √(x-x`)². That is the variable z = b - β/σ.y.x / √(x-x`)² is a standard normal variable. But σ.y.x  is not known so we use S.y.x and also student's t distribution rather than normal distribution.

t= b - β/S.y.x / √(x-x`)² = b - β/Sb    [Sb = S.y.x / √(x-x`)²]

with v= n-2 degrees of freedom.

Consequently

P [ - t α/2<  b - β/Sb  < t α/2] = 1- α

or

P [ b- t α/2 Sb<   β  < b+ t α/2 Sb] = 1- α

Hence a 100( 1-α) percent confidence for β the population regression coefficient for a particular sample size n <30  is given by

b± t α/2 Sb

Using the same statistic a confidence interval for α can be constructed in the same way for  β  replacing  a with b and Sa with Sb.

a± t α/2 Sa

Using the t statistic  we may construct the confidence interval for U.y.x for the given value X0 in the same manner

Y~0 ± t α/2(n-2) SY~

Y~0= a+b X0