Question

True or False, explain your answer:

a) An observation with a studentized residual of more than 10 is probably an outlier.

b) If assumptions are met, least squares residuals are not correlated with the fitted values.

c) It is possible to reject a null hypothesis when the null hypothesis is true.

Answer 1

a) TRUE

To check whether a observation is an outlier with the help of studentized residual we need to compare the value of the studentized residual with t dsitribution with (n-k-1) degrees of freedom where n is number of observations and k is number of predictors, this is because the studentized t-dsitribution is given by

where ei is ith error

MSE is mean square error

hii is (i,i)th element of hat matrix.

and ei is standard normally distributed and denominator has chi square distribution with (n-k-1) degree of freedom, hence ti has t distribution with (n-k-1) degree of freedom.

so depending upon this value we can detect whether a point is outlier or not.

Usually if a value of studentized residual is beyond the tail of t distribution we take that point as an outlier, so in most of the t distribution cases 10 is far beyond the tails.

Hence value greater than 10 can b considered as outlier.

b) TRUE

I have attached the proof below

c) TRUE

Yes it is true that it is possible to reject the null hypothesis even if it is true because type 1 error is defined as the probability to reject the null hypothesis even if it is true, so at some level of significance it is possible to reject the null hypothesis.