MATH 376 -- Probability and Statistics 2 

A hypothesis test on a regression coefficient  

April 15, 2010 

>	with(Statistics): with(plots):

 

Using the median housing price data from class a few days ago.  

We enter the lists of x  and  y  coordinates of the data points separately. 

>	XList:=[0.,1,2,3,4,5,6,7];

 

(1)

 

>	YList:=[27.6,32.5,35.9,39.3,44.8,48.8,55.7,62.9];

 

(2)

 

Means:
 

>	Xbar:=Mean(XList);

 

(3)

 

>	Ybar:=Mean(YList);

 

(4)

 

Organizing the computation of the least squares estimators of the regression coefficients
as we described in class: 

>	SXY:=add((XList[i]-Xbar)*(YList[i]-Ybar),i=1..8);

 

(5)

 

>	SXX:=add((XList[i]-Xbar)*(XList[i]-Xbar),i=1..8);

 

(6)

 

>	hatbeta[1]:=SXY/SXX;

 

(7)

 

>	hatbeta[0]:=Ybar-hatbeta[1]*Xbar;

 

(8)

 

We can ask whether there is evidence to say that the median house price was 

increasing at a rate less than 5 thousand dollars per year.


That is, we set up for testing 

     :    (or perhaps ≥ 5 -- essentially equivalent)   versus the alternative 

     :   

To compute the test statistic we need the estimator  for the variance  

>	SYY:=add((YList[i]-Ybar)*(YList[i]-YBar),i=1..8);

 

(9)

 

>	S2:=(1/(8-2))*(SYY-hatbeta[1]*SXY);

 

(10)

 

Then the test statistic is  ,  where  

>	t:=(hatbeta[1] - 5)/sqrt(S2/SXX);

 

(11)

 

The test statistic has a  t-distribution with 8 - 2 = 6 d.f.  so the 

p-value for the lower tail test is: 

>

 

(12)

 

>

CDF(T,t);

 

(13)

 

Note:  1 - CDF(T,-t)  gives the same result by symmetry of the t  

density function. 

This value is much too large to indicate rejection of .  There is  

not sufficient evidence to suggest that