ENGI 3423 Additional Exercises
Question 4

A particular type of motor is known to have an output torque whose range in normal operation follows a normal distribution. Seven motors are chosen at random and are tested with the old and new methods of controlling the range of torque values. The results of the tests are as follows:
Motor:	  1	  2	  3	  4	  5	  6	  7	
New method:	 5.25	 3.16	 4.43	 6.12	 5.75	 2.21	 6.01	
Old method:	 7.83	 6.22	 7.46	 8.83	 8.19	 5.64	 8.88	

(a) [Justify the use of a paired two sample t test.]
(b) Conduct an appropriate hypothesis test to determine whether there is sufficient evidence to conclude that the range of torques with the new method is at least 2 units less than with the old method. 
(c) Use the simple linear regression model on these data to find the equation of the line of best fit to these data.
(d) Find the coefficient of determination R2 and use it to comment on your answer to part (a) above.
(b) Using Stat > Basic Statistics > Paired t ... ,  the Session window produces 
Paired T-Test and CI: New, Old

Paired T for New - Old

                  N      Mean     StDev   SE Mean
New               7     4.704     1.516     0.573
Old               7     7.579     1.246     0.471
Difference        7    -2.874     0.334     0.126

99% upper bound for mean difference: -2.477
T-Test of mean difference = -2 (vs < -2): T-Value = -6.92  P-Value = 0.000

(c) Using Stat > Regression > Fitted Line Plot ... ,  the Session window produces
Regression Analysis: Old versus New

The regression equation is                             
Old = 3.75102 + 0.813631 New                           
                                                       
S = 0.195488      R-Sq = 97.9 %      R-Sq(adj) = 97.5 %

Analysis of Variance

Source            DF         SS         MS         F      P
Regression         1    9.12361    9.12361   238.740  0.000
Error              5    0.19108    0.03822                 
Total              6    9.31469                            

and the graph is 

(d)  The high  R2 value of 97.9% (and therefore high correlation) justify the choice of a paired t test.   The unpaired test relies on an assumption of zero correlation.