a researcher wishes to see whether there is any difference i

a researcher wishes to see whether there is any difference in the weight gains of three athletes following one of three special diets. Athletes are randomly assigned to three groups and placed on the diet for 6 weeks. The weight gains (in pounds) are shown here. At alpha equal to 0.05, can the researcher conclude that there is a difference in the diets? *Manually* construct an ANOVA and perform Scheffe\'s test. Diet A: 3, 6, 7, 4. Diet B: 10, 12, 11, 14, 8, 6. Diet C: 8, 3, 2, 5.

Solution

Step 1                  
   Null Hypothesis Ho :   µ1 =µ2 =µ3   
   Alternative Hypothesis : µ1 µ2 µ3   
Step 2                              
   Degrees of freedom between = k - 1 = 3 - 1 = 2                          
   Degrees of freedom Within = n - k = 14 - 3 = 11                      
                              
   Degrees of freedom Total F( k-1,n - k,) at 0.05 is = F Crit = 3.9822
Step 3
Grand Mean = G / N = 7.1                              
SST = ( Xi - GrandMean)^2 = (3-5.5)^2 + (6-5.5)^2 + (7-5.5)^2 + ……..& so on = 101.10                      
SS Within = (Xi - Mean of Xi ) ^2 =,(3-3.333)^2 + (6-3.333)^2 + (7-3.333)^2 + ……..& so on = 71.83

Step 4.
Mean Square Between = SS Between / df Between = 101.10/2 = 50.547
Mean Square Within = SS Within / df Within = 71.83/11 = 6.53

Step 5                          
   F Cal = MS Between / Ms Within = 50.547/6.53 = 7.740
   We got |F cal| = 7.740 & |F Crit| =3.9822                      
                          
MAKE DECISION                          
   Hence Value of |F cal| > |F Crit|and Here We Reject Ho                      

conclude that    there is a difference in the diets