Question

Answer the one-way ANOVA questions using the data below. Use α = 0.01.

1	2	3	4
53 49 47 44 51 34 44	44 39 36 36 39 30 12	46 42 39 37 36 34 30	25 35 30 40 33 28 31

a) Compute the preliminary statistics below.
SS_BG =  ;   df_BG =  
SS_WG =  ;   df_WG =  
SS_T =  ; df_T =  

b) Compute the appropriate test statistic(s) to make a decision about H₀.
critical value =  ; test statistic =  
Decision:  ---Select--- Reject H0 Fail to reject H0

c) Compute the corresponding effect size(s) and indicate magnitude(s).
η² =  ;  ---Select--- na trivial effect small effect medium effect large effect

d) Make an interpretation based on the results.

At least one mean is different from another.None of the means differed from one another.    

Answer 1

a)

Let T_i be the total of observations for group i, n_i be number of observations of group i.

Let G be the total of all observations and N be total number of observations.

ΣX² is sum of squares of all observations.

T1 = 322, T2 = 236 , T3 = 264, T4 = 222

G = 322 + 236 + 264 + 222 = 1044

ΣX² = 15048 + 8614 + 10122 + 7184 = 40968

SST = ΣX² - G²/N = 40968 - 1044²/28 = 2041.714

SS_BG = ΣT²/n - G²/N = (322² /7 + 236² /7 + 264² /7 + 222² /7 ) - 1044²/28 = 839.4286

SS_WG = 2041.714 - 839.4286 = 1202.285

dfT = N - 1 = 28 - 1 = 27

df_BG = Number of groups - 1 = 4 - 1 = 3

df_WG  = Number of observations (N) - Number of groups = 28 - 4 = 24

b)

For α = 0.01, df = df_BG, df_WG = 3,24 , critical value of F is 4.718

MS_BG = SS_BG / df_BG = 839.4286 / 3 = 279.8095

MS_WG = SS_WG / df_WG   = 1202.285 / 24 = 50.09521

Test statistic, F =  MS_BG / MS_WG = 279.8095 / 50.09521 = 5.585

Since, F > critical value of F , Decision: Reject H0

c)

η² = SS_BG / SST= 839.4286 / 2041.714 = 0.41

Since, η² < 0.5, small effect

d)

Since we reject H0, we conclude that At least one mean is different from another.