Compute a one-way ANOVA - Give the source table, test null hypothesis at .05 level. Please explain the steps to solving this.
Group 1: 26, 34, 46, 48, 42, 49, 74, 61, 51, 53
EX 1 : 484
Ex 1^2 : 25,024
n = 10
Group 2: 51, 50, 33, 28, 47, 50, 48, 60, 71, 42
EX 2 : 480
EX 2^2 : 24,392
n = 10
Group 3: 52, 64, 39, 54, 58, 53, 77, 56, 63, 59
EX 3 : 575
EX 3^2 : 33,925
n = 10
Group 4: 41, 49, 56, 64, 72, 65, 63, 87, 77, 62
EX 4 : 636
EX 4^2: 42,034
n = 10
Let's use an example, and perhaps you can use this as a guideline for similar ANOVA problems.
Suppose we wish to establish if a difference exists between three different levels of something (it doesn't matter what it is...this is just for purposes of illustration).
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Here are the values for each level:
Level 1: 0, 1, 1, 2, 2
Level 2: 1, 2, 1, 3, 3
Level 3: 3, 3, 2, 4, 3
Totals for each level are:
Level 1: 6 -->add each value in the level for a total.
Level 2: 10
Level 3: 15
�Total of squared values for each level are:
Level 1: 10 -->square each value in the level, then add for total.
Level 2: 24
Level 3: 47
To calculate SS within for the ANOVA table, take (sum of total squared values) minus (sum of totals divided by number of values per level). It would look like this:
(10 + 24 + 47) - [(6^2 + 10^2 + 15^2)/5] = 8.8
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SS between would be calculated this way:
(6^2 + 10^2 + 15^2)/5 - (6 + 10 + 15)^2/15 = 8.13
Note: 15 = total number of values in all levels.
SS total = SS between + SS within = 16.93
To calculate df between:
k - 1 = 3 - 1 = 2
Note: k = number of levels.
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To calculate df within:
N - k = 15 - 3 = 12
Note: N = total number of values in all levels.
df total = df between + df within = 14
To calculate MS between:
SS between/df between = 8.13/2 = 4.065
To calculate MS within:
�SS within/df within = 8.8/12 = 0.73
To calculate F-ratio:
MS between/MS within = 4.065/0.73 = 5.57
Therefore, the ANOVA table would look like this:
Source......SS......df....MS......F
Between....8.13......2....4.065...5.57
Within.....8.8......12....0.73
Totals....16.93.....14
I hope this example will help. It looks like you already have the values needed to do the calculations for each of your 4 groups. Remember to check the appropriate table for your critical value to compare to your F-ratio. Once you do the comparison, you will either fail to reject the null or you will reject the null. You can then form your conclusions.
To compute a one-way ANOVA and test the null hypothesis at a .05 level, you can follow these steps:
Step 1: Calculate the sum of each group.
Group 1: Sum of X1 = 26 + 34 + 46 + 48 + 42 + 49 + 74 + 61 + 51 + 53 = 484
Group 2: Sum of X2 = 51 + 50 + 33 + 28 + 47 + 50 + 48 + 60 + 71 + 42 = 480
Group 3: Sum of X3 = 52 + 64 + 39 + 54 + 58 + 53 + 77 + 56 + 63 + 59 = 575
Group 4: Sum of X4 = 41 + 49 + 56 + 64 + 72 + 65 + 63 + 87 + 77 + 62 = 636
Step 2: Calculate the sum of squares (SS) for each group.
SS1 = Σ(X1i - EX1)^2 = 25024 - (484)^2/10 = 2502.4 - 23696.96 = -21194.56
SS2 = Σ(X2i - EX2)^2 = 24392 - (480)^2/10 = 2439.2 - 23040 = -20600.8
SS3 = Σ(X3i - EX3)^2 = 33925 - (575)^2/10 = 3392.5 - 33062.5 = -29670
SS4 = Σ(X4i - EX4)^2 = 42034 - (636)^2/10 = 4203.4 - 40461.6 = -36258.2
Step 3: Calculate the total sum of squares (SST).
SST = SS1 + SS2 + SS3 + SS4
Step 4: Calculate the degrees of freedom (df).
df_between = number_of_groups - 1
df_within = total_number_of_observations - number_of_groups
df_total = total_number_of_observations - 1
Step 5: Calculate the mean square between (MSB) and mean square within (MSW).
MSB = SS_between / df_between
MSW = SS_within / df_within
Step 6: Calculate the F-statistic.
F = MSB / MSW
Step 7: Determine the critical value of F for a significance level of .05 using the F-distribution table or calculator.
Step 8: Compare the calculated F-statistic with the critical value of F. If the calculated F-statistic is greater than the critical value, reject the null hypothesis; otherwise, fail to reject the null hypothesis.
By following these steps, you can compute a one-way ANOVA and test the null hypothesis at a .05 level using the provided data.