An ANOVA procedure is applied to data obtained from 6 samples where each sample contains 20 observations. The degrees of freedom

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An ANOVA procedure is applied to data obtained from 6 samples where each sample contains 20 observations. The degrees of freedom

for the critical value of F areSelect one:A. 6 numerator and 20 denominator degrees of freedomB. 5 numerator and 14 denominator degrees of freedomC. 5 numerator and 114 denominator degrees of freedomD. 6 numerator and 20 denominator degrees of freedom

Mathematics

1 answer:

PIT_PIT [3.9K] · Answer 1 · 2026-03-03T12:26:24+03:00

Answer:

C. 5 degrees of freedom for the numerator and 114 for the denominator

Step-by-step explanation:

Analysis of variance (ANOVA) is utilized to examine the variations among group means within a sample.

The sum of squares represents the cumulative square of variation, which refers to the deviation of each individual value from the grand mean.

Assuming there are $6$ groups and each group contains $j=1,\dots,20$ individuals, the variation can be calculated using the following formulas:

$SS_{total}=\sum_{j=1}^p \sum_{i=1}^{n_j} (x_{ij}-\bar x)^2$

$SS_{between}=SS_{model}=\sum_{j=1}^p n_j (\bar x_{j}-\bar x)^2$

$SS_{within}=SS_{error}=\sum_{j=1}^p \sum_{i=1}^{n_j} (x_{ij}-\bar x_j)^2$

This also has the property

$SST=SS_{between}+SS_{within}$

The numerator's degrees of freedom in this case is given by $df_{num}=df_{within}=k-1=6-1=5$ where k = 6 represents the number of groups.

The denominator's degrees of freedom in this scenario is indicated by $df_{den}=df_{between}=N-K=6*20-6=114$ .

The total degrees of freedom would be $df=N-1=6*20 -1 =119$ .

Thus, the appropriate answer would be 5 degrees of freedom for the numerator and 119 degrees of freedom for the denominator.

C. 5 numerator and 114 denominator degrees of freedom