Question

1. a researcher uses an anova to compare six treatment conditions with a sample of ( n = 12 ) in each treatment. for this analysis, find ( d f_{\text {total }}, d f_{\text {between }} ), and ( d f_{\text {within }} ).

Explanation:

Step1: Calculate \( df_{\text{total}} \)

The formula for \( df_{\text{total}} \) in ANOVA is \( df_{\text{total}} = N - 1 \), where \( N \) is the total number of observations. There are 6 treatment conditions, each with \( n = 12 \), so \( N = 6\times12 = 72 \). Then \( df_{\text{total}} = 72 - 1 = 71 \).

Step2: Calculate \( df_{\text{between}} \)

The formula for \( df_{\text{between}} \) is \( df_{\text{between}} = k - 1 \), where \( k \) is the number of treatment conditions. Here, \( k = 6 \), so \( df_{\text{between}} = 6 - 1 = 5 \).

Step3: Calculate \( df_{\text{within}} \)

The formula for \( df_{\text{within}} \) is \( df_{\text{within}} = N - k \). We know \( N = 72 \) and \( k = 6 \), so \( df_{\text{within}} = 72 - 6 = 66 \).

Answer:

\( df_{\text{total}} = 71 \), \( df_{\text{between}} = 5 \), \( df_{\text{within}} = 66 \)