Question

For each of the following situations, give the degrees of freedom for the group (DFG), for error (DFE), and for the total (DFT). State the null and alternative hypotheses, H₀ and H_a, and give the numerator and denominator degrees of freedom for the F statistic.

(a) A poultry farmer is interested in reducing the cholesterol level in his marketable eggs. He wants to compare two different cholesterol-lowering drugs added to the hen's standard diet as well as an all-vegetarian diet. He assigns 30 of his hens to each of the three treatments.

DFG	=
DFE	=
DFT	=

numerator df    =

denominator df =

(b) A researcher is interested in students' opinions regarding an additional annual fee to support non-income-producing varsity sports. Students were asked to rate their acceptance of this fee on a seven-point scale. She received 95 responses, of which 31 were from students who attend varsity football or basketball games only, 17 were from students who also attend other varsity competitions, and 47 who did not attend any varsity games.

DFG	=
DFE	=
DFT	=

numerator df    =
denominator df =

(c) A professor wants to evaluate the effectiveness of his teaching assistants. In one class period, the 42 students were randomly divided into three equal-sized groups, and each group was taught power calculations from one of the assistants. At the beginning of the next class, each student took a quiz on power calculations, and these scores were compared.

DFG	=
DFE	=
DFT	=

numerator df    =
denominator df =

Answer 1

DFG is given by the number of columns - 1, DFE = Total sample size - number of columns and DFT = DFG + DFE

The df Numerator = DFG and df denominator = DFE

(a) Number of columns = number of treatments = 3

Total sample size = 90 (30 hens for each treatment)

DFG = 3 - 1 = 2

DFE = 90 - 3 = 87

DFT = 87 + 2 = 89

df numerator = DFG = 2

df denominator = DFE = 87

(b) Number of columns = number of categories = 3

Total sample size = 95

DFG = 3 - 1 = 2

DFE = 95 - 3 = 92

DFT = 92 + 2 = 94

df numerator = DFG = 2

df denominator = DFE = 92

(c) Number of columns = number of tgroups = 3

Total sample size = 42

DFG = 3 - 1 = 2

DFE = 42 - 3 = 39

DFT = 39 + 2 = 41

df numerator = DFG = 2

df denominator = DFE = 39