Chapter 12.1, Problem 19E

(a)

Section 1:

To determine

To find: The degrees of freedom for group, for error, and for the total.

Answer to Problem 19E

Solution: The required degrees of freedom are $\underset{\_}{\text{DFG}=2,\text{DFE =}72,\text{ and DFT}=74}$.

Explanation of Solution

Calculation: The variable I represents the number of groups and the variable N represents the total number of observations. According to the information given in the question

$\begin{array}{c}I=3\\ N=3\times 25\\ =75\end{array}$

If DFG, DFE and DFT represent the group, error and total degrees of freedom respectively then they are calculated as shown below,

$\begin{array}{c}\text{DFG}=I-1\\ =3-1\\ =2\end{array}$

$\begin{array}{c}\text{DFE}=N-I\\ =75-3\\ =72\end{array}$

$\begin{array}{c}\text{DFT}=N-1\\ =75-1\\ =74\end{array}$

Section 2:

To determine

To find: The degrees of freedom for group, for error, and for the total.

Answer to Problem 19E

Solution: The required degrees of freedom are $\underset{\_}{\text{DFG}=2,\text{ DFE}=91,\text{ and DFT}=93}$.

Explanation of Solution

Calculation: The variable I represents the number of groups and the variable N represents the total number of observations. According to the information given in the question

$\begin{array}{c}I=3\\ N=94\end{array}$

If DFG, DFE and DFT represent the group, error and total degrees of freedom respectively then they are calculated as shown below,

$\begin{array}{c}\text{DFG}=I-1\\ =3-1\\ =2\end{array}$

$\begin{array}{c}\text{DFE}=N-I\\ =94-3\\ =91\end{array}$

$\begin{array}{c}\text{DFT}=N-1\\ =94-1\\ =93\end{array}$

Section 3:

To determine

To find: The degrees of freedom for group, for error, and for the total.

Answer to Problem 19E

Solution: The required degrees of freedom are $\underset{\_}{\text{DFG}=2,\text{DFE}=39,\text{and}\text{DFT}=41}$.

Explanation of Solution

Calculation: The variable I represents the number of groups and the variable N represents the total number of observations. According to the information given in the question

$\begin{array}{c}I=3\\ N=42\end{array}$

If DFG, DFE and DFT represent the group, error and total degrees of freedom respectively then they are calculated as shown below,

$\begin{array}{c}\text{DFG}=I-1\\ =3-1\\ =2\end{array}$

$\begin{array}{c}\text{DFE}=N-I\\ =42-3\\ =39\end{array}$

$\begin{array}{c}\text{DFT}=N-1\\ =42-1\\ =41\end{array}$

(b)

Section 1:

To determine

To explain: The Null and alternative hypotheses for cholesterol level in egg.

Answer to Problem 19E

Solution: The required hypotheses are,

Null Hypothesis: All the treatment means are equal.

Alternative Hypothesis: At least two of the treatment means differ significantly.

Explanation of Solution

The null hypothesis is that there is no difference in the effect of three treatments on egg against the alternative hypothesis that not all the effects are equal that is, at least one effect will differ significantly from the other at the provided confidence level.

Section 2:

To determine

To explain: The Null and alternative hypotheses for rating on a seven point scale.

Answer to Problem 19E

Solution: The required hypotheses are,

Null Hypothesis: All the three types of students gave same mean ratings.

Alternative Hypothesis: The average ratings of at least two sources of response differ significantly.

Explanation of Solution

The null hypothesis assumes that there is no significant difference in ratings against the alternative hypothesis that not all the ratings are equal that is, at least one rating will differ significantly from other at the desired level of significance.

Section 3:

To determine

The Null and alternative hypotheses for quiz score.

Answer to Problem 19E

Solution: The required hypotheses are,

Null Hypothesis: All the three groups of students have same mean scores.

Alternative Hypothesis: The mean score of at least two groups differ significantly.

Explanation of Solution

The null hypothesis assumes that there is no difference in quiz score of the three groups against the alternative hypothesis that not all the scores are equal that is, at least one score will differ significantly from other at the provided confidence level.

(c)

Section 1:

To determine

To find: The degrees of freedom for numerator and denominator of F statistic.

Answer to Problem 19E

Solution: The required degrees of freedom are $\underset{\_}{\left(2,72\right)}$.

Explanation of Solution

Calculation: In F statistic, the degrees of freedom for numerator are given by $\text{DFG}=I-1$ and denominator degrees of freedom are given by $\text{DFE}=N-I$. From part (a) the degrees of freedom are,

$\text{DFG}=2\text{and DFE}=72$

Thus, the F statistic is represented as,

$F\left(I-1,N-I\right)=F\left(2,72\right)$

Section 2:

To determine

To find: The degrees of freedom for numerator and denominator of F statistic.

Answer to Problem 19E

Solution: The required degrees of freedom are $\underset{\_}{\left(2,91\right)}$.

Explanation of Solution

Calculation: In F statistic, the degrees of freedom for numerator are given by $DFG=I-1$ and denominator degrees of freedom are given by $\text{DFE}=N-I$. From part (a) the degrees of freedom are,

$\text{DFG}=2\text{and DFE}=91$

Hence F statistic is represented as

$F\left(I-1,N-I\right)=F\left(2,91\right)$

Section 3:

To determine

To find: The degrees of freedom for numerator and denominator of F statistic.

Answer to Problem 19E

Solution: The required degrees of freedom are $\underset{\_}{\left(2,39\right)}$.

Explanation of Solution

Calculation: In F statistic, the degrees of freedom for numerator are given by $\text{DFG}=I-1$ and denominator degrees of freedom are given by $\text{DFE}=N-I$. From part (a) the degrees of freedom are,

$\text{DFG}=2\text{and DFE}=39$

Hence, F statistic is represented as,

$F\left(I-1,N-I\right)=F\left(2,39\right)$