Chapter 24, Problem 24.11AYK

a.

To determine

To find out whether the standard deviations satisfy the rule of thumb for safe use of ANOVA.

Explanation of Solution

Given:

The following is the given table-

  

Calculation:

The largest standard deviation is 54 and smallest standard deviation is 30. The ratio of the largest standard deviation to smallest standard deviation is:

  $\frac{54}{30}=1.8$

The ratio of the largest standard deviation to smallest standard deviation is less than 2. The standard deviations satisfy the rule of thumb for safe use of ANOVA.

b.

To determine

To find out whether ANOVA is safe for data.

Explanation of Solution

Given:

The following is the given table-

  

Calculation:

All samples are greater than 30. Sample sizes are large enough to assume that sampling distribution is normally distributed.

c.

To determine

To calculate the overall mean response $\overline{x}$ , the mean square for groups (MSG), and the mean square for error (MSE).

Explanation of Solution

Given:

The following is the given table-

  

Calculation:

Here

  $\begin{array}{l}k=3\\ n=36+39+35=110\end{array}$

Compute combined sample mean:

  $\begin{array}{l}\overline{x}=\frac{{\displaystyle \sum _{i=1}^k{n}_i{x}_i}}{{\displaystyle \sum _{i=1}^k{n}_i}}\\ =\frac{36\times 25.1+39\times 8.2+35\times 28.7}{110}\\ =20.25\end{array}$

Compute the sum of squares:

  $\begin{array}{l}SSG={\displaystyle \sum _{i=1}^k{n}_i{({\overline{x}}_i-\overline{x})}^2}\\ $ =36{(25.1-20.25)}^2+39{(8.2-20.25)}^2+35{(38.7-20.25)}^2$=9008.79\\ SSE={\displaystyle \sum _{i=1}^k(n_i-1)s_i{}^2}\\ =(36-1){30}^2+(39-1){54}^2+(35-1){37}^2\\ =188854\end{array}$

The sample size $n=110$ and Number of treatments $k=3$ .

  $\begin{array}{l}d{f}_1=k-1\\ =3-1\\ =2\end{array}$

Compute mean sum of squares:

  $\begin{array}{l}d{f}_2=n-k\\ =110-3\\ =107\end{array}$

The ANOVA

Use MSEXCEL formula, to get.

The completed table is as shown below:

Since P-value is greater than 0.05, fail to reject the null hypothesis of equal means. Hence, conclude that there is insufficient evidence that the mean subcutaneous fat reductions of overweight adults who follow the three exercise programs differ.

d.

To determine

To obtain the ANNOVA f- static and P-test value also find out the evidence.

Explanation of Solution

Given:

The following is the given table-

  

Calculation:

The ANOVA F-static:

  $\begin{array}{l}F=\frac{MSG}{MSE}\\ =\frac{4504.4}{1764.99}\end{array}$

Use MSEXCEL formula, $FDDIST(2.55,2,107)$ to get $P-Value=0.0827$

The completed table is as shown below:

  

Since P-value is greater than 0.05, fail to reject the null hypothesis of equal means. Hence, conclude that there is insufficient evidence that the mean subcutaneous fat reductions of overweight adults who follow the three exercise programs differ.