Chapter 12, Problem 1P

Nutrition

Researchers compared protein intake among three groups of postmenopausal women: (1) women eating a standard American diet (STD), (2) women eating a lacto-ovo-vegetarian diet (LAC), and (3) women eating a strict vegetarian diet (VEG). The mean ± 1 sd for protein intake (mg) is presented in Table 12.29.

Perform a statistical procedure to compare the means of the three groups using the critical-value method.

Table 12.29 Protein intake (mg) among three dietary groups of postmenopausal women

To determine

Conduct the test to compare the means of the three groups using critical value method.

Answer to Problem 1P

There is sufficient evidence to conclude that there is a significant difference between the means of the three groups at 5% level of significance.

Explanation of Solution

Hypotheses for the test is given below:

Null hypothesis:

$H_0:{\mu }_1={\mu }_2={\mu }_3$

Alternative hypothesis:

$H_1:\text{At least one mean is different}$

There are 3 groups and 26 observations.

That is, $n=26,k=3.\text{ Thus, }n-k=23,k-1=2.$.

The means sum of squares are as follows:

$\begin{array}{c}M{S}_{Between}=\frac{S{S}_{Between}}{k-1}\\ M{S}_{Within}=\frac{S{S}_{Within}}{n-k}\end{array}$

The sum of squares is calculated as follows:

$\begin{array}{c}S{S}_{Between}={\displaystyle \sum _{i=1}^k{n}_i{\overline{y}}_i^2}-\frac{{\left({\displaystyle \sum _{i=1}^k{n}_i{\overline{y}}_i}\right)}^2}{n}\\ =10{\left(75\right)}^2+10{\left(57\right)}^2+6{\left(47\right)}^2-\frac{{\left[10\left(75\right)+10\left(57\right)+6\left(47\right)\right]}^2}{26}\\ =101,994-98,707.85\\ =3286.15\\ k-1=2\\ M{S}_{Between}=\frac{S{S}_{Between}}{k-1}\\ =\frac{3286.15}{2}\\ =1643.075\\ S{S}_{Within}={\displaystyle \sum _{i=1}^k\left(n_i-1\right)s_i^2}\\ =\left(10-1\right)9^2+\left(10-1\right){13}^2+\left(6-1\right){17}^2\\ =3695\\ n-k=23\\ M{S}_{Within}=\frac{S{S}_{Within}}{n-k}\\ =\frac{3695}{23}\\ =160.6522\end{array}$

The test statistic for the test is calculated as given below:

$\begin{array}{c}F=\frac{M{S}_{Between}}{M{S}_{Within}}\\ =\frac{1643.075}{160.6522}\\ =10.2275\\ \approx 10.23\end{array}$

Critical value:

Consider the level of significance as 0.05.

The numerator degrees of freedom is 2 and the denominator degrees of freedom is 23.

Software procedure:

Step-by-step procedure to obtain critical value using Excel software:

• Open Excel sheet.
• Enter the formula, “=F.INV(0.95,2,23)” in cell A1.
• Click Enter.

Output using Excel software is given below:

Thus, the F-critical value is 3.42.

Decision based on critical value:

Reject the null hypothesis H₀ if, if F> F-critical value;

Otherwise fail to reject H₀.

Conclusion:

Here, F(=10.23)> F-critical value (=3.42).

Therefore, the null hypothesis is rejected.

Thus, there is sufficient evidence to conclude that there is a significant difference between the means of the three groups at 5% level of significance.