Chapter 14.1, Problem 18E

(a)

To determine

Derive the data.

Explanation of Solution

The null and alternative hypothesis can be written as follows:

$\begin{array}{l}{\text{H}}_0:{\mu }_1={\mu }_2={\mu }_3\\ {\text{H}}_1:\text{At least two means differ}\end{array}$

Age Rejection region can be written as follows:

${\text{F>F}}_{\alpha ,\text{k-1,n-k}}={\text{F}}_{0.05,2,147}\approx 2.61$

SST can be calculated as follows:

$\begin{array}{c}\text{SST}=\sum {\text{n}}_{\text{j}}\left(\overline{{\text{x}}_{\text{j}}}-\overline{\text{x}}\right)\\ =\left(63{\left(31.3-36.23\right)}^2+81{\left(34.42-36.23\right)}^2+40{\left(37.38-36.23\right)}^2+111{\left(39.93-36.23\right)}^2\right)\\ =3,366\end{array}$

SSE can be calculated as follows:

$\begin{array}{c}\text{SSE}=\sum \left({\text{n}}_{\text{j}}-1\right){\text{s}}_{\text{j}}^2\\ =\left(63-1\right)28.34+\left(81-1\right)23.2+\left(40-1\right)31.16+\left(111-1\right)72.03\\ =12,752\end{array}$

Thus, the value of SSE is 12,752

Table -1 shows the ANOVA Table as follows:

Table -1
Source	Degree of freedom	Sum of square	Mean square	F
Treatments	k-1 = 36	SST = 3,366	1,122	25.60
Error	n- k = 291	SSE = 12,752	43.82

Thus, the estimated F- value is 25.6 and p-value is 0. Here, there is an adequate evidence to infer that the age of the four group of cereal buyers differ.

(b)

To determine

Explain the required conditions for the test conducted.

Explanation of Solution

The null and alternative hypothesis can be written as follows:

$\begin{array}{l}{\text{H}}_0:{\mu }_1={\mu }_2={\mu }_3\\ {\text{H}}_1:\text{At least two means differ}\end{array}$

Income rejection region can be written as follows:

${\text{F>F}}_{\alpha ,\text{k-1,n-k}}={\text{F}}_{0.05,3,291}\approx 2.61$

SST can be calculated as follows:

$\begin{array}{c}\text{SST}=\sum {\text{n}}_{\text{j}}\left(\overline{{\text{x}}_{\text{j}}}-\overline{\text{x}}\right)\\ =\left(63{\left(37.22-39.97\right)}^2+81{\left(38.91-39.97\right)}^2+40{\left(41.48-39.97\right)}^2+111{\left(41.75-39.97\right)}^2\right)\\ =1,008\end{array}$

SSE can be calculated as follows:

$\begin{array}{c}\text{SSE}=\sum \left({\text{n}}_{\text{j}}-1\right){\text{s}}_{\text{j}}^2\\ =\left(63-1\right)39.8+\left(81-1\right)40.85+\left(40-1\right)61.38+\left(111-1\right)46.59\\ =13,256\end{array}$

Thus, the value of SSE is 13,256

Table -1 shows the ANOVA Table as follows:

Table -2
Source	Degree of freedom	Sum of square	Mean square	F
Treatments	k-1 = 3	SST = 1,008	336	7.37
Error	n- k = 291	SSE = 13,256	45.55

Thus, the estimated F- value is 7.37 and p-value is 0.0001. Here, there is an adequate evidence to conclude that theincome differ between he four groups of cereal buyers.

(c)

To determine

Explain the required conditions for the test conducted.

Explanation of Solution

The null and alternative hypothesis can be written as follows:

$\begin{array}{l}{\text{H}}_0:{\mu }_1={\mu }_2={\mu }_3\\ {\text{H}}_1:\text{At least two means differ}\end{array}$

Education rejection region can be written as follows:

${\text{F>F}}_{\alpha ,\text{k-1,n-k}}={\text{F}}_{0.05,3,291}\approx 2.61$

SST can be calculated as follows:

$\begin{array}{c}\text{SST}=\sum {\text{n}}_{\text{j}}\left(\overline{{\text{x}}_{\text{j}}}-\overline{\text{x}}\right)\\ =\left(63{\left(11.75-11.98\right)}^2+81{\left(12.41-11.98\right)}^2+40{\left(11.73-11.98\right)}^2+111{\left(11.89-11.98\right)}^2\right)\\ =21.71\end{array}$

SSE can be calculated as follows:

$\begin{array}{c}\text{SSE}=\sum \left({\text{n}}_{\text{j}}-1\right){\text{s}}_{\text{j}}^2\\ =\left(63-1\right)3.93+\left(81-1\right)3.39+\left(40-1\right)4.26+\left(111-1\right)4.3\\ =1,154\end{array}$

Thus, the value of SSE is 1,154

Table -1 shows the ANOVA Table as follows:

Table -2
Source	Degree of freedom	Sum of square	Mean square	F
Treatments	k-1 = 3	SST = 21.71	7.24	1.82
Error	n- k = 291	SSE = 1,154	3.697

Thus, the estimated F- value is 1.82 and p-value is 0.1428. Here, there is not an adequate evidence to conclude that the educationdiffer between he four groups of cereal buyers.

(d)

To determine

Summarize the findings.

Explanation of Solution

Using F – tests and descriptive statistics, the mean age and the mean household income are in ascending order. Example, sweet smacks buyers are younger and earn less than the buyers of the other three cereals. Thus, individual ‘C’ purchases are older and earn the most.