Chapter 4.1, Problem 24E

(a)

To determine

Determine the variance.

Explanation of Solution

Table–1 shows the sample as follows:

Table-1
People	Sample
1	17
2	29
3	12
4	16
5	11

$\begin{array}{c}\text{Mean}=\frac{\sum {\text{x}}_{\text{i}}}{\text{n}}\\ =\frac{17+29+12+16+11}{5}\\ =\frac{85}{5}\\ =17\end{array}$

Thus, the value of mean is 17.

Variance can be calculated as follows:

$\begin{array}{c}{\text{S}}^2=\frac{\sum {\left({\text{x}}_{\text{i}}-\overline{\text{x}}\right)}^2}{\text{n}-\text{1}}\\ =\frac{{\left(17-17\right)}^2+{\left(29-17\right)}^2+{\left(12-17\right)}^2+{\left(16-17\right)}^2+{\left(11-17\right)}^2}{5-1}\\ =\frac{0+144+25+1+36}{4}\\ =\frac{206}{4}\\ =51.5\end{array}$

Thus, the value of variance for sample is 51.5.

(b)

To determine

Determine the variance.

Explanation of Solution

Table–2 shows the sample as follows:

Table-2
People	Sample
1	22
2	18
3	23
4	20
5	17

$\begin{array}{c}\text{Mean}=\frac{\sum {\text{x}}_{\text{i}}}{\text{n}}\\ =\frac{22+18+23+20+17}{5}\\ =\frac{100}{5}\\ =20\end{array}$

Thus, the value of mean is 20.

Variance can be calculated as follows:

$\begin{array}{c}{\text{S}}^2=\frac{\sum {\left({\text{x}}_{\text{i}}-\overline{\text{x}}\right)}^2}{\text{n}-\text{1}}\\ =\frac{{\left(22-20\right)}^2+{\left(18-20\right)}^2+{\left(23-20\right)}^2+{\left(20-20\right)}^2+{\left(17-20\right)}^2}{5-1}\\ =\frac{4+4+9+0+9}{4}\\ =\frac{26}{4}\\ =6.5\end{array}$

Thus, the value of variance for sample is 6.5.

(c)

To determine

Determine the variance.

Explanation of Solution

Table–3 shows the sample as follows:

Table-3
People	Sample
1	24
2	37
3	6
4	39
5	29

$\begin{array}{c}\text{Mean}=\frac{\sum {\text{x}}_{\text{i}}}{\text{n}}\\ =\frac{24+37+6+39+29}{5}\\ =\frac{135}{5}\\ =27\end{array}$

Thus, the value of mean is 27.

Variance can be calculated as follows:

$\begin{array}{c}{\text{S}}^2=\frac{\sum {\left({\text{x}}_{\text{i}}-\overline{\text{x}}\right)}^2}{\text{n}-\text{1}}\\ =\frac{{\left(24-27\right)}^2+{\left(37-27\right)}^2+{\left(6-27\right)}^2+{\left(39-27\right)}^2+{\left(29-27\right)}^2}{5-1}\\ =\frac{9+100+441+144+4}{4}\\ =\frac{698}{4}\\ =174.5\end{array}$

Thus, the value of variance for sample is 174.5.