Chapter 7.1, Problem 24E

(a)

To determine

To find:

The standard deviation of the monthly home run totals in the player’s rookie season.

Answer to Problem 24E

  $2$

Explanation of Solution

Given:

The table

  $x,\overline{x},x-\overline{x},{\left(x-\overline{x}\right)}^2$

Concept used:

The mean of the squared deviation or variance

  $=\frac{{\left(x_1-\overline{x}\right)}^2+{\left(x_2-\overline{x}\right)}^2+\mathrm{.............}+{\left(x_n-\overline{x}\right)}^2}{n}$

The square root of the variance

  $\sigma =\sqrt{\frac{{\left(x_1-\overline{x}\right)}^2+{\left(x_2-\overline{x}\right)}^2+\mathrm{.............}+{\left(x_n-\overline{x}\right)}^2}{n}}$

Calculation:

The mean is

  $\begin{array}{l}\overline{x}=\frac{12}{6}\\ \overline{x}=2\end{array}$

The mean of the squared deviation or variance

  $\begin{array}{l}=\frac{{\left(x_1-\overline{x}\right)}^2+{\left(x_2-\overline{x}\right)}^2+\mathrm{.............}+{\left(x_n-\overline{x}\right)}^2}{n}\\ =\frac{1+4+\mathrm{...........}+1}{6}\\ =4.33\end{array}$

$x$	$\overline{x}$	$x-\overline{x}$	${\left(x-\overline{x}\right)}^2$
$1$	$2$	$-1$	$1$
$0$	$2$	$-2$	$4$
$6$	$2$	$4$	$16$
$2$	$2$	$0$	$0$
$0$	$2$	$-2$	$4$
$3$	$2$	$1$	$1$

The square root of the variance

  $\begin{array}{l}\sigma =\sqrt{\frac{{\left(x_1-\overline{x}\right)}^2+{\left(x_2-\overline{x}\right)}^2+\mathrm{.............}+{\left(x_n-\overline{x}\right)}^2}{n}}\\ \sigma =\sqrt{\frac{1+4+\mathrm{...........}+1}{6}}\\ \sigma =\sqrt{\frac{26}{6}}\\ \sigma \approx 2\end{array}$

The standard deviation of the monthly home run totals in the player’s rookie season $2$ points

(b)

To determine

To find:

The standard deviation of the monthly home run totals in this player’s rookie season.

Answer to Problem 24E

  $3$

Explanation of Solution

Given:

The table

  $x,\overline{x},x-\overline{x},{\left(x-\overline{x}\right)}^2$

Concept used:

The mean of the squared deviation or variance

  $=\frac{{\left(x_1-\overline{x}\right)}^2+{\left(x_2-\overline{x}\right)}^2+\mathrm{.............}+{\left(x_n-\overline{x}\right)}^2}{n}$

The square root of the variance

  $\sigma =\sqrt{\frac{{\left(x_1-\overline{x}\right)}^2+{\left(x_2-\overline{x}\right)}^2+\mathrm{.............}+{\left(x_n-\overline{x}\right)}^2}{n}}$

Calculation:

The mean of the squared deviation or variance

  $\begin{array}{l}=\frac{{\left(x_1-\overline{x}\right)}^2+{\left(x_2-\overline{x}\right)}^2+\mathrm{.............}+{\left(x_n-\overline{x}\right)}^2}{n}\\ =\frac{9+1+\mathrm{...........}+36}{6}\\ =9.33\end{array}$

$x$	$\overline{x}$	$x-\overline{x}$	${\left(x-\overline{x}\right)}^2$
$4$	$7$	$-3$	$9$
$6$	$7$	$-1$	$1$
$4$	$7$	$-3$	$9$
$8$	$7$	$1$	$1$
$7$	$7$	$0$	+   $0$
$13$	+   $7$	$6$	$36$

The square root of the variance

  $\begin{array}{l}\sigma =\sqrt{\frac{{\left(x_1-\overline{x}\right)}^2+{\left(x_2-\overline{x}\right)}^2+\mathrm{.............}+{\left(x_n-\overline{x}\right)}^2}{n}}\\ \sigma =\sqrt{\frac{9+1+\mathrm{...........}+36}{6}}\\ \sigma =\sqrt{\frac{56}{6}}\\ \sigma \approx 3\end{array}$

The standard deviation of the monthly home run totals in this player’s rookie season $3$ .

(c)

To determine

To explain:

The number of homeruns in this season and the number of hone runs in rookie season.

Answer to Problem 24E

The number of homeruns in this season is more spread out than the number of hone runs in rookie season

Explanation of Solution

Given:

The table

  $x,\overline{x},x-\overline{x},{\left(x-\overline{x}\right)}^2$

Concept used:

The mean of the squared deviation or variance

  $=\frac{{\left(x_1-\overline{x}\right)}^2+{\left(x_2-\overline{x}\right)}^2+\mathrm{.............}+{\left(x_n-\overline{x}\right)}^2}{n}$

The square root of the variance

  $\sigma =\sqrt{\frac{{\left(x_1-\overline{x}\right)}^2+{\left(x_2-\overline{x}\right)}^2+\mathrm{.............}+{\left(x_n-\overline{x}\right)}^2}{n}}$

Calculation:

The mean is

  $\begin{array}{l}\overline{x}=\frac{12}{6}\\ \overline{x}=2\end{array}$

The mean of the squared deviation or variance

  $\begin{array}{l}=\frac{{\left(x_1-\overline{x}\right)}^2+{\left(x_2-\overline{x}\right)}^2+\mathrm{.............}+{\left(x_n-\overline{x}\right)}^2}{n}\\ =\frac{1+4+\mathrm{...........}+1}{6}\\ =4.33\end{array}$

The square root of the variance

  $\begin{array}{l}\sigma =\sqrt{\frac{{\left(x_1-\overline{x}\right)}^2+{\left(x_2-\overline{x}\right)}^2+\mathrm{.............}+{\left(x_n-\overline{x}\right)}^2}{n}}\\ \sigma =\sqrt{\frac{1+4+\mathrm{...........}+1}{6}}\\ \sigma =\sqrt{\frac{26}{6}}\\ \sigma \approx 2\end{array}$

The mean of the squared deviation or variance

+ $\begin{array}{l}=\frac{{\left(x_1-\overline{x}\right)}^2+{\left(x_2-\overline{x}\right)}^2+\mathrm{.............}+{\left(x_n-\overline{x}\right)}^2}{n}\\ =\frac{9+1+\mathrm{...........}+36}{6}\\ =9.33\end{array}$

The square root of the variance

++ $\begin{array}{l}\sigma =\sqrt{\frac{{\left(x_1-\overline{x}\right)}^2+{\left(x_2-\overline{x}\right)}^2+\mathrm{.............}+{\left(x_n-\overline{x}\right)}^2}{n}}\\ \sigma =\sqrt{\frac{9+1+\mathrm{...........}+36}{6}}\\ \sigma =\sqrt{\frac{56}{6}}\\ \sigma \approx 3\end{array}$

The number of homeruns in this season is more spread out than the number of hone runs in rookie season.