Chapter 3.2, Problem 35E

a.

To determine

Find the population standard deviation of the rating for 2007-2008.

Answer to Problem 35E

• The population standard deviation of the ratting for 2007-2008 is 2.36.

Explanation of Solution

Calculation:

The ratings for the top 20 prime-time show for 2007-2008 and 2012-2013 seasons are given.

Standard deviation:

Let $x_1,x_2\mathrm{...}{x}_N$ be the population observations of size N. the population mean is denoted by $\mu$. Hence, the population standard deviation can be expressed as,

$\begin{array}{c}\sigma =\sqrt{\frac{{\displaystyle \sum {\left(x-\mu \right)}^2}}{N}}\\ =\sqrt{\frac{{\displaystyle \sum x^2}-N{\mu }^2}{N}}\end{array}$

Population standard deviation:

Finding mean:

Software procedure:

Step-by-step procedure to find mean using the MINITAB software:

• Choose Stat > Basic Statistics > Display Descriptive Statistics.
• In Variables enter the columns Rating (2007-08).
• Choose option statistics, and select Mean.
• Click OK.

Output using the MINITAB software is given below:

• 
• From the MINITAB output the mean is 10.76.

The deviation, deviation’s squares are calculated below:

x	$\left(x-\mu \right)$	${\left(x-\mu \right)}^2$
16.1	5.34	28.5156
15.9	5.14	26.4196
14	3.24	10.4976
12.7	1.94	3.7636
12.6	1.84	3.3856
11.8	1.04	1.0816
11.6	0.84	0.7056
10.6	–0.16	0.0256
10.5	–0.26	0.0676
10.4	–0.36	0.1296
9.7	–1.06	1.1236
9.2	–1.56	2.4336
9.2	–1.56	2.4336
9.2	–1.56	2.4336
9	–1.76	3.0976
8.8	–1.96	3.8416
8.8	–1.96	3.8416
8.5	–2.26	5.1076
8.4	–2.36	5.5696
8.2	–2.56	6.5536
Total		${\displaystyle \sum {\left(x-\mu \right)}^2}=111.028$

Substituting ${\displaystyle \sum {\left(x-\mu \right)}^2}=111.028$ and N = 20 in the standard deviation formula,

$\begin{array}{c}\sigma =\sqrt{\frac{{\displaystyle \sum {\left(x-\mu \right)}^2}}{N}}\\ =\sqrt{\frac{111.028}{20}}\\ =2.36\end{array}$

• Hence, the population standard deviation of the ratting for 2007-2008 is 2.36.

b.

To determine

Find the population standard deviation of the ratting for 2012-2013.

Answer to Problem 35E

• The population standard deviation of the ratting for 2012-2013 is 1.05.

Explanation of Solution

Calculation:

Population standard deviation:

Finding mean:

Software procedure:

Step-by-step procedure to find mean using the MINITAB software:

• Choose Stat > Basic Statistics > Display Descriptive Statistics.
• In Variables enter the columns Rating (2012-13).
• Choose option statistics, and select Mean.
• Click OK.

Output using the MINITAB software is given below:

• 
• From the MINITAB output the mean is 4.35.

The deviation, deviation’s squares are calculated below:

x	$\left(x-\mu \right)$	${\left(x-\mu \right)}^2$
7.9	3.55	12.6025
6.2	1.85	3.4225
5.1	0.75	0.5625
4.9	0.55	0.3025
4.6	0.25	0.0625
4.6	0.25	0.0625
4.3	–0.05	0.0025
4.3	–0.05	0.0025
4.1	–0.25	0.0625
4.1	–0.25	0.0625
4	–0.35	0.1225
4	–0.35	0.1225
3.9	–0.45	0.2025
3.7	–0.65	0.4225
3.7	–0.65	0.4225
3.6	–0.75	0.5625
3.6	–0.75	0.5625
3.5	–0.85	0.7225
3.5	–0.85	0.7225
3.4	–0.95	0.9025
Total		${\displaystyle \sum {\left(x-\mu \right)}^2}=21.91$

Substituting ${\displaystyle \sum {\left(x-\mu \right)}^2}=21.91$ and N = 20 in the standard deviation formula,

$\begin{array}{c}\sigma =\sqrt{\frac{{\displaystyle \sum {\left(x-\mu \right)}^2}}{N}}\\ =\sqrt{\frac{21.91}{20}}\\ =1.05\end{array}$

• Hence, the population standard deviation of the ratting for 2012-2013 is 1.05.

c.

To determine

Find the range for the rating of both seasons.

Answer to Problem 35E

• The range for the rating of 2007-08 is 7.9 and the range for the rating of 2012-13 seasons is 4.5.

Explanation of Solution

Calculation:

Range:

The range of a data is defined as, $\text{Largest value}-\text{smallest value}$.

Range:

• From the data of the season 2007-08 the largest rating is 16.1 and the smallest value is 8.2.
• Therefore the range is $16.1-8.2=7.9$.
• From the data of the season 2012-13 the largest rating is 7.9 and the smallest value is 3.4.
• Therefore the range is $7.9-3.4=4.5$.
• Thus, the range for the rating of 2007-08 is 7.9 and the range for the rating of 2012-13 seasons is 4.5.

d.

To determine

Explain whether the spread in ratings increase or decrease over the seasons based on the standard deviation.

Answer to Problem 35E

• It can be concluded that the spreads in rating has decreased from 2007-08 to 2012-13 season.

Explanation of Solution

• From part (a), the population standard deviation of the ratting for 2007-2008 is 2.36.
• From part (b), the population standard deviation of the ratting for 2012-2013 is 1.05.
• The standard deviation of the rating for 2007-2008 is more than the population standard deviation of the ratting for 2012-2013.
• Thus, it can be concluded that the spreads in rating has decreased from 2007-08 to 2013-13 season.

e.

To determine

Explain whether the spread in ratings increase or decrease over the seasons based on the range.

Answer to Problem 35E

• It can be concluded that the spreads in rating has decreased from 2007-08 to 2012-13 season.

Explanation of Solution

• From part (c), the range of the ratting for 2007-2008 is 7.9.
• From part (b), the range of the ratting for 2012-2013 is 4.5.
• The range of the ratting for 2007-2008 is more than the range of the ratting for 2012-2013.
• Thus, it can be concluded that the spreads in rating has decreased from 2007-08 to 2012-13 season.