Chapter 3.2, Problem 39E

What’s your favorite TV show? The following tables present the numbers of viewers, in millions for the top 15 prime-time shows for the 2012—2013 and 2015—2016 seasons. The numbers of viewers include those who watched the program on any platform, including time-shifting up to seven days after the original telecast.

1. Find the population standard deviation of the ratings for 20 12—2013.
2. Find the population standard deviation of the ratings for 2015—2016.
3. Compute the range for the ratings for both seasons.
4. Based on the standard deviations, did the spread in ratings increase or decrease over the two seasons?
5. Based on the ranges, did the spread in ratings increase or decrease over the two seasons?

To determine

To find: the sample standard deviation of rating 2012 − 2013.

Answer to Problem 39E

Standard deviation = 2.74

Explanation of Solution

Given:

Top Rated TV Programs: 2012−2013		Top Rated TV Programs: 2015−2016
Program	Rating	Program	Rating
NCIS	21.6	NFL Sunday Night Football	22
Sunday Night Football	21	The Big Bang Theory	20.6
The Big Bang Theory	19	NCIS	20.5
NCIS: Los Angeles	17.5	The Walking Dead	18.9
Person of Interest	16.2	NFL Thursday Night Football	17.7
American Idol (Wednesday)	15.1	Empire	16.2
Dancing with the Stars (Monday)	15	NCIS: New Orleans	16.1
American Idol (Thursday)	14.8	Blue Bloods	14.6
The Voice (Monday)	14.4	Dancing with the Stars	13.7
The Walking Dead	14.3	The X-Files	13.6
Dancing with the Stars (Tuesday)	14	The Voice (Monday)	13.5
Two and a Half Men	13.9	Scorpion	13.3
The Voice (Tuesday)	13.5	The Voice (Tuesday)	13.2
Blue Bloods	13.3	Little Big Shots	13
Elementary	13.3	NFL Monday Night Football	12.9

Formula used:

  $\text{Standard}\text{Deviation = }\sqrt{\frac{\sum {\left(x_i-\overline{x}\right)}^2}{n-1}}$

Calculation:

Top Rated TV Programs: 2012−2013
Program	Rating	${\left(x_i-\overline{x}\right)}^2$
NCIS	21.6	33.72
Sunday Night Football	21	27.11
The Big Bang Theory	19	10.28
NCIS: Los Angeles	17.5	2.91
Person of Interest	16.2	0.17
American Idol (Wednesday)	15.1	0.48
Dancing with the Stars (Monday)	15	0.63
American Idol (Thursday)	14.8	0.99
The Voice (Monday)	14.4	1.94
The Walking Dead	14.3	2.23
Dancing with the Stars (Tuesday)	14	3.22
Two and a Half Men	13.9	3.58
The Voice (Tuesday)	13.5	5.26
Blue Bloods	13.3	6.22
Elementary	13.3	6.22
Sum	236.90	104.95
average	15.79
Standard deviation	2.74

  $\begin{array}{c}\text{Standard}\text{Deviation = }\sqrt{\frac{104.95}{14}}\\ =2.74\end{array}$

To determine

To find: the sample standard deviation for the rating in year 2015 − 2016.

Answer to Problem 39E

Standard deviation = 3.18

Explanation of Solution

Calculation:

Top Rated TV Programs: 2015−2016
Program	Rating	${\left(y_i-\overline{y}\right)}^2$
NFL Sunday Night Football	22	36.16
The Big Bang Theory	20.6	21.28
NCIS	20.5	20.37
The Walking Dead	18.9	8.49
NFL Thursday Night Football	17.7	2.94
Empire	16.2	0.05
NCIS: New Orleans	16.1	0.01
Blue Bloods	14.6	1.92
Dancing with the Stars	13.7	5.23
The X-Files	13.6	5.70
The Voice (Monday)	13.5	6.18
Scorpion	13.3	7.22
The Voice (Tuesday)	13.2	7.77
Little Big Shots	13	8.92
NFL Monday Night Football	12.9	9.53
Sum	239.8	141.76
average	15.99
Standard deviation	3.18

  $\begin{array}{c}\text{Standard}\text{Deviation = }\sqrt{\frac{141.76}{14}}\\ =3.18\end{array}$

To determine

To find: the range of ratings for both years.

Answer to Problem 39E

Range:

2012 − 13 = 8.3

2015 − 15 = 9.1

Explanation of Solution

Formula used:

Range = Highest Value − lowest value

Calculation:

  $\begin{array}{c}\text{Range for }2012–2013=21.6–13.3 \\ =8.3\end{array}$

  $\begin{array}{c}\text{Range for }2015–2016=22-12.9\\ =\text{9}\text{.1}\end{array}$

To determine

To explain: whether the spread has increase or decrease based on standard deviation of both years.

Answer to Problem 39E

Increased

Explanation of Solution

Since the standard deviation in 2012 − 2013 is 2.74 and in year 2015 − 2016 it is 3.18, which shows that the spread of rating has increased over the given time periods.

To determine

To explain: whether the spread has increase or decrease based on standard deviation of both years.

Answer to Problem 39E

Increased

Explanation of Solution

Since the Range in 2012 − 2013 is 8.3 and in year 2015 − 2016 it is 9.1, which shows that the spread of rating has increased over the given time periods.