Chapter 3.2, Problem 35E

What’s your favorite TV show? The following table presents the rating for the top 20 prime time shows for the 2007 -2008 season and for the 2012 -2013 season. The rating is the percentage of households with TV sets watched the program

1. Find the population standard deviation of the ratings for 2007—2008.
2. Find the population standard deviation of the ratings for 2012—2013.
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 population standard deviation for 2007 − 2008.

Answer to Problem 35E

The population standard deviation of the ratings for $2007-2008\text{ is 2}\text{.356}$.

Explanation of Solution

Given information:

Top Rated TV programs: $2007-2008$

Programs	Rating
American Idol −− Tuesday	$16.1$
American Idol −− Wednesday	$15.9$
Dancing With the Star −− Monday $(2007)$	$14.0$
Dancing With the Star −− Tuesday $(2007)$	$12.7$
Dancing With the Star −− Monday $(2008)$	$12.6$
Dancing With the Star −− Tuesday $(2008)$	$11.8$
Desperate Housewives	$11.6$
CIS	$10.6$
House −− Tuesday	$10.5$
Grey’s Anatomy	$10.4$
Sunday Night Football	$9.7$
CIS: Miami	$9.2$
House −− Monday	$9.2$
NCIS	$9.2$
Survivor: China	$9.0$
Without a Trace	$8.8$
The Moment of Truth	$8.8$
Two and a Half Men	$8.5$
$60$ Minutes	$8.4$
Criminal Minds	$8.2$

Top Rated TV Programs: 2012-2013

Programs	Rating
NBC Sunday Night Football	7.9
Bing Bang Theory	6.2
The Voice	5.1
Modern Family	4.9
The Voice- Tuesday	4.6
American Idol -Wednesday	4.6
American Idol -Thursday	4.3
The Following	4.3
Two and a Half Men	4.1
Grey's Anatomy	4.1
NCIS	4.0
Football Night in American Part 3	4.0
Revolution	3.9
2 Broke Girls	3.7
How I Met Your Mother	3.7
Family Guy	3.6
Once Upon a Time	3.6
Survivor- Philippines	3.5
X-Factor - Wednesday	3.5
NCIS : LA	3.4

Concept used:

Following steps of TI-83 calculator are used to calculate the population standard deviation.

1. Click on STAT > Edit
2. Enter the provided data in L1.
3. Press STAT > CALC > 1-Var Stats
4. Enter L1 by pressing 2nd and 1.
5. Press Enter.

Calculation:

Using the above mentioned step of TI-83 calculator, the obtained output is:

Thus, the population standard deviation $\left(\sigma \right)$ of the ratings for $2007-2008$ is $2.356$.

To determine

To calculate the population standard deviation for 2012 − 2013.

Answer to Problem 35E

The population standard deviation of the ratings for year $2012-2013$ is $1.047$

Explanation of Solution

Calculation:

Using the TI-83 calculator, the obtained output is:

Thus, the population standard deviation of the ratings for $2012-2013$ is $1.047$.

To determine

To calculate the range for both years.

Answer to Problem 35E

The range of the ratings for $2007-2008\text{ is 7}\text{.9}$

The range of the ratings for $2012-2013\text{ is 4}\text{.5}$

Explanation of Solution

The range of the set of data is the difference between the largest and smallest values of the data.

The range is calculated as:

$\text{Range = Maximum value}-\text{Minimum value}$

The range of the ratings for the year 2007-2008 can be calculated as:

$\begin{array}{c}\text{Range = Maximum value}-\text{Minimum value}\\ \text{=16}\text{.1-8}\text{.2}\\ \text{=7}\text{.9}\end{array}$

Similarly, the range of the ratings for the year 2012-2013 can be calculated as:

$\begin{array}{c}\text{Range = Maximum value}-\text{Minimum value}\\ \text{=7}\text{.9}-3.4\\ \text{=4}\text{.5}\end{array}$

Thus, the range of the ratings for the years 2007-2008 and 2012-2013 are 7.9 and 4.5 respectively.

To determine

To explain whether the spread increases or decrease over the two seasons based on standard deviation.

Answer to Problem 35E

Yes, there is a decrease in spread ove.r the two seasons

Explanation of Solution

The population standard deviation of the ratings for $2007-2008\text{ is 2}\text{.356}\text{.}$.

The population standard deviation of the ratings for $2012-2013$ is $1.047$.

Based on the population standard deviation, the spread in ratings has decreased over the two seasons.

To determine

To explain whether the spread increase or decrease over the two seasons based on the range.

Answer to Problem 35E

Based on the ranges, the spread in ratings decrease over the two seasons

Explanation of Solution

The range of the ratings for $2007-2008\text{ is 7}\text{.9}$.

The range of the ratings for $2012-2013\text{ is 4}\text{.5}$.

Based on the ranges, the spread in ratings decrease over the two seasons