Chapter 3.1, Problem 47E

Commercial break: Following are the amounts spent (in millions of dollars) on media advertising in the United States by a sample of 10 companies.

1. Find the mean amount spent on advertising.
2. Find the median amount spent on advertising.
3. If the amount of 722.5 was incorrectly listed as 7225, how would this affect the mean? How would it affect the median?

To determine

To Find: To Find the mean amount spent on advertising.

Answer to Problem 47E

Mean amount spent for media advertising $=373.49$

Explanation of Solution

Given:

  $257.0,280.3,286.9,295.9,331.2,340.1,362.9,394.6,463.5,722.5$

Formula used:

  $\begin{array}{l}\text{Mean}(\overline{x})=\frac{{\displaystyle \sum x_i}}{n}\\ \\ \text{Median}=\left\{\begin{array}{l}\frac{{\left(n+1\right)}^{\text{th }}\text{term}}{2},\text{When}\text{n}\text{is}\text{odd}\\ \frac{{\left(\frac{n}{2}+1\right)}^{\text{th }}\text{term + }{\left(\frac{n}{2}\right)}^{\text{th}}\text{term}}{2},\text{When n is even}\end{array}\right\}\end{array}$

Calculation:

Mean amount spent for media advertising, Mean, $\overline{x}$

  $\begin{array}{l}=\frac{x_1+x_2+------x_n}{n}\\ =\frac{295.9+722.5+280.3+286.9+362.9+331.2+394.6+257.0+463.5+340.1}{10}\\ =\frac{3734.9}{10}=373.49\end{array}$

To determine

To Find: To Find the median amount spent on advertising.

Answer to Problem 47E

Median amount spent for media advertising $=335.65$

Explanation of Solution

Calculation:

Median amount spent for media advertising, For Median, first arrange data in sequential order,

  $257.0,280.3,286.9,295.9,331.2,340.1,362.9,394.6,463.5,722.5$

  $\begin{array}{c}\text{Median =}\left(\frac{{\left(\frac{\text{n}}{\text{2}}\right)}^{\text{th}}\text{term +}{\left(\frac{\text{n}}{\text{2}}\text{+1}\right)}^{\text{th}}\text{term}}{\text{2}}\right)\\ \text{=}\left(\frac{{\left(\frac{\text{10}}{\text{2}}\right)}^{\text{th}}\text{term+}{\left(\frac{\text{10}}{\text{2}}\text{+1}\right)}^{\text{th}}\text{term}}{\text{2}}\right)\\ \text{=}\left(\frac{{\left(\text{5}\right)}^{\text{th}}\text{term+}{\left(\text{6}\right)}^{\text{th}}\text{term}}{\text{2}}\right)\\ \text{=}\left(\frac{\text{331}\text{.2+340}\text{.1}}{\text{2}}\right)\\ \text{=}\frac{\text{671}\text{.3}}{\text{2}}\\ \text{=335}\text{.65}\end{array}$

Therefore, amount spent on media advertising, Mean amount $=373.49$ and Median amount $=335.65$

To determine

To Find: the mean and median, if $722.5$ was incorrectly listed as $7225$ .

Answer to Problem 47E

Mean will become $1023.74$ instead of $373.49$ and median will remain same $335.65$

Explanation of Solution

Calculation- If an amount of $722.5$ was incorrectly listed as $7225$ , then for new mean and new median, it is needed to apply formula on new set of data,

  $\begin{array}{c}\overline{x}=\frac{{\displaystyle \sum x_i}}{n}\\ =\frac{295.9+7225+280.3+286.9+362.9+331.2+394.6+257.0+463.5+340.1}{10}\\ =\frac{10237.4}{10}=1023.74\end{array}$

For Median, first arrange data in sequential order,

  $257.0,280.3,286.9,295.9,331.2,340.1,362.9,394.6,463.5,7225$

  $\begin{array}{c}\text{Median =}\left(\frac{{\left(\frac{\text{n}}{\text{2}}\right)}^{\text{th}}\text{term+}{\left(\frac{\text{n}}{\text{2}}\text{+1}\right)}^{\text{th}}\text{term}}{\text{2}}\right)\\ \text{=}\left(\frac{{\left(\frac{\text{10}}{\text{2}}\right)}^{\text{th}}\text{term+}{\left(\frac{\text{10}}{\text{2}}\text{+1}\right)}^{\text{th}}\text{term}}{\text{2}}\right)\\ \text{=}\left(\frac{{\left(\text{5}\right)}^{\text{th}}\text{term+}{\left(\text{6}\right)}^{\text{th}}\text{term}}{\text{2}}\right)\\ \text{=}\left(\frac{\text{331}\text{.2+340}\text{.1}}{\text{2}}\right)\\ \text{=}\frac{\text{671}\text{.3}}{\text{2}}\\ \text{=335}\text{.65}\end{array}$

Thus, if an amount of $722.5$ was incorrectly listed as $7225$ , then mean will become $1023.74$ instead of $373.49$ and median will remain same $335.65$ .