Chapter 3, Problem 12RE

Advertising costs: The amounts spent (in billions) on media advertising in the United States for a sample of categories in a recent year are presented in the following table.

1. a. Find the mean amount spent on advertising.
2. b. Find the median amount spent on advertising.
3. c. Find the sample variance of the advertising amounts.
4. d. Find the sample standard deviation of the advertising amounts.
5. e. Find the first quartile of the advertising amounts.
6. f. Find the third quartile of the advertising amounts.
7. g. Find the 40th percentile of the advertising amounts.
8. h. Find the 65th percentile of the advertising amounts.

To determine

Find the mean amount spent on advertising.

Answer to Problem 12RE

The mean amount spent on advertising is $8.75.

Explanation of Solution

Calculation:

The amount spent on advertising of U.S. is given.

Mean:

Computation of mean:

If $x_1,x_2\mathrm{...}{x}_n$ be n values.

The mean value is calculated as,

$\begin{array}{c}\text{Mean}=\frac{\text{sum of all values}}{\text{number of values}}\\ =\frac{{\displaystyle \sum _{i=1}^n{x}_i}}{n}\end{array}$

Software procedure:

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

• Choose Stat > Basic Statistics > Display Descriptive Statistics.
• In Variables enter the columns Amount.
• Choose option statistics, and select N total, Mean.
• Click OK.

Output using the MINITAB software is given below:

• 

Thus, the mean amount spent on advertising is $8.75.

To determine

Find the median amount spent on advertising.

Answer to Problem 12RE

The median amount spent on advertising is 7.37.

Explanation of Solution

Calculation:

Median:

Let $x_1,x_2\mathrm{...}{x}_n$ be n values.

The steps for finding the median:

• The all data values should be arranged in ascending order.
• If the total number of data values, n is odd, then the median will be the middle value or if n is even, then the median will be the average of middle two values.

Software procedure:

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

• Choose Stat > Basic Statistics > Display Descriptive Statistics.
• In Variables enter the columns Amount.
• Choose option statistics, and select N total, Median.
• Click OK.

Output using the MINITAB software is given below:

• 

Thus, the median amount spent on advertising is $7.37.

To determine

Find the sample variance of amount spent on advertising.

Answer to Problem 12RE

The sample variance of amount spent on advertising is 14.62.

Explanation of Solution

Calculation:

Sample variance:

Let $x_1,x_2\mathrm{...}{x}_n$ be the sample observations of size n. the sample mean is denoted by $\mu$. Hence, the sample variance can be expressed as,

${\sigma }^2=\frac{{\displaystyle \sum {\left(x-\mu \right)}^2}}{n-1}$

Software procedure:

Step-by-step procedure to find the sample variance and using the MINITAB software:

• Choose Stat > Basic Statistics > Display Descriptive Statistics.
• In Variables enter the columns process.
• Choose option statistics, and select N total, variance.
• Click OK.

Output using the MINITAB software is given below:

• 

Thus, the sample variance of amount spent on advertising is 14.62.

To determine

Find the sample standard deviation of amount spent on advertising.

Answer to Problem 12RE

The sample standard deviation of amount spent on advertising is $3.82.

Explanation of Solution

Calculation:

Sample standard deviation:

The sample standard deviation is expressed as,

$\begin{array}{c}SD=\sqrt{{\sigma }^2}\\ =\sigma \end{array}$

Software procedure:

Step-by-step procedure to find the sample standard deviation and using the MINITAB software:

• Choose Stat > Basic Statistics > Display Descriptive Statistics.
• In Variables enter the columns Amount.
• Choose option statistics, and select N total, standard deviation.
• Click OK.

Output using the MINITAB software is given below:

• 

Thus, the sample standard deviation of amount spent on advertising is $3.82.

To determine

Find the first quartile.

Answer to Problem 12RE

The first quartile is 6.34.

Explanation of Solution

Calculation:

First quartile:

• The first quartile separates the lowest 25% of the observations from the other 75% of the observations. The first quartile is denoted by $Q_1$.
• Procedure for finding the first quartile:
• Step 1: The observations should be arranged in increasing order.
• Step 2: The size of the data is n.
•                                     For finding first quartile, $L=0.25n$
• Step 3: The quartile will be the average of the observation of the position L and the observation in position $L+1$, if L is a whole number. If the L value is not a whole number, the next higher whole number will be considered. The quartile is the observation in the position of rounded-up value.
• The observations are arranged in increasing order:
• $\left\{4.86,6.19,6.34,6.57,6.84,7.89,8.66,8.98,14.84,16.35\right\}$
• The size of the data is $n=10$.
• For finding first quartile,
• $\begin{array}{c}L=0.25\times 10\\ =2.5\end{array}$
• Here, 2.5 is not a whole number, hence the 1^st quartile will be the observation in the 3^rd position.
• From the arranged observations the first quartile is 6.34.

Hence, the first quartile is 6.34.

To determine

Find the third quartile.

Answer to Problem 12RE

The third quartile is 8.98.

Explanation of Solution

Calculation:

Third quartile:

• The third quartile separates the lowest 75% of the observations from the other 25% of the observations. The third quartile is denoted by $Q_3$.
• Procedure for finding the third quartile:
• Step 1: The observations should be arranged in increasing order.
• Step 2: The size of the data is n.
•                                     For finding third quartile, $L=0.75n$
• Step 3: The quartile will be the average of the observation of the position L and the observation in position $L+1$, if L is a whole number. If the L value is not a whole number, the next higher whole number will be considered. The quartile is the observation in the position of rounded-up value.
• The observations are arranged in increasing order:
• $\left\{4.86,6.19,6.34,6.57,6.84,7.89,8.66,8.98,14.84,16.35\right\}$
• The size of the data is $n=10$.
• For finding third quartile,
• $\begin{array}{c}L=0.75\times 10\\ =7.5\end{array}$
• Here, 7.5 is not a whole number, hence the 1^st quartile will be the observation in the 8^th position.
• From the arranged observations the third quartile is 8.98.

Hence, the third quartile is 8.98.

To determine

Find the 40^th percentile of the advertising amount.

Answer to Problem 12RE

The 40^th percentile of the advertising amount is 6.705.

Explanation of Solution

Calculation:

p^th percentile:

The p^th percentile separates the lowest p% of the observations from the highest $\left(100-p\right)\%$ of the observations.

Procedure for finding p^th percentile:

• Step 1: The observations should be arranged in increasing order.
• Step 2: The size of the data is n.

                              For finding p^th percentile, $L=\frac{p}{100}n$

• Step 3: The p^th percentile will be the average of the observation of the position L and the observation in position $L+1$, if L is a whole number. If the L value is not a whole number, the next higher whole number will be considered. The p^th percentile is the observation in the position of rounded-up value.
• The observations are arranged in increasing order:
• $\left\{4.86,6.19,6.34,6.57,6.84,7.89,8.66,8.98,14.84,16.35\right\}$
• The size of the data is $n=10$.
• For finding 40^th percentile,
• $\begin{array}{c}L=\frac{40}{100}\times 10\\ =4\end{array}$
• Here, 4 is a whole number, hence the 40^th percentile will be the average of the observations in the position of 4^th and 5^th .
• From the arranged observations the 40^th percentile is $\frac{6.57+6.84}{2}=6.705$.

Hence, the 40^th percentile is 6.705.

To determine

Find the 65^th percentile of the advertising amount.

Answer to Problem 12RE

The 65^th percentile of the advertising amount is 8.66.

Explanation of Solution

Calculation:

p^th percentile:

The p^th percentile separates the lowest p% of the observations from the highest $\left(100-p\right)\%$ of the observations.

Procedure for finding p^th percentile:

• Step 1: The observations should be arranged in increasing order.
• Step 2: The size of the data is n.

                              For finding p^th percentile, $L=\frac{p}{100}n$

• Step 3: The p^th percentile will be the average of the observation of the position L and the observation in position $L+1$, if L is a whole number. If the L value is not a whole number, the next higher whole number will be considered. The p^th percentile is the observation in the position of rounded-up value.
• The observations are arranged in increasing order:
• $\left\{4.86,6.19,6.34,6.57,6.84,7.89,8.66,8.98,14.84,16.35\right\}$
• The size of the data is $n=10$.
• For finding 65^th percentile,
• $\begin{array}{c}L=\frac{65}{100}\times 10\\ =6.5\end{array}$
• Here, 6.5 is not a whole number, hence the 65^th percentile will be the observations in the position of 7^th.
• From the arranged observations the 65^th percentile is 8.66.

Hence, the 65^th percentile is 8.66.