Chapter 1.4, Problem 48E

Exercise 34 presented the following data on endotoxin concentration in settled dust both for a sample of urban homes and for a sample of farm homes:

U:	6.0	5.0	11.0	33.0	4.0	5.0	80.0	18.0	35.0	17.0	23.0
F:	4.0	14.0	11.0	9.0	9.0	8.0	4.0	20.0	5.0	8.9	21.0
	9.2	3.0	2.0	0.3

1. a. Determine the value of the sample standard deviation for each sample, interpret these values, and then contrast variability in the two samples. [Hint: Σx_i = 237.0 for the urban sample and = 128.4 for the farm sample, and ${\displaystyle \sum x_i^2\text{ = 10,079}}$ for the urban sample and 1617.94 for the farm sample.]
2. b. Compute the fourth spread for each sample and compare. Do the fourth spreads convey the same message about variability that the standard deviations do? Explain.
3. c. The authors of the cited article also provided endotoxin concentrations in dust bag dust:

U:	34.0	49.0	13.0	33.0	24.0	24.0	35.0	104.0	34.0	40.0	38.0	1.0
F:	2.0	64.0	6.0	17.0	35.0	11.0	17.0	13.0	5.0	27.0	23.0
	28.0	10.0	13.0	0.2

Construct a comparative boxplot (as did the cited paper) and compare and contrast the four samples.

To determine

Find and interpret the sample standard deviations of endotoxin concentration of urban homes and farm homes.

Compare the two standard deviations.

Answer to Problem 48E

The sample standard deviation of endotoxin concentration urban homes is $\underset{\_}{22.3\frac{EU}{mg}}$.

The sample standard deviation of endotoxin concentration of farm homes is $\underset{\_}{6.0877\frac{EU}{mg}}$.

Urban homes have greater variability than farm homes.

Explanation of Solution

Given info:

The data represents the values of endotoxin concentration in settled dust for a sample of urban homes and farm homes. Additional information is that the sum of observations and sum of squares of observations is given for both urban homes and farm homes. The values are ${\displaystyle \sum _{i=1}^{11}{x}_{i\left(U\right)}}=237$, ${\displaystyle \sum _{i=1}^{15}{x}_{i\left(F\right)}}=128.4$, ${\displaystyle \sum _{i=1}^{11}{x}_{i\left(U\right)}{}^2}=10,079$ and ${\displaystyle \sum _{i=1}^{15}{x}_{i\left(F\right)}{}^2}=1,617.94$.

Calculation:

Urban homes:

Variance:

Both the variance and the standard deviation are based on how much each observation deviates from a central point represented by the mean. In general, the greater the distances between the individual observations and the mean, the greater the variability of the data set.

Because the mean acts as a balancing point for observations larger and smaller than it, the sum of the deviations around the mean is always zero.

The shortcut formula for variance is,

$\text{Variance}\left(s^2\right)=\frac{{\displaystyle \sum _{i=1}^{11}{x}_i{}^2-\frac{{\left({\displaystyle \sum _{i=1}^{11}{x}_i}\right)}^2}{n}}}{n-1}$.

The values of sum of observations and sum of squares of observations for urban homes is ${\displaystyle \sum _{i=1}^{11}{x}_{i\left(U\right)}}=237$ and ${\displaystyle \sum _{i=1}^{11}{x}_{i\left(U\right)}{}^2}=10,079$.

The variance of endotoxin concentration of urban homes is:

$\begin{array}{c}\text{Variance}\left(s^2\right)=\frac{{\displaystyle \sum _{i=1}^{11}{x}_{i\left(U\right)}{}^2-\frac{{\left({\displaystyle \sum _{i=1}^{11}{x}_{i\left(U\right)}}\right)}^2}{n}}}{n-1}\\ =\frac{10,097-\frac{{\left(237\right)}^2}{11}}{10}\\ =\frac{10,097-5,106.27}{10}\\ =499.073\end{array}$

Thus, the variance of the endotoxin concentration of urban homes is 499.073.

Standard deviation:

The general formula for standard deviation is,

$\text{Standard}\text{deviation}=\sqrt{\text{Variance}}$

The sample standard deviation of endotoxin concentration of urban homes is:

$\begin{array}{c}\text{Standard}\text{deviation}=\sqrt{\text{Variance}}\\ =\sqrt{499.073}\\ =22.3\end{array}$

The standard deviation of endotoxin concentration of urban homes is 22.3.

Interpretation:

The standard deviation is based on how much each observation deviates from a central point represented by the mean.

The standard deviation of endotoxin concentration of urban homes is $\underset{\_}{22.3\frac{EU}{mg}}$

From this it can be said that, among 11 observations of endotoxin concentration of urban homes, the approximate deviation between observed endotoxin concentration and mean endotoxin concentration for urban homes is $\underset{\_}{22.3\frac{EU}{mg}}$.

Farm homes:

The values of sum of observations and sum of squares of observations for farm homes is ${\displaystyle \sum _{i=1}^{15}{x}_{i\left(F\right)}}=128.4$ and ${\displaystyle \sum _{i=1}^{15}{x}_{i\left(F\right)}{}^2}=1,617.94$.

The variance of endotoxin concentration of farm homes is:

$\begin{array}{c}\text{Variance}\left(s^2\right)=\frac{{\displaystyle \sum _{i=1}^{15}{x}_{i\left(F\right)}{}^2-\frac{{\left({\displaystyle \sum _{i=1}^{15}{x}_{i\left(F\right)}}\right)}^2}{n}}}{n-1}\\ =\frac{1,617.94-\frac{{\left(128.4\right)}^2}{15}}{14}\\ =\frac{1,617.94-1,099.104}{14}\\ =37.0597\end{array}$

Thus, the variance of the endotoxin concentration of farm homes is 37.0597.

Standard deviation:

The general formula for standard deviation is,

$\text{Standard}\text{deviation}=\sqrt{\text{Variance}}$.

The sample standard deviation of endotoxin concentration of farm homes is:

$\begin{array}{c}\text{Standard}\text{deviation}=\sqrt{\text{Variance}}\\ =\sqrt{37.0597}\\ =6.0877\end{array}$

The standard deviation of endotoxin concentration of farm homes is 6.0877.

Interpretation:

The standard deviation is based on how much each observation deviates from a central point represented by the mean.

The standard deviation of endotoxin concentration of farm homes is $\underset{\_}{6.0877\frac{EU}{mg}}$

From this it can be said that among 15 observations of endotoxin concentration of farm homes, the approximate deviation between observed endotoxin concentration and mean endotoxin concentration for farm homes is $\underset{\_}{6.0877\frac{EU}{mg}}$.

Comparison:

The sample standard deviation of endotoxin concentration urban homes is $\underset{\_}{22.3\frac{EU}{mg}}$.

The sample standard deviation of endotoxin concentration of farm homes is $\underset{\_}{6.0877\frac{EU}{mg}}$.

From the standard deviation of the endotoxin concentration of urban homes and farm homes, it can be concluded that, the variation in observed and mean endotoxin concentration is greater for urban homes than for farm homes.

To determine

Obtain the fourth spread of endotoxin concentration in urban homes and farm homes.

Compare the two fourth spreads.

Check whether the interpretation about relative variability of the two different samples is same using standard deviations and fourth spreads are same.

Answer to Problem 48E

The fourth spread of the endotoxin concentration in urban homes is $\underset{\_}{22.5\frac{\text{EU}}{\text{mg}}}$.

The fourth spread of the endotoxin concentration in farm homes is $\underset{\_}{6.1\frac{\text{EU}}{\text{mg}}}$.

The variation in the endotoxin concentration is higher for urban homes than farm homes.

Yes, standard deviation and fourth spreads convey same message regarding relative variability of endotoxin concentration of urban and farm homes.

Explanation of Solution

Justification:

Fourth spreads:

In contrast to the range, which measures only differences between the extremes, the fourth spread (also called mid spread) is the difference between the upper fourth and the lower fourth. Thus, it measures the variation in the middle 50 percent of the data, and, unlike the range, is not affected by extreme values.

The general formula for fourth spread is,

$f_s=\text{Upper fourth}-\text{Lower}\text{fourth}$

Fourth spread of urban homes:

The lower fourth, upper fourth and fourth spreads are obtained as follows:

Step1:

The data set is arranged in ascending order as follows:

S.no	Urban homes
1	4
2	5
3	5
4	6
5	11
6	17
7	18
8	23
9	33
10	35
11	80

Step2:

The lower fourth of the data is the median of the lower half of the ordered data set.

Here, the values of lower half of the ordered data set are 4, 5, 5, 6, 11and 17.

The number of observations in the lower half of the ordered data set is 6 and is even.

Hence, the median is the average of 3^rd and 4^th observation of the lower half of the ordered data set.

The lower fourth is obtained below as,

$\text{Lower fourth}=\frac{5+6}{2}=5.5$

Thus, the lower fourth is 5.5.

Step4:

The upper fourth of the data is the median of the upper half of the ordered data set.

Here, the values of upper half of the ordered data set are 17, 18, 23, 33, 35 and 80.

The number of observations in the upper half of the ordered data set is 6 and is even.

Hence, the median is the average of 3^rd and 4^th observation of the upper half of the ordered data set.

The upper fourth is obtained below as,

$\text{Upper fourth}=\frac{23+33}{2}=28$

Thus, the upper fourth is 2.015.

Step5:

The lower fourth is 5.5 and the upper fourth is 28.

The fourth spread is obtained below as,

$\begin{array}{c}{f}_s=\text{upper fourth}-\text{lower fourth}\\ =28-5.5\\ =22.5\frac{\text{EU}}{\text{mg}}\end{array}$

Thus, the fourth spread of urban homes is $\underset{\_}{22.5\frac{\text{EU}}{\text{mg}}}$.

Fourth spread of farm homes:

The lower fourth, upper fourth and fourth spreads are obtained as follows:

Step1:

The data set is arranged in ascending order as follows:

S.no	Farm homes
1	0.3
2	2
3	3
4	4
5	4
6	5
7	8
8	8.9
9	9
10	9
11	9.2
12	11
13	14
14	20
15	21

Step2:

The lower fourth of the data is the median of the lower half of the ordered data set.

Here, the values of lower half of the ordered data set are 0.3, 2, 3, 4, 4, 5, 8 and 8.9.

The number of observations in the lower half of the ordered data set is 8 and is even.

Hence, the median is the average of 4^th and 5^th observation of the lower half of the ordered data set.

The lower fourth is obtained below as,

$\text{Lower fourth}=\frac{4+4}{2}=4$

Thus, the lower fourth is 4.

Step4:

The upper fourth of the data is the median of the upper half of the ordered data set.

Here, the values of upper half of the ordered data set are 8.9, 9, 9, 9.2, 11, 14, 20 and 21.

The number of observations in the upper half of the ordered data set is 8 and is even.

Hence, the median is the average of 4^th and 5^th observation of the upper half of the ordered data set.

The upper fourth is obtained below as,

$\text{Upper fourth}=\frac{9.2+11}{2}=10.1$

Thus, the upper fourth is 10.1.

Step5:

The lower fourth is 4 and the upper fourth is 10.1.

The fourth spread is obtained below as,

$\begin{array}{c}{f}_s=\text{upper fourth}-\text{lower fourth}\\ =10.1-4\\ =6.1\frac{\text{EU}}{\text{mg}}\end{array}$

Thus, the fourth spread of farm homes is $\underset{\_}{6.1\frac{\text{EU}}{\text{mg}}}$.

Comparison:

The fourth spread of the urban homes is $22.5\frac{\text{EU}}{\text{mg}}$.

The fourth spread of the farm homes is $6.1\frac{\text{EU}}{\text{mg}}$.

From the fourth spread of the endotoxin concentration of urban homes and farm homes it can be concluded that, the variation in observed and mean endotoxin concentration is greater for urban homes than for farm homes.

Conclusion:

The standard deviations obtained in part (a) conveyed that, the variability in endotoxin concentration is greater for urban homes than for farm homes.

From these two conclusions, it can be said that the information conveyed by standard deviations and fourth spreads issimilar regarding the variability in endotoxin concentration.

To determine

Draw and interpret the comparative boxplot for the four different samples.

Answer to Problem 48E

Box plot:

Output obtained from MINITAB is given below:

Explanation of Solution

Given info:

The data represents the values of endotoxin concentration in dust bag for a sample of urban homes and farm homes.

Calculation:

Box plot:

Software Procedure:

Step-by-step procedure to draw the box plot using the MINITAB software:

• Choose Graph > Box plot.
• Choose Multiple, and then click OK.
• In Graph variables, enter the column of urban-set, farm-set, urban-bag and farm-bag.
• Click OK.

Interpretation:

From the comparativeboxplots, the following points are observed.

• The median endotoxin concentration in bag is higher for urban homes.
• The data sets of settled dust in urban homes, bag dust in urban homes and bag dust in farm homes consist of one outlier each.
• The variation in the entire distribution is lower for the data set of settled dust in urban homes and higher for the data set of bag dust in urban homes.
• Overall it can be concluded that, endotoxin concentration is higher for urban homes than in farm homes for both settled dust and bag dust.
• The overall variation is less for farm homes.
• The bag dust is higher than the settled dust in both urban and farm homes.