Question

“radon: the problem no one wants to face” is the title of an article appearing in consumer reports. radon is a gas emitted from the ground that can collect in houses and buildings. at certain levels it can cause lung cancer. radon concentrations are measured in picocuries per liter (pci/l). a radon level of 4 pci/l is considered “acceptable.” radon levels in a house vary from week to week. in one house, a sample of 8 weeks had the following readings for radon level (in pci/l): 1.9 2.7 5.7 4.9 1.9 8.7 3.9 7 (a) find the mean, median, and mode. what is the mean? what is the median? what is the mode? (b) find the standard deviation. (round your answer to four decimal places.) find the coefficient of variation (in percent). (round your answer to two decimal places.) find the range.

Explanation:

Step1: Calculate the mean

The mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$, where $n = 8$ and $x_{i}$ are the data - points $1.9,2.7,5.7,4.9,1.9,8.7,3.9,7$.
$\sum_{i=1}^{8}x_{i}=1.9 + 2.7+5.7+4.9+1.9+8.7+3.9+7=36.7$
$\bar{x}=\frac{36.7}{8}=4.5875$

Step2: Calculate the median

First, order the data: $1.9,1.9,2.7,3.9,4.9,5.7,7,8.7$. Since $n = 8$ (an even number), the median is the average of the $\frac{n}{2}$ - th and $(\frac{n}{2}+1)$ - th ordered values.
The $\frac{n}{2}=4$ - th value is $3.9$ and the $(\frac{n}{2}+1)=5$ - th value is $4.9$.
Median $=\frac{3.9 + 4.9}{2}=4.4$

Step3: Calculate the mode

The mode is the value that appears most frequently. The value $1.9$ appears twice, and all other values appear once, so the mode is $1.9$.

Step4: Calculate the standard deviation

The formula for the sample standard deviation $s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}$.
$(1.9-4.5875)^{2}=(-2.6875)^{2}=7.22265625$
$(1.9-4.5875)^{2}=7.22265625$
$(2.7-4.5875)^{2}=(-1.8875)^{2}=3.56265625$
$(3.9-4.5875)^{2}=(-0.6875)^{2}=0.47265625$
$(4.9-4.5875)^{2}=(0.3125)^{2}=0.09765625$
$(5.7-4.5875)^{2}=(1.1125)^{2}=1.23765625$
$(7-4.5875)^{2}=(2.4125)^{2}=5.82015625$
$(8.7-4.5875)^{2}=(4.1125)^{2}=16.91265625$
$\sum_{i = 1}^{8}(x_{i}-\bar{x})^{2}=7.22265625+7.22265625+3.56265625+0.47265625+0.09765625+1.23765625+5.82015625+16.91265625 = 42.54875$
$s=\sqrt{\frac{42.54875}{7}}\approx2.4644$

Step5: Calculate the coefficient of variation

The coefficient of variation $CV=\frac{s}{\bar{x}}\times100\%$.
$CV=\frac{2.4644}{4.5875}\times100\%\approx53.72\%$

Step6: Calculate the range

The range is the difference between the maximum and minimum values.
Range $=8.7 - 1.9=6.8$

Answer:

Mean: $4.5875$
Median: $4.4$
Mode: $1.9$
Standard Deviation: $2.4644$
Coefficient of Variation: $53.72\%$
Range: $6.8$