Question

the following data represent the concentration of dissolved organic carbon (mg/l) collected from 20 samples of organic soil. assume that the population is normally distributed. complete parts (a) through (c) on the right. 22.49 29.80 27.10 16.51 15.72 8.81 5.20 20.46 14.90 33.67 30.91 14.86 15.42 15.35 9.72 19.80 14.86 8.09 11.40 18.30 (a) find the sample mean. the sample mean is 17.67 (round to two decimal places as needed.) (b) find the sample standard deviation. the sample standard deviation is (round to two decimal places as needed.)

Explanation:

Step1: Calculate the sample variance

The formula for sample variance \(s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}\), where \(n = 20\) (sample size) and \(\bar{x}=17.67\) (sample mean).
First, calculate \((x_{i}-\bar{x})^{2}\) for each \(x_{i}\):
For \(x_{1}=22.49\): \((22.49 - 17.67)^{2}=(4.82)^{2}=23.2324\)
For \(x_{2}=29.80\): \((29.80 - 17.67)^{2}=(12.13)^{2}=147.1369\)
For \(x_{3}=27.10\): \((27.10 - 17.67)^{2}=(9.43)^{2}=88.9249\)
For \(x_{4}=16.51\): \((16.51 - 17.67)^{2}=(-1.16)^{2}=1.3456\)
For \(x_{5}=15.72\): \((15.72 - 17.67)^{2}=(-1.95)^{2}=3.8025\)
For \(x_{6}=8.81\): \((8.81 - 17.67)^{2}=(-8.86)^{2}=78.4996\)
For \(x_{7}=5.20\): \((5.20 - 17.67)^{2}=(-12.47)^{2}=155.5009\)
For \(x_{8}=20.46\): \((20.46 - 17.67)^{2}=(2.79)^{2}=7.7841\)
For \(x_{9}=14.90\): \((14.90 - 17.67)^{2}=(-2.77)^{2}=7.6729\)
For \(x_{10}=33.67\): \((33.67 - 17.67)^{2}=(16)^{2}=256\)
For \(x_{11}=30.91\): \((30.91 - 17.67)^{2}=(13.24)^{2}=175.2976\)
For \(x_{12}=14.86\): \((14.86 - 17.67)^{2}=(-2.81)^{2}=7.8961\)
For \(x_{13}=15.42\): \((15.42 - 17.67)^{2}=(-2.25)^{2}=5.0625\)
For \(x_{14}=15.35\): \((15.35 - 17.67)^{2}=(-2.32)^{2}=5.3824\)
For \(x_{15}=9.72\): \((9.72 - 17.67)^{2}=(-7.95)^{2}=63.2025\)
For \(x_{16}=19.80\): \((19.80 - 17.67)^{2}=(2.13)^{2}=4.5369\)
For \(x_{17}=14.86\): \((14.86 - 17.67)^{2}=(-2.81)^{2}=7.8961\)
For \(x_{18}=8.09\): \((8.09 - 17.67)^{2}=(-9.58)^{2}=91.7764\)
For \(x_{19}=11.40\): \((11.40 - 17.67)^{2}=(-6.27)^{2}=39.3129\)
For \(x_{20}=18.30\): \((18.30 - 17.67)^{2}=(0.63)^{2}=0.3969\)

Sum up all \((x_{i}-\bar{x})^{2}\):
\(\sum_{i = 1}^{20}(x_{i}-\bar{x})^{2}=23.2324+147.1369+88.9249+1.3456+3.8025+78.4996+155.5009+7.7841+7.6729+256+175.2976+7.8961+5.0625+5.3824+63.2025+4.5369+7.8961+91.7764+39.3129+0.3969 = 1193.93\)

Then \(s^{2}=\frac{1193.93}{20 - 1}=\frac{1193.93}{19}\approx62.849\)

Step2: Calculate the sample standard deviation

The formula for sample standard deviation \(s=\sqrt{s^{2}}\)
Since \(s^{2}\approx62.849\), then \(s=\sqrt{62.849}\approx7.93\)

Answer:

\(7.93\)