Question

find the sample variance and standard deviation.
5, 47, 14, 50, 34, 25, 28, 31, 35, 28
choose the correct answer below. fill in the answer box to complete your choice.
(round to two decimal places as needed.)
○ a. ( s^{2}= )
○ b. ( sigma^{2}= )
choose the correct answer below. fill in the answer box to complete your choice.
(round to one decimal place as needed.)
○ a. ( s= )
○ b. ( sigma= )

Explanation:

Step1: Calculate the sample mean

The formula for the sample mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\).
Here \(n = 10\) and \(x=\{5,47,14,50,34,25,28,31,35,28\}\).
\(\sum_{i=1}^{10}x_{i}=5 + 47+14 + 50+34+25+28+31+35+28=297\).
\(\bar{x}=\frac{297}{10}=29.7\).

Step2: Calculate the sum of squared deviations

The formula for the sum of squared deviations \(\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\).
\((5 - 29.7)^{2}=(- 24.7)^{2}=610.09\), \((47-29.7)^{2}=(17.3)^{2}=299.29\), \((14 - 29.7)^{2}=(-15.7)^{2}=246.49\), \((50 - 29.7)^{2}=(20.3)^{2}=412.09\), \((34 - 29.7)^{2}=(4.3)^{2}=18.49\), \((25 - 29.7)^{2}=(-4.7)^{2}=22.09\), \((28 - 29.7)^{2}=(-1.7)^{2}=2.89\), \((31 - 29.7)^{2}=(1.3)^{2}=1.69\), \((35 - 29.7)^{2}=(5.3)^{2}=28.09\), \((28 - 29.7)^{2}=(-1.7)^{2}=2.89\).
\(\sum_{i = 1}^{10}(x_{i}-\bar{x})^{2}=610.09+299.29+246.49+412.09+18.49+22.09+2.89+1.69+28.09+2.89 = 1644.1\).

Step3: Calculate the sample variance

The formula for the sample variance \(s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}\).
Since \(n = 10\), \(s^{2}=\frac{1644.1}{9}\approx182.68\).

Step4: Calculate the sample standard deviation

The formula for the sample standard deviation \(s=\sqrt{s^{2}}\).
\(s=\sqrt{182.68}\approx13.5\).

Answer:

A. \(s^{2}=182.68\)
A. \(s = 13.5\)