IDNLearn.com: Where your questions meet expert answers and community support. Ask any question and get a detailed, reliable answer from our community of experts.
Sagot :
Let's solve this step by step.
### Step 1: Calculate the Mean ([tex]\(\bar{x}\)[/tex])
The given [tex]\( x \)[/tex] values are:
[tex]\[ 6, 15, 13, 2, 11 \][/tex]
The mean ([tex]\(\bar{x}\)[/tex]) is calculated as follows:
[tex]\[ \bar{x} = \frac{\sum x_i}{n} \][/tex]
Where [tex]\( n \)[/tex] is the number of values.
[tex]\[ \bar{x} = \frac{6 + 15 + 13 + 2 + 11}{5} = \frac{47}{5} = 9.4 \][/tex]
### Step 2: Calculate [tex]\((x - \bar{x})\)[/tex] for each [tex]\(x\)[/tex] value
[tex]\[ x - \bar{x} = \begin{cases} 6 - 9.4 = -3.4 \\ 15 - 9.4 = 5.6 \\ 13 - 9.4 = 3.6 \\ 2 - 9.4 = -7.4 \\ 11 - 9.4 = 1.6 \\ \end{cases} \][/tex]
### Step 3: Calculate [tex]\((x - \bar{x})^2\)[/tex] for each [tex]\(x\)[/tex] value
[tex]\[ (x - \bar{x})^2 = \begin{cases} (-3.4)^2 = 11.56 \\ 5.6^2 = 31.36 \\ 3.6^2 = 12.96 \\ (-7.4)^2 = 54.76 \\ 1.6^2 = 2.56 \\ \end{cases} \][/tex]
### Step 4: Calculate [tex]\(\sum (x - \bar{x})^2\)[/tex]
[tex]\[ \sum (x - \bar{x})^2 = 11.56 + 31.36 + 12.96 + 54.76 + 2.56 = 113.2 \][/tex]
### Step 5: Calculate the Sample Standard Deviation [tex]\(s\)[/tex]
The standard deviation for a sample is calculated as follows:
[tex]\[ s = \sqrt{\frac{\sum (x - \bar{x})^2}{n - 1}} \][/tex]
Where [tex]\( n - 1 \)[/tex] is the degrees of freedom, in this case, [tex]\( 5 - 1 = 4 \)[/tex].
[tex]\[ s = \sqrt{\frac{113.2}{4}} = \sqrt{28.3} \approx 5.319774431308154 \][/tex]
### Step 6: Round to Two Decimal Places
The rounded standard deviation [tex]\( s \)[/tex] is:
[tex]\[ s \approx 5.32 \][/tex]
### Summary of Results
[tex]\[ \bar{x} = 9.4 \\ (x - \bar{x}) = [-3.4, 5.6, 3.6, -7.4, 1.6] \\ (x - \bar{x})^2 = [11.56, 31.36, 12.96, 54.76, 2.56] \\ \sum (x - \bar{x})^2 = 113.2 \\ s = 5.32 \][/tex]
### Final Answer:
The sample standard deviation is approximately 5.32.
### Step 1: Calculate the Mean ([tex]\(\bar{x}\)[/tex])
The given [tex]\( x \)[/tex] values are:
[tex]\[ 6, 15, 13, 2, 11 \][/tex]
The mean ([tex]\(\bar{x}\)[/tex]) is calculated as follows:
[tex]\[ \bar{x} = \frac{\sum x_i}{n} \][/tex]
Where [tex]\( n \)[/tex] is the number of values.
[tex]\[ \bar{x} = \frac{6 + 15 + 13 + 2 + 11}{5} = \frac{47}{5} = 9.4 \][/tex]
### Step 2: Calculate [tex]\((x - \bar{x})\)[/tex] for each [tex]\(x\)[/tex] value
[tex]\[ x - \bar{x} = \begin{cases} 6 - 9.4 = -3.4 \\ 15 - 9.4 = 5.6 \\ 13 - 9.4 = 3.6 \\ 2 - 9.4 = -7.4 \\ 11 - 9.4 = 1.6 \\ \end{cases} \][/tex]
### Step 3: Calculate [tex]\((x - \bar{x})^2\)[/tex] for each [tex]\(x\)[/tex] value
[tex]\[ (x - \bar{x})^2 = \begin{cases} (-3.4)^2 = 11.56 \\ 5.6^2 = 31.36 \\ 3.6^2 = 12.96 \\ (-7.4)^2 = 54.76 \\ 1.6^2 = 2.56 \\ \end{cases} \][/tex]
### Step 4: Calculate [tex]\(\sum (x - \bar{x})^2\)[/tex]
[tex]\[ \sum (x - \bar{x})^2 = 11.56 + 31.36 + 12.96 + 54.76 + 2.56 = 113.2 \][/tex]
### Step 5: Calculate the Sample Standard Deviation [tex]\(s\)[/tex]
The standard deviation for a sample is calculated as follows:
[tex]\[ s = \sqrt{\frac{\sum (x - \bar{x})^2}{n - 1}} \][/tex]
Where [tex]\( n - 1 \)[/tex] is the degrees of freedom, in this case, [tex]\( 5 - 1 = 4 \)[/tex].
[tex]\[ s = \sqrt{\frac{113.2}{4}} = \sqrt{28.3} \approx 5.319774431308154 \][/tex]
### Step 6: Round to Two Decimal Places
The rounded standard deviation [tex]\( s \)[/tex] is:
[tex]\[ s \approx 5.32 \][/tex]
### Summary of Results
[tex]\[ \bar{x} = 9.4 \\ (x - \bar{x}) = [-3.4, 5.6, 3.6, -7.4, 1.6] \\ (x - \bar{x})^2 = [11.56, 31.36, 12.96, 54.76, 2.56] \\ \sum (x - \bar{x})^2 = 113.2 \\ s = 5.32 \][/tex]
### Final Answer:
The sample standard deviation is approximately 5.32.
Your participation means a lot to us. Keep sharing information and solutions. This community grows thanks to the amazing contributions from members like you. Discover the answers you need at IDNLearn.com. Thanks for visiting, and come back soon for more valuable insights.