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Answer :
Sure! Let's walk through the process of calculating the sample standard deviation for the given data set. Here are the steps we will follow:
1. List the Data: [tex]$86, 83, 76, 82, 67, 69, 82, 82, 69$[/tex]
2. Calculate the Sample Mean:
[tex]\[
\text{Mean (\(\bar{x}\))} = \frac{\text{Sum of all data points}}{\text{Number of data points}}
\][/tex]
[tex]\[
\text{Mean (\(\bar{x}\))} = \frac{86 + 83 + 76 + 82 + 67 + 69 + 82 + 82 + 69}{9} = \frac{696}{9} = 77.333
\][/tex]
3. Calculate Each Data Point's Deviation from the Mean and Square It:
[tex]\[
\text{Squared Differences} = (86 - 77.333)^2, (83 - 77.333)^2, \ldots, (69 - 77.333)^2
\][/tex]
[tex]\[
(86 - 77.333)^2 = 75.111 \\
(83 - 77.333)^2 = 31.361 \\
(76 - 77.333)^2 = 1.778 \\
(82 - 77.333)^2 = 21.861 \\
(67 - 77.333)^2 = 106.778 \\
(69 - 77.333)^2 = 69.444 \\
(82 - 77.333)^2 = 21.861 \\
(82 - 77.333)^2 = 21.861 \\
(69 - 77.333)^2 = 69.444
\][/tex]
4. Sum of Squared Differences:
[tex]\[
\text{Sum of Squared Differences} = 75.111 + 31.361 + 1.778 + 21.861 + 106.778 + 69.444 + 21.861 + 21.861 + 69.444 = 420.000
\][/tex]
5. Calculate the Sample Variance:
[tex]\[
\text{Sample Variance} = \frac{\text{Sum of Squared Differences}}{\text{Number of data points} - 1}
\][/tex]
[tex]\[
\text{Sample Variance} = \frac{420.000}{9 - 1} = \frac{420.000}{8} = 52.500
\][/tex]
6. Calculate the Sample Standard Deviation:
[tex]\[
\text{Sample Standard Deviation} = \sqrt{\text{Sample Variance}} = \sqrt{52.500} = 7.246
\][/tex]
So, the sample standard deviation for the given data set, rounded to the nearest thousandth, is 7.246.
1. List the Data: [tex]$86, 83, 76, 82, 67, 69, 82, 82, 69$[/tex]
2. Calculate the Sample Mean:
[tex]\[
\text{Mean (\(\bar{x}\))} = \frac{\text{Sum of all data points}}{\text{Number of data points}}
\][/tex]
[tex]\[
\text{Mean (\(\bar{x}\))} = \frac{86 + 83 + 76 + 82 + 67 + 69 + 82 + 82 + 69}{9} = \frac{696}{9} = 77.333
\][/tex]
3. Calculate Each Data Point's Deviation from the Mean and Square It:
[tex]\[
\text{Squared Differences} = (86 - 77.333)^2, (83 - 77.333)^2, \ldots, (69 - 77.333)^2
\][/tex]
[tex]\[
(86 - 77.333)^2 = 75.111 \\
(83 - 77.333)^2 = 31.361 \\
(76 - 77.333)^2 = 1.778 \\
(82 - 77.333)^2 = 21.861 \\
(67 - 77.333)^2 = 106.778 \\
(69 - 77.333)^2 = 69.444 \\
(82 - 77.333)^2 = 21.861 \\
(82 - 77.333)^2 = 21.861 \\
(69 - 77.333)^2 = 69.444
\][/tex]
4. Sum of Squared Differences:
[tex]\[
\text{Sum of Squared Differences} = 75.111 + 31.361 + 1.778 + 21.861 + 106.778 + 69.444 + 21.861 + 21.861 + 69.444 = 420.000
\][/tex]
5. Calculate the Sample Variance:
[tex]\[
\text{Sample Variance} = \frac{\text{Sum of Squared Differences}}{\text{Number of data points} - 1}
\][/tex]
[tex]\[
\text{Sample Variance} = \frac{420.000}{9 - 1} = \frac{420.000}{8} = 52.500
\][/tex]
6. Calculate the Sample Standard Deviation:
[tex]\[
\text{Sample Standard Deviation} = \sqrt{\text{Sample Variance}} = \sqrt{52.500} = 7.246
\][/tex]
So, the sample standard deviation for the given data set, rounded to the nearest thousandth, is 7.246.
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