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Answer :
To solve this problem and find out the range and standard deviation of the given data set, let's go through the steps:
### Data Set
The given data set is:
[tex]\[ 79.4, 81.6, 116.4, 99.8, 98, 107.8, 90.3, 82.9, 116.4, 116.4 \][/tex]
### Step 1: Calculate the Range
The range of a data set is the difference between the maximum and minimum values.
- Maximum value in the data set: 116.4
- Minimum value in the data set: 79.4
Range = Maximum value - Minimum value
[tex]\[ \text{Range} = 116.4 - 79.4 = 37.0 \][/tex]
### Step 2: Calculate the Sample Standard Deviation
The standard deviation is a measure of how spread out the numbers in a data set are. Since we're dealing with a sample (not the entire population), we will calculate the sample standard deviation.
1. Find the mean (average) of the data set:
[tex]\[
\text{Mean} = \frac{79.4 + 81.6 + 116.4 + 99.8 + 98 + 107.8 + 90.3 + 82.9 + 116.4 + 116.4}{10}
\][/tex]
[tex]\[
\text{Mean} = \frac{989.0}{10} = 98.9
\][/tex]
2. Calculate each data point's deviation from the mean, square it, and find the sum:
[tex]\[
\sum (\text{Data point} - \text{Mean})^2 = (79.4 - 98.9)^2 + (81.6 - 98.9)^2 + \ldots + (116.4 - 98.9)^2
\][/tex]
3. Divide by (n - 1) where n is the number of data points to get the variance:
[tex]\[
\text{Variance} = \frac{\sum (\text{Data point} - \text{Mean})^2}{n - 1}
\][/tex]
- Here, [tex]\(n = 10\)[/tex], so use [tex]\(n - 1 = 9\)[/tex].
4. Take the square root of the variance to find the standard deviation:
[tex]\[
\text{Standard Deviation} = \sqrt{\text{Variance}}
\][/tex]
Upon calculating, the sample standard deviation rounds to:
[tex]\[ \text{Standard Deviation} = 14.9 \][/tex]
### Final Result
- The range is [tex]\(37.0\)[/tex].
- The standard deviation is [tex]\(14.9\)[/tex].
These calculations provide the complete solution for the problem!
### Data Set
The given data set is:
[tex]\[ 79.4, 81.6, 116.4, 99.8, 98, 107.8, 90.3, 82.9, 116.4, 116.4 \][/tex]
### Step 1: Calculate the Range
The range of a data set is the difference between the maximum and minimum values.
- Maximum value in the data set: 116.4
- Minimum value in the data set: 79.4
Range = Maximum value - Minimum value
[tex]\[ \text{Range} = 116.4 - 79.4 = 37.0 \][/tex]
### Step 2: Calculate the Sample Standard Deviation
The standard deviation is a measure of how spread out the numbers in a data set are. Since we're dealing with a sample (not the entire population), we will calculate the sample standard deviation.
1. Find the mean (average) of the data set:
[tex]\[
\text{Mean} = \frac{79.4 + 81.6 + 116.4 + 99.8 + 98 + 107.8 + 90.3 + 82.9 + 116.4 + 116.4}{10}
\][/tex]
[tex]\[
\text{Mean} = \frac{989.0}{10} = 98.9
\][/tex]
2. Calculate each data point's deviation from the mean, square it, and find the sum:
[tex]\[
\sum (\text{Data point} - \text{Mean})^2 = (79.4 - 98.9)^2 + (81.6 - 98.9)^2 + \ldots + (116.4 - 98.9)^2
\][/tex]
3. Divide by (n - 1) where n is the number of data points to get the variance:
[tex]\[
\text{Variance} = \frac{\sum (\text{Data point} - \text{Mean})^2}{n - 1}
\][/tex]
- Here, [tex]\(n = 10\)[/tex], so use [tex]\(n - 1 = 9\)[/tex].
4. Take the square root of the variance to find the standard deviation:
[tex]\[
\text{Standard Deviation} = \sqrt{\text{Variance}}
\][/tex]
Upon calculating, the sample standard deviation rounds to:
[tex]\[ \text{Standard Deviation} = 14.9 \][/tex]
### Final Result
- The range is [tex]\(37.0\)[/tex].
- The standard deviation is [tex]\(14.9\)[/tex].
These calculations provide the complete solution for the problem!
Thank you for reading the article A sample of size tex n 10 tex is drawn from a population The data is shown below tex begin array ccccc 79 4 81. We hope the information provided is useful and helps you understand this topic better. Feel free to explore more helpful content on our website!
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Rewritten by : Jeany
