Find the sample variance s² for the following sample data. Round your answer to the nearest hundredth. X: O 64.20 O 30.56 23 38.20 O 77.64 80.25 20 15 31 27
Find the sample variance s² for the following sample data. Round your answer to the nearest hundredth. X: O 64.20 O 30.56 23 38.20 O 77.64 80.25 20 15 31 27
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter10: Statistics
Section: Chapter Questions
Problem 25SGR
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![### Sample Variance Calculation
**Problem:**
Find the sample variance \( s^2 \) for the following sample data. Round your answer to the nearest hundredth.
**Data:**
\( x: \) 23, 20, 15, 31, 27
**Options:**
- A. \( 64.20 \)
- B. \( 30.56 \)
- C. \( 38.20 \)
- D. \( 77.64 \)
- E. \( 80.25 \)
**Solution Explanation:**
To calculate the sample variance \( s^2 \), follow these steps:
1. **Find the mean \( \bar{x} \):**
\[
\bar{x} = \frac{\sum x}{n} = \frac{23 + 20 + 15 + 31 + 27}{5} = \frac{116}{5} = 23.2
\]
2. **Calculate each data point's deviation from the mean, square it, and sum those squares:**
\[
\begin{aligned}
& (23 - 23.2)^2 = (-0.2)^2 = 0.04 \\
& (20 - 23.2)^2 = (-3.2)^2 = 10.24 \\
& (15 - 23.2)^2 = (-8.2)^2 = 67.24 \\
& (31 - 23.2)^2 = (7.8)^2 = 60.84 \\
& (27 - 23.2)^2 = (3.8)^2 = 14.44 \\
\end{aligned}
\]
3. **Sum these squared deviations:**
\[
0.04 + 10.24 + 67.24 + 60.84 + 14.44 = 152.8
\]
4. **Divide by \( n - 1 \) (where \( n \) is the number of data points) to get the sample variance:**
\[
s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1} = \frac{152.8}{4} = 38.2
\]
Thus, the correct answer is:
- C. \( 38.20 \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1f8a2f38-e81c-44ad-a4de-e9efb4a606bb%2Fe1720772-66ea-4990-86aa-470eff0dc2a8%2F4ctb4sc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Sample Variance Calculation
**Problem:**
Find the sample variance \( s^2 \) for the following sample data. Round your answer to the nearest hundredth.
**Data:**
\( x: \) 23, 20, 15, 31, 27
**Options:**
- A. \( 64.20 \)
- B. \( 30.56 \)
- C. \( 38.20 \)
- D. \( 77.64 \)
- E. \( 80.25 \)
**Solution Explanation:**
To calculate the sample variance \( s^2 \), follow these steps:
1. **Find the mean \( \bar{x} \):**
\[
\bar{x} = \frac{\sum x}{n} = \frac{23 + 20 + 15 + 31 + 27}{5} = \frac{116}{5} = 23.2
\]
2. **Calculate each data point's deviation from the mean, square it, and sum those squares:**
\[
\begin{aligned}
& (23 - 23.2)^2 = (-0.2)^2 = 0.04 \\
& (20 - 23.2)^2 = (-3.2)^2 = 10.24 \\
& (15 - 23.2)^2 = (-8.2)^2 = 67.24 \\
& (31 - 23.2)^2 = (7.8)^2 = 60.84 \\
& (27 - 23.2)^2 = (3.8)^2 = 14.44 \\
\end{aligned}
\]
3. **Sum these squared deviations:**
\[
0.04 + 10.24 + 67.24 + 60.84 + 14.44 = 152.8
\]
4. **Divide by \( n - 1 \) (where \( n \) is the number of data points) to get the sample variance:**
\[
s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1} = \frac{152.8}{4} = 38.2
\]
Thus, the correct answer is:
- C. \( 38.20 \)
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