Certainly! Let's work through the problem step-by-step.
### Given Values:
Our sample data consists of the following values: 7, 2, 6, 2, and 3.
### Step a: Compute the Sample Variance
1. Calculate the Sample Mean:
[tex]\[
\text{Sample Mean} (\bar{x}) = \frac{\sum_{i=1}^{n} x_i}{n}
\][/tex]
Where [tex]\( n \)[/tex] is the number of sample values.
[tex]\[
\text{Mean} = \frac{7 + 2 + 6 + 2 + 3}{5} = \frac{20}{5} = 4
\][/tex]
2. Calculate the Sum of Squared Differences from the Mean:
[tex]\[
\sum (x_i - \bar{x})^2
\][/tex]
[tex]\[
(7 - 4)^2 = 3^2 = 9
\][/tex]
[tex]\[
(2 - 4)^2 = (-2)^2 = 4
\][/tex]
[tex]\[
(6 - 4)^2 = 2^2 = 4
\][/tex]
[tex]\[
(2 - 4)^2 = (-2)^2 = 4
\][/tex]
[tex]\[
(3 - 4)^2 = (-1)^2 = 1
\][/tex]
[tex]\[
\sum (x_i - \bar{x})^2 = 9 + 4 + 4 + 4 + 1 = 22
\][/tex]
3. Calculate the Sample Variance:
[tex]\[
\text{Sample Variance} = \frac{\sum (x_i - \bar{x})^2}{n - 1}
\][/tex]
Since we are dealing with a sample, we use [tex]\( n-1 \)[/tex] as the denominator (Bessel's correction).
[tex]\[
\text{Sample Variance} = \frac{22}{5-1} = \frac{22}{4} = 5.5
\][/tex]
4. Round the Sample Variance to 1 Decimal Place:
[tex]\[
\text{Sample Variance} = 5.5
\][/tex]
### Step b: Determine the Sample Standard Deviation
1. Calculate the Sample Standard Deviation:
[tex]\[
\text{Sample Standard Deviation} = \sqrt{\text{Sample Variance}}
\][/tex]
[tex]\[
\text{Sample Standard Deviation} = \sqrt{5.5} \approx 2.345
\][/tex]
2. Round the Sample Standard Deviation to 3 Decimal Places:
[tex]\[
\text{Sample Standard Deviation} = 2.345
\][/tex]
### Final Answers:
a. Sample Variance = 5.5 (rounded to 1 decimal place)
b. Sample Standard Deviation = 2.345 (rounded to 3 decimal places)
