To calculate the variance of the weekly salaries of the employees, we follow these steps:
### Step 1: List the salaries and calculate the mean
First, we list the salaries given:
- \[tex]$245
- \$[/tex]300
- \[tex]$325
- \$[/tex]465
- \$100
To find the mean salary ([tex]\(\bar{x}\)[/tex]), sum up the salaries and divide by the number of employees.
[tex]\[
\bar{x} = \frac{245 + 300 + 325 + 465 + 100}{5}
\][/tex]
[tex]\[
\bar{x} = \frac{1435}{5} = 287
\][/tex]
### Step 2: Calculate the squared differences from the mean
Next, we subtract the mean salary from each individual salary and square the result.
[tex]\[
(245 - 287)^2 = (-42)^2 = 1764
\][/tex]
[tex]\[
(300 - 287)^2 = (13)^2 = 169
\][/tex]
[tex]\[
(325 - 287)^2 = (38)^2 = 1444
\][/tex]
[tex]\[
(465 - 287)^2 = (178)^2 = 31684
\][/tex]
[tex]\[
(100 - 287)^2 = (-187)^2 = 34969
\][/tex]
### Step 3: Sum the squared differences
Now, we sum all these squared differences:
[tex]\[
1764 + 169 + 1444 + 31684 + 34969 = 70030
\][/tex]
### Step 4: Divide by the number of observations minus one (n-1) to get the sample variance
Since we have a sample size of 5 (n = 5), we divide by 4 (n-1):
[tex]\[
s^2 = \frac{70030}{5-1} = \frac{70030}{4} = 17507.5
\][/tex]
So, the sample variance [tex]\(s^2\)[/tex] is 17507.5.
