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The weekly salaries of a sample of employees at the local bank are given in the table below:

| Employee | Weekly Salary |
|----------|----------------|
| Anja | [tex]$245 |
| Raz | $[/tex]300 |
| Natalie | [tex]$325 |
| Mic | $[/tex]465 |
| Paul | $100 |

What is the variance for the data?

[tex]\[ \text{Variance: } s^2 = \frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \ldots + (x_n - \bar{x})^2}{n - 1} \][/tex]

A. 118.35

B. 132.32


Sagot :

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.
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