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To determine the variance for the given data, we will follow these steps:
1. List the weekly salaries:
[tex]\[ 245, 300, 325, 465, 100 \][/tex]
2. Calculate the mean (average) salary:
The mean [tex]\(\bar{x}\)[/tex] is calculated by adding all the salaries and dividing by the number of salaries.
[tex]\[ \bar{x} = \frac{245 + 300 + 325 + 465 + 100}{5} = \frac{1435}{5} = 287.0 \][/tex]
3. Determine the deviations from the mean for each salary:
[tex]\[ 245 - 287.0 = -42.0 \][/tex]
[tex]\[ 300 - 287.0 = 13.0 \][/tex]
[tex]\[ 325 - 287.0 = 38.0 \][/tex]
[tex]\[ 465 - 287.0 = 178.0 \][/tex]
[tex]\[ 100 - 287.0 = -187.0 \][/tex]
4. Square each deviation:
[tex]\[ (-42.0)^2 = 1764.0 \][/tex]
[tex]\[ (13.0)^2 = 169.0 \][/tex]
[tex]\[ (38.0)^2 = 1444.0 \][/tex]
[tex]\[ (178.0)^2 = 31684.0 \][/tex]
[tex]\[ (-187.0)^2 = 34969.0 \][/tex]
5. Sum the squared deviations:
[tex]\[ 1764.0 + 169.0 + 1444.0 + 31684.0 + 34969.0 = 70130.0 \][/tex]
6. Divide by the number of salaries minus 1 (to get the sample variance):
[tex]\[ s^2 = \frac{70130.0}{5 - 1} = \frac{70130.0}{4} = 17507.5 \][/tex]
Therefore, the variance for the weekly salaries is:
[tex]\[ s^2 = 17507.5 \][/tex]
So, the correct answer from the given choices is [tex]$17,507.5$[/tex].
1. List the weekly salaries:
[tex]\[ 245, 300, 325, 465, 100 \][/tex]
2. Calculate the mean (average) salary:
The mean [tex]\(\bar{x}\)[/tex] is calculated by adding all the salaries and dividing by the number of salaries.
[tex]\[ \bar{x} = \frac{245 + 300 + 325 + 465 + 100}{5} = \frac{1435}{5} = 287.0 \][/tex]
3. Determine the deviations from the mean for each salary:
[tex]\[ 245 - 287.0 = -42.0 \][/tex]
[tex]\[ 300 - 287.0 = 13.0 \][/tex]
[tex]\[ 325 - 287.0 = 38.0 \][/tex]
[tex]\[ 465 - 287.0 = 178.0 \][/tex]
[tex]\[ 100 - 287.0 = -187.0 \][/tex]
4. Square each deviation:
[tex]\[ (-42.0)^2 = 1764.0 \][/tex]
[tex]\[ (13.0)^2 = 169.0 \][/tex]
[tex]\[ (38.0)^2 = 1444.0 \][/tex]
[tex]\[ (178.0)^2 = 31684.0 \][/tex]
[tex]\[ (-187.0)^2 = 34969.0 \][/tex]
5. Sum the squared deviations:
[tex]\[ 1764.0 + 169.0 + 1444.0 + 31684.0 + 34969.0 = 70130.0 \][/tex]
6. Divide by the number of salaries minus 1 (to get the sample variance):
[tex]\[ s^2 = \frac{70130.0}{5 - 1} = \frac{70130.0}{4} = 17507.5 \][/tex]
Therefore, the variance for the weekly salaries is:
[tex]\[ s^2 = 17507.5 \][/tex]
So, the correct answer from the given choices is [tex]$17,507.5$[/tex].
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