To compute the sample variance for City A's ATM fees data, we follow these steps:
1. List the given ATM fees:
[tex]\[
1.25, 1.00, 1.50, 1.50, 1.00
\][/tex]
2. Calculate the mean (average) of the data:
[tex]\[
\text{Mean} = \frac{1.25 + 1.00 + 1.50 + 1.50 + 1.00}{5} = \frac{6.25}{5} = 1.25
\][/tex]
3. Find the squared differences from the mean for each data point:
[tex]\[
(1.25 - 1.25)^2 = 0.00
\][/tex]
[tex]\[
(1.00 - 1.25)^2 = (-0.25)^2 = 0.0625
\][/tex]
[tex]\[
(1.50 - 1.25)^2 = (0.25)^2 = 0.0625
\][/tex]
[tex]\[
(1.50 - 1.25)^2 = (0.25)^2 = 0.0625
\][/tex]
[tex]\[
(1.00 - 1.25)^2 = (-0.25)^2 = 0.0625
\][/tex]
4. Sum these squared differences:
[tex]\[
0.00 + 0.0625 + 0.0625 + 0.0625 + 0.0625 = 0.25
\][/tex]
5. Calculate the sample variance (noting that we divide by [tex]\(n-1 = 5-1 = 4\)[/tex]):
[tex]\[
s^2 = \frac{0.25}{4} = 0.0625
\][/tex]
6. Round the variance to the nearest cent:
[tex]\[
0.0625 \approx 0.06
\][/tex]
So, the variance for ATM fees in City A is [tex]\(\$0.06\)[/tex].
