Respuesta :
Answer: σ² = 2.90
Step-by-step explanation: For a frequency table, population variance is calculated by:
[tex]\sigma^{2}=\frac{\Sigma f(x-\mu)^{2}}{\Sigma f}[/tex]
where
f is frequency
μ is expected value or mean of the distribution
In this case, mean of the distribution for a grouped frequency table is calculated by:
[tex]\mu=\Sigma x.f(x)[/tex]
where
x is (as the data is grouped) the mid value of the response time
So, the table is
x | f
6.45 | 18
7.45 | 21
8.45 | 34
9.45 | 46
10.45 | 46
11.45 | 25
12.45 | 21
And mean will be:
[tex]\mu=6.45(18)+7.45(21)+8.45(34)+9.45(46)+10.45(46)+11.45(25)+12.45(21)[/tex]
μ = 9.5874
To calculate the variance, the formula above is transformed into
[tex]\sigma^{2}=\frac{\Sigma f.x^{2}-\frac{(\Sigma f.x)^{2}}{\Sigma f} }{\Sigma f}[/tex]
So, to facilitate the calculations, divide each term in a table:
f | x | f.x | f.x²
18 | 6.45 | 116.1 | 748.845
21 | 7.45 | 156.45 | 1165.5525
34 | 8.45 | 287.3 | 2427.685
46 | 9.45 | 434.7 | 4107.915
46 | 10.45 | 480.7 | 5023.315
25 | 11.45 | 286.25 | 3277.5625
21 | 12.45 | 261.45 | 3255.0525
∑ 211 | | 2022.95 | 20005.9275
Population variance will be:
[tex]\frac{(\Sigma f.x)^{2}}{\Sigma f} =[/tex] [tex]\frac{(2022.95)^{2}}{211}[/tex] = 19394.9133
[tex]\sigma^{2}=\frac{\Sigma f.x^{2}-\frac{(\Sigma f.x)^{2}}{\Sigma f} }{\Sigma f}[/tex]
[tex]\sigma^{2}=\frac{20005.9275-19394.9133}{211}[/tex]
σ² = 2.90
The population variance for the response time for EMTs is 2.90.
