Answer:
The mean = 16
The standard deviation = 7.19
Explanation:
N1 = 200 X1 = 25 σ1 = 3
N2= 250 X2 = 10 σ2 = 4
N3 = 300 X3= 15 σ3 = 5
The mean of a combined distribution is given by:
[tex]X = \frac{X_1N_1+X_2N_2+X_3N_3}{N_1+N_2+N_3}\\X = \frac{25*200+10*250+15*300}{200+250+300}\\X=16[/tex]
The differences from the mean for each component are:
[tex]D_1 = 25-16=9\\D_2=10-16=-6\\D_3=15-16=-1[/tex]
The standard deviation of a combined distribution is given by:
[tex]\sigma=\sqrt{\frac{N_1(\sigma_1^2+D_1^2)+N_2(\sigma_2^2+D_2^2)+N_3(\sigma_3^2+D_3^2)}{N_1+N_2+N_3}}\\\sigma=\sqrt{\frac{200(3^2+9^2)+250(4^2+(-6)^2)+300(5^2+(-1)^2)}{200+250+300}}\\\sigma=\sqrt{\frac{18000+13000+7800}{750} }\\\sigma=7.19[/tex]
The mean = 16
The standard deviation = 7.19
