Question

6. A distribution consists of three components with frequencies 200, 250 and 300 having means

Answer 1

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:

$X = \frac{X_1N_1+X_2N_2+X_3N_3}{N_1+N_2+N_3}\\X = \frac{25*200+10*250+15*300}{200+250+300}\\X=16$

The differences from the mean for each component are:

$D_1 = 25-16=9\\D_2=10-16=-6\\D_3=15-16=-1$

The standard deviation of a combined distribution is given by:

$\sigma=\sqrt{\frac{N_1(\sigma_1^2+D_1^2)+N_2(\sigma_2^2+D_2^2)+N_3(\sigma_3^2+D_3^2)}{N_1+N_2+N_3}}\\\sigma=\sqrt{\frac{200(3^2+9^2)+250(4^2+(-6)^2)+300(5^2+(-1)^2)}{200+250+300}}\\\sigma=\sqrt{\frac{18000+13000+7800}{750} }\\\sigma=7.19$

The mean  = 16

The standard deviation = 7.19