Two sample of size 100 and 150 respectively have means 15 and 16 and standard deviations 3 and 4 respectively. Find the combined mean and standard deviation of Size 250.

Explanation

Given: `N_(1)`=100, `barX_(1)`=15, `sigma_(1)`=3
`N_(2)`=150, `barX_(2)`=16, `sigma_(2)`=4
Now,
`barX_(12)=(N_(1)barX_(1)+N_(2)barX_(2))/(N_(1)+N_(2))`
`=(100xx15+150xx16)/(100+150)`
`=(1,500+2,400)/(250)=(3,900)/(250)=15.6`
`d_(1)=barX_(1)-barX_(12)=15-15.6=-0.6`
`d_(2)=barX_(2)-barX_(12)=16-15.6=0.4`
`sigma_(12)=sqrt(N_(1)sigma_(1)^(2)+N_(2)sigma_(2)^(2)+N_(1)d_(1)^(2)+N_(2)d_(2)^(2))/(N_(1)+N_(2))`
`=sqrt(100xx(3)^(2)+150xx(4)^(2)+100xx(-0.6)^(2)+150xx(0.4)^(2))/(100+150)`
`=sqrt(100xx9+150xx16+100xx0.36+150xx0.16)/(250)`
`=sqrt((900+2,400+36+24)/(250))`
`=sqrt((3,360)/(250))=sqrt(13.44)`
3.67
Hence, the combined mean is 15.6 and combined standard deviation is 3.67.