Question

The mean life of a sample of 60 bulbs was 650 hours and the standard deviation was 8 hours, a second sample of 80 bulbs has a mean life of 660 hours and standard deviation of 7 hours. find the overall standard deviation.

Answer 1

We have,

$n_1=60,{\overline{x}}_1=650,s_1=8$

$n_2=80,{\overline{x}}_2=660,s_2=7$

$\sigma =\sqrt{\frac{n_1{s}_1^2+n_2{s}_2^2}{n_1+n_2}+\frac{n_1{n}_2({\overline{x}}_1-{\overline{x}}_2{)}^2}{(n_1+n_2{)}^2}}$

$\sigma =\sqrt{\frac{60\left(64\right)+80\left(49\right)}{60+80}+\frac{60\times 80(650-660{)}^2}{(60+80{)}^2}}$

$\sigma =\sqrt{\frac{3840+3920}{140}+\frac{4800\left(100\right)}{(140{)}^2}}$

$\sigma =\sqrt{\frac{192+196}{7}+\frac{1200}{49}}$

$\sigma =\sqrt{\frac{3916}{49}}$

$\sigma =\sqrt{79.9}$

$\sigma =8.9$

Hence, the overall standard deviation is 8.9