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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 7 hours. Find the overall standard deviation.

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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 7 hours. We have to find the overall standard deviation.

As per the given criteria. In the first set of samples number of sample bulbs , \\\\ n_{1}=60 \\\\

\\ \\ \text{ Standard deviation }s_{1}=8 \text{hrs~ Mean life, }\overline{x_{1}}=650 \\\\ \\ \text{ And in second set of samples, number of sample bulbs }n_{2}=80 \text{ Standard deviation }s_{2}=7 hrs \\\\ \\ \text{~ Mean life }\overline{x_{2}}=660~ \\\\ \\ \text{We know the standard deviation for combined two series is } \\\\ \\ \sigma =\sqrt {\frac{n_{1}s_{1}^{2}+n_{2}s_{2}^{2}}{n_{1}+n_{2}}+ \left( n_{1}n_{2} \right) \frac{ \left( \overline{x_{1}}-\overline{x_{2}} \right) ^{2}}{ \left( n_{1}+n_{2} \right) ^{2}}} \\ \text{Substituting the corresponding values we get} \sigma =\sqrt {\frac{60 \left( 8 \right) ^{2}+80 \left( 7 \right) ^{2}}{60+80}+\frac{ \left( 60\ast80 \right) \left( 650-660 \right) ^{2}}{ \left( 20+80 \right) ^{2}}}= \sqrt {\frac{388}{7}+\frac{1200}{49}} = \\\\ =\sqrt {\frac{3916}{49}} = 8.9\\

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