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The mean life of a sample of 60 bulbs was 650 h and the standard deviation was 8 h. A second sample of 80 bulbs has a mean life of 660 h and standard deviation 7 h. Find the over all standard deviation.

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a

8.97

b

8.98

c

8.94

d

None of these

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detailed solution

Correct option is C

Given, $\begin{aligned} n_{1} = 60, {\bar{x}}_{1} = 650, σ_{1} = 8 \\ n_{2} = 80, {\bar{x}}_{2} = 660, σ_{2} = 7 \end{aligned}$

$∴$ Combined $\begin{array}{l} SD = \sqrt{\frac{n_{1} σ_{1}^{2} + n_{2} σ_{2}^{2}}{n_{1} + n_{2}} + \frac{n_{1} n_{2} {({\bar{x}}_{1} - {\bar{x}}_{2})}^{2}}{{(n_{1} + n_{2})}^{2}}} \\ = \sqrt{\frac{60 \times 64 + 80 \times 49}{60 + 80} + \frac{60 \times 80 (650 - 660)^{2}}{(60 + 80)^{2}}} \\ = \sqrt{\frac{3840 + 3920}{140} + \frac{(4800 \times 100)}{(140)^{2}}} \\ = \sqrt{\frac{7760}{140} + \frac{480000}{19600}} = \sqrt{\frac{776}{14} + \frac{4800}{196}} \\ = \sqrt{55 .42 + 24 .49} \\ = \sqrt{79 .91} = 8 .94 \end{array}$

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