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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 7 hours. Find the overall standard deviation.
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Solution
Given that `n_1 = 60, barx_1 = 650, s_1 = 8`
And `n_2 = 80, barx_2 = 660, s_2 = 7`
We know that for a combined series.
`sigma = sqrt((n_1s_1^2 + n_2s_2^2)/(n_1 + n_2) + (n_1n_2(barx_1 - barx_2)^2)/(n_1 + n_2)^2`
= `sqrt((60 xx (8)^2 + 80 xx (7)^2)/(60 + 80) + (60 xx 80(650 - 660)^2)/(60 + 80)^2`
= `sqrt((6 xx 64 + 8 xx 49)/14 + (60 xx 80 xx 100)/(140 xx 140)`
= `sqrt((192 + 196)/7 + 1200/49)`
= `sqrt(388/7 + 1200/49)`
= `sqrt((2716 + 1200)/49)`
= `sqrt(3916/49)`
= `62.58/7`
= 8.9
Hence, the required S.D. = 8.9
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