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  • Two sets each of 20 observations, have the same standard derivation 5. The first set has a mean 17 and the second a mean 22. Determine the standard deviation of the set obtained by combining the given two sets.

Two sets each of 20 observations, have the same standard derivation 5. The first set has a mean 17 and the second a mean 22. Determine the standard deviation of the set obtained by combining the given two sets.

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\\\\ \text{Given that two sets each of 20 observations, have the same standard derivation 5.} \\\\ \\ \text{ The first set has a mean 17 and the second a mean 22.} \\\\ \\ \text{ We have to show that the standard deviation of the set obtained by combining the given two sets } \\\\ \\ \text{ As per given criteria, for first set Number of observations , }n_{1}=20~ \\\\


\\ \text{Standard deviation }s_{1}=5~ \text{And mean } \overline{x_{1}}=17 \\\\ \\ \text{~ For second set Number of observations, }n_{2}=20~ \text{Standard deviation } s_{2}=5~ \\\\ \\ \text{and mean }\overline{x_{2}}=22 \\\\ \\ \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{20 \left( 5 \right) ^{2}+20 \left( 5 \right) ^{2}}{20+20}+\frac{ \left( 20\ast20 \right) \left( 17-22 \right) ^{2}}{ \left( 20+20 \right) ^{2}}}= \sqrt {\frac{1000}{40}+\frac{10000}{1600}} = \\\\ \\\\ \sqrt {25+\frac{25}{4}}=\sqrt {\frac{100+25}{4}}=\sqrt {\frac{125}{4}} = 5.59\\

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