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JEE Main 2010
Exam
JEE Main
Year
2010
Subject
Mathematics
Chapter
Statistics

JEE Main 2010 · Mathematics · Statistics

For two data sets, each of size 5 , the variances are given to be 4...

For two data sets, each of size 5 , the variances are given to be 4 and 5 and the corresponding means are given to be 2 and 4 , respectively. The variance of the combined data set is

Options

  1. A112\frac{11}{2}
  2. B6
  3. C132\frac{13}{2}
  4. D52\frac{5}{2}

Correct answer

A. 112\frac{11}{2}

Step-by-step solution

σx2=4σy2=5xˉ=2yˉ=4\sigma_x^2=4 \sigma_y^2=5 \bar{x}=2 \bar{y}=4∑xi5=2σx2=(12∑xi2)−(xˉ)2=15(∑yi2)−16∑xi2=40∑yi2=105σz2=110(∑xi2+∑yi2)−(xˉ+yˉ2)2=110(40+105)−9=145−9010=5510=112\begin{aligned} \frac{\sum x_i}{5} & =2 \\ \sigma_x^2 & =\left(\frac{1}{2} \sum x_i^2\right)-(\bar{x})^2=\frac{1}{5}\left(\sum y_i^2\right)-16 \\ \sum x_i^2 & =40 \\ \sum y_i^2 & =105 \\ \sigma_z^2 & =\frac{1}{10}\left(\sum x_i^2+\sum y_i^2\right)-\left(\frac{\bar{x}+\bar{y}}{2}\right)^2=\frac{1}{10}(40+105)-9=\frac{145-90}{10}=\frac{55}{10}=\frac{11}{2}\end{aligned}$$
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