Statistics And Probability Question 377
Question: The variance of 20 observations is 5. If each observation is multiplied by 2, then what is the new variance of the resulting observations?
Options:
A) 5
B) 10
C) 20
D) 40
Show Answer
Answer:
Correct Answer: C
Solution:
[c] Let $ x_1,x_2,…,x_{20} $ be the given observations. Given, $ \frac{1}{20}\sum\limits_{i=1}^{20}{{{(x_{i}-\bar{x})}^{2}}=5} $ To find variance of $ 2x_1,2x_2,2x_3,…2x_{20}, $ Let $ \bar{x} $ denotes the mean of new observation, Clearly, $ \bar{x}=\frac{\sum\limits_{i=1}^{20}{2x_{i}}}{20}=\frac{2\sum\limits_{i=1}^{20}{x_{i}}}{20}=2\bar{x} $ Now, variance of new observation $ =\frac{1}{20}\sum\limits_{i=1}^{20}{{{(2x_{i}-\bar{x})}^{2}}=\frac{1}{20}\sum\limits_{i=1}^{20}{{{(2x_{i}-2\bar{x})}^{2}}}} $ $ =\frac{1}{20}\sum\limits_{i=1}^{20}{4{{(x_{i}-\bar{x})}^{2}}=4( \frac{1}{20}\sum\limits_{i=1}^{20}{{{(x_{i}-\bar{x})}^{2}}} )} $ $ =4\times 5=20 $
BETA
