Let $x_1 , x_2,...., x_n$ be n observations, and let $\bar{x}$ be their arithmetic mean and $\sigma^2$ be the variance.
Variance of $2x_1, 2x_2, ..., 2x_n$ is $4 \sigma^2$.
Arithmetic mean $2x_1, 2x_2, ..., 2x_n $ is 4 $\bar{x}$ .
Statement-1 is true, statement-2 is true; statement-2 is a correct explanation for Statement-1
Statement-1 is true, statement-2 is true; statement-2 is not a correct explanation for Statement-1
Statement-1 is true, statement-2 is false
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The Correct Option isD
Approach Solution - 1
If each observation is multiplied by k, mean gets multiplied by k and variance gets multiplied by $k^2$. Hence the new mean should be $2 \bar{x}$ and new variance should be $k^2 \sigma^2$. So statement-1 is true and statement-2 is false.
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Approach Solution -2
\(\sigma^2=\sum\frac{x_i^2}{n}-(\sum\frac{x_i}{n})^2\) and variance of \(2x_1.......2x_n\) \(\sum\frac{(2x_i)^2}{n}-(\sum\frac{(2x_i)}{n})^2=4\sigma^2\) \(\therefore\) Statement 1: variance of \(2x_1,2x_2.......2x_n\) is \(4\sigma^2 \space{is} \space true\) \(Then\space A.M\space of 2x_1......2x_n\) \(=\frac{2x_1+2x_2+......+2x_n}{n}\) \(=2\bar{x}\) \(\therefore\space statement\space 2\space is \space false\)
Set is the collection of well defined objects. Sets are represented by capital letters, eg. A={}. Sets are composed of elements which could be numbers, letters, shapes, etc.
Example of set: Set of vowels A={a,e,i,o,u}
Representation of Sets
There are three basic notation or representation of sets are as follows:
Statement Form: The statement representation describes a statement to show what are the elements of a set.
For example, Set A is the list of the first five odd numbers.
Roster Form: The form in which elements are listed in set. Elements in the set is seperatrd by comma and enclosed within the curly braces.
For example represent the set of vowels in roster form.
A={a,e,i,o,u}
Set Builder Form:
The set builder representation has a certain rule or a statement that specifically describes the common feature of all the elements of a set.
The set builder form uses a vertical bar in its representation, with a text describing the character of the elements of the set.
For example, A = { k | k is an even number, k ≤ 20}. The statement says, all the elements of set A are even numbers that are less than or equal to 20.
