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 false, Statement-2 is true
- 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
The Correct Option is D
Approach Solution - 1
Approach Solution -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
\(4\sigma^2 \space{is} \space true\)
\(Then\space A.M\space of 2x_1......2x_n\)
\(=\frac{2x_1+2x_2+......+2x_n}{n}\)
\(=2\bar{x}\)
\(\therefore\space statement\space 2\space is \space false\)
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Concepts Used:
Sets
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.
- Sometimes a ":" is used in the place of the "|".