The mean of a data is 9. If each observation is multiplied by 3 and th

1.	The mean of a data is 9. If each observation is multiplied by 3 and then 1 is added to each result, then the mean of new observation is………A) 19 B) 20 C) 28 D) 30
Answer» Correct option is: C) 28 Let \(x_1\)'s are observations of the original data . Given that mean of data is 9. Let total no. of observation be n. \(\therefore\) Sum of observations = n\(\overline x\) = 9n \(\Rightarrow\) \(\sum \limits _{i=1}^n x_i\) = 9n ....(1) Since, each observation ( \(x_1\)'s) are multiplied by 3 and then 1 is added to each result. \(\therefore\) (\(3x_i + 1\))'s are new observations. \(\therefore\) Mean of formed data = \(\frac {sum\, of \,new \,observations}n\) = \(\frac {\sum \limits_{i=1}^n (3x_i+1)}{n}\) = \(\frac {\sum \limits_{i=1}^n 3x_i + \sum \limits_{i=1}^n 1}{n}\) = \(\frac {3\sum \limits_{i=1}^n x_i +n}{n}\) (\(\because\) \(\sum \limits_{i=1}^n 1 = 1 + 1+1+...+1 (n \, times) = n\)) = \(\frac {3\times 9n+n}{n} \) (From (i)) = \(\frac {28n}{n} = 28\) Hence, the mean of new observations is 28. Correct option is: C) 28

The mean of a data is 9. If each observation is multiplied by 3 and then 1 is added to each result, then the mean of new observation is………A) 19 B) 20 C) 28 D) 30

Answer»

Correct option is: C) 28

Let \(x_1\)'s are observations of the original data .

Given that mean of data is 9.

Let total no. of observation be n.

\(\therefore\) Sum of observations = n\(\overline x\) = 9n

\(\Rightarrow\) \(\sum \limits _{i=1}^n x_i\) = 9n ....(1)

Since, each observation ( \(x_1\)'s) are multiplied by 3 and then 1 is added to each result.

\(\therefore\) (\(3x_i + 1\))'s are new observations.

\(\therefore\) Mean of formed data = \(\frac {sum\, of \,new \,observations}n\)

= \(\frac {\sum \limits_{i=1}^n (3x_i+1)}{n}\)

= \(\frac {\sum \limits_{i=1}^n 3x_i + \sum \limits_{i=1}^n 1}{n}\)

= \(\frac {3\sum \limits_{i=1}^n x_i +n}{n}\) (\(\because\) \(\sum \limits_{i=1}^n 1 = 1 + 1+1+...+1 (n \, times) = n\))

= \(\frac {3\times 9n+n}{n} \) (From (i))

= \(\frac {28n}{n} = 28\)

Hence, the mean of new observations is 28.

Correct option is: C) 28