Question

The mean of 10 observations is 15. Find the 11th observation such that the mean of these
observations is 2 more than the original mean = 15.​

Answer 1

Answer:

• The 11th observation is 37.

Step-by-step explanation:

Given that:

• The mean of 10 observations is 15.

To Find:

• The 11th observation such that the mean of these observations is 2 more than the original.

We know that:

• Mean = Sum of observations ÷ No. of observations

Let us assume:

• The 11th observation be x.

Finding the sum of observations:

• When, Mean = 15
• No. of observations = 10

→ 15 = Sum of observations/10

→ Sum of observations = 15 × 10

→ Sum of observations = 150

∴ Original sum = 150

Finding the sum of observations:

• When, Mean = 15 + 2 = 17
• No. of observations = 10 + 1 = 11

→ 17 = Sum of observations/11

→ Sum of observations = 17 × 11

→ Sum of observations = 187

∴ New sum = 187

Finding the 11th observation:

11th observation = New sum - Original sum

• 11th observation = 187 - 150
• 11th observation = 37

Hence,

• The 11th observation is 37.

Answer 2

Given : The mean of 10 observations is 15 & the 11th observation such that the mean of these

observations is 2 more than the original mean .

Need To Find : The 11 th observation.

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀

❍ Let's Consider the 11 th observation be a .

⠀⠀⠀⠀⠀Given that ,

⠀⠀⠀⠀⠀⠀⠀▪︎⠀⠀The mean of 10 observations is 15 .

⠀⠀⠀⠀⠀Finding Original Sum of Observation :

As , We know that ,

⠀⠀⠀⠀⠀⠀Formula for MEAN :

$\qquad \star \qquad \boxed {\pink{\pmb{ \:\:\: Mean \:\:: \:\: \dfrac{ Sum \:of \:Observation \:\:}{Number \:\:of\:Observation \:} \:\:\:}}}$

⠀⠀⠀⠀⠀Here , Number of Observations is 10 and Mean is 15 .

$\qquad :\implies \sf Mean \:\:= \:\: \dfrac{ Sum \:of \:Observation \:\:}{Number \:\:of\:Observation \:} \:$

⠀⠀⠀⠀⠀⠀$\underline {\boldsymbol{\star\:Now \: By \: Substituting \: the \: known \: Values \::}}$

$\qquad :\implies \sf Mean \:\:= \:\: \dfrac{ Sum \:of \:Observation \:\:}{Number \:\:of\:Observation \:} \:$

$\qquad :\implies \sf 15 \:\:= \:\: \dfrac{ Sum \:of \:Observation \:\:}{10 \:} \:$

$\qquad :\implies \sf 15 \:\times 10 \:= \:\: Sum \:of \:Observation \:\:\:$

$\qquad :\implies \sf 150 \:= \:\: Sum \:of \:Observation \:\:\:$

$\qquad \therefore \underline{\purple{\:\pmb{Original \:Sum \:of \:Observation\:=\:150\: }} }\:\:\bigstar$

• Original Sum of Observation is 150 .

Now ,

⠀⠀⠀⠀⠀⠀$\underline {\boldsymbol{\star\:According \: to \: \: the \: Question \::}}$

⠀⠀☆⠀The 11th observation such that the mean of these observations is 2 more than the original mean is 15 .

⠀⠀⠀⠀⠀Therefore,

• Mean of the New Observation will be : 2 + Original mean = 2 + 15 = 17
• Number of New Observation is 11 .

⠀⠀⠀⠀⠀Finding New Sum of Observation :

$\qquad :\implies \sf Mean \:\:= \:\: \dfrac{ Sum \:of \:Observation \:\:}{Number \:\:of\:Observation \:} \:$

⠀⠀⠀⠀⠀⠀$\underline {\boldsymbol{\star\:Now \: By \: Substituting \: the \: known \: Values \::}}$

$\qquad :\implies \sf Mean \:\:= \:\: \dfrac{ Sum \:of \:Observation \:\:}{Number \:\:of\:Observation \:} \:$

$\qquad :\implies \sf 17 \:\:= \:\: \dfrac{ Sum \:of \:Observation \:\:}{11 \:} \:$

$\qquad :\implies \sf 17 \:\times 11 \:= \:\: Sum \:of \:Observation \:\:\:$

$\qquad :\implies \sf 187 \:= \:\: Sum \:of \:Observation \:\:\:$

$\qquad \therefore \underline{\purple{\:\pmb{New \:Sum \:of \:Observation\:=\:187\: }} }\:\:\bigstar$

• New Sum of Observation is 187 .

⠀⠀⠀⠀⠀⠀⠀Finding 11 th Observation : ⠀⠀

As , We know that,

$\qquad \star \qquad \boxed {\pink{\pmb{ \:\:\:11^{th}\:\:Observation \:\:: \:\: Sum \:_{(New \:Observation)}- Sum \:_{(Original \:Observation)}\:\:\:}}}$

⠀⠀⠀⠀⠀⠀⠀Here Sum of News Observation is 187 & Sum of Original Observation is 150 .

$\qquad \dashrightarrow \sf \:\:\:11^{th}\:\:Observation \:\:= \:\: Sum \:_{(New \:Observation)}- Sum \:_{(Original \:Observation)}\:\:$

⠀⠀⠀⠀⠀⠀$\underline {\boldsymbol{\star\:Now \: By \: Substituting \: the \: known \: Values \::}}$

$\qquad \dashrightarrow \sf \:a \:or\:\:11^{th}\:\:Observation \:\:= \:\: Sum \:_{(New \:Observation)}- Sum \:_{(Original \:Observation)}\:\:$

$\qquad \dashrightarrow \sf \:\:a \:or\:11^{th}\:\:Observation \:\:= \:\: 187 - 150\:\:$

$\qquad \dashrightarrow \sf \:a\:or\:11^{th}\:\:Observation \:\:= \:\: 37\:\:$

$\qquad \therefore \underline{\purple{\pmb{\:a\:or\:11^{th}\:\:Observation \:\:= \:\: 37\:\:\: }}}\:\:\bigstar$

• Here a signifies 11 th Observation which is 37 .

$\therefore \:\: \underline { \sf Hence, \:The \:11^{th} \:Observation \;is \:\bf 37 \:\:. }$