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I don't know how to solve in my statistics homework problem

Sample

In a class of 6 students, their average age is 21. When the teacher joins the class, the avg age becomes 26. What's the age of the teacher ?

Is there a formula to do this or a series of substitutions ?

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    yes I think there is one: $\frac{1}{N}\sum_{i=1}^N x_i$. In your problem $N=6$ when the teacher joins then $N=7$.2012-09-18
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    @SeyhmusGungoren could you explain your formula ? I don't understand very well2012-09-18
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    In the formula $x_i$ is the age of each person and $N$ is the total number of persons. For example if there are $6$ persons, then you have $\frac{1}{6}(x_1+x_2+...+x_6)=21$ becase $N=6$. If a new person comes $N$ will be $7$ and the result will be $26$. Can you do the rest? I think you can!!2012-09-18
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    @SeyhmusGungoren Thanks a lot! it helps a lot.2012-09-18

4 Answers 4

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If the teacher's age was $21$, the average would be $21$. But the average is $26$, so the teacher's age must be $(7)(26-21)$ more than $21$.

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If you have the sample average based on $n$ samples, $\overline{x}_n$, and add a new observation, $x$, then the new sample average is

$$ \overline{x}_{n+1} = \frac{ n \overline{x}_n + x }{n + 1} $$

using the data you provided, $\overline{x}_n = 21$, $n = 6$ and $\overline{x}_{n+1} = 26$, so you know 3 out of 4 unknowns:

$$ 26 = \frac{ 6 \cdot 21 + x }{6 + 1} $$

and you can solve for the last unknown, $x$, which I'll leave to you. I hope this helps!!

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Hint: Do you know the relationship between the average age, the total age and the number of people? How could you use this in two steps to get from the average age and number of people in one case to the age of an additional person that gets added to the total?

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    No, could explain it or give me a reference ?2012-09-18
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    "No", just like that? No you do not know any of these?2012-09-18
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    @Lucas: I was referring to the formula that Seyhmus gave in a comment shortly thereafter.2012-09-18
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You have this relationship between sample means of size n and n+1.

∑X$_i$/(n+1)= n(∑X$_i$/n)/(n+1) +X$_n$$_+$$_1$/(n+1)

In your case:

26= 6(21)/7 + X$_7$/7.

So solve for X$_7$ =7[26-6(21)/7]=7 (26) -6(21).