Finding out missing observation $a$ and $b$

Question

The mean of $a,b,8,5,10$ is $6$ and the variance is $6.8$. find out value of $a$ and $b$

I tried something like this:

$$\bar{x}=6=\frac{a+b+8+5+10}{5}$$ $$7=a+b$$

Then i have put the value in the formula of variance

$$variance\ \sigma^2={\frac{{\Sigma x_i^2}}{n}-(\bar{x}^2)}$$ $$6.8={\frac{{a^2+b^2+189}}{5}-(36)}$$ $$25=a^2+b^2$$

I don't this my second equation is correct.If it correct then can anyone one guide me what do further?

You found $a+b$ wrong....it should be $7$, whereas your $a^{2}+b^{2}$ is correct. — Indrayudh Roy, Jun 10 '14 at 17:59

Ross Millikan · Accepted Answer · 2014-06-10T18:10:51.497

1

You wrote the correct equation for $a^2+b^2$, but I get $$6.8={\frac{{a^2+b^2+189}}{n}-(36)}\\34=a^2+b^2+189-180\\25=a^2+b^2$$ Coupled with your correct value for $a+b$, you should be able to get home.

Added: Now you have two equations in two unknowns. You might be able to solve them by inspection. Otherwise, you can write $b=7-a$ in substitute that into $25=a^2+b^2=a^2+(7-a)^2$, getting a quadratic in $a$.

edited Jun 10 '14 at 18:10

answered Jun 10 '14 at 18:05

Ross Millikan

374,822

you are right i haven't multiplied 36 with 5 – Freddy Jun 10 '14 at 18:08
ok so i got answer as a=3, b=4 – Freddy Jun 10 '14 at 18:14
Actually they can go either way around, but you are correct. The "by inspection" refers to the fact that when you see the sum of two squares, it might make you think of Pythagorean triples, and $3,4,5$ in particular. – Ross Millikan Jun 10 '14 at 18:16

score -1 · Answer 2 · answered Jun 10 '14 at 18:17

-1

Your second equation is wrong as well.

$$variance\ \sigma^2={\frac{{\Sigma (x_i - \bar{x})^2} }{n}}$$

That reduces to $$13 = (a-6)^2 + (b-6)^2$$

You can then substitute for either a or b from the first equation a + b = 7 to get a fairly simple quadtratic equation.

answered Jun 10 '14 at 18:17

benleis

123

I think second equation is correct.The method which You are telling by that also finally you will get same answer it's just you are using different formula for variance i haven't tried out your finding answer by your method if it is different then what i got then please inform me – Freddy Jun 10 '14 at 18:26

Finding out missing observation $a$ and $b$

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