Consider the statistics of two sets of observations as follows: Size, Mean, Variance for t…

Question

Consider the statistics of two sets of observations as follows: Size, Mean, Variance for two groups (group I: $n_1=10$, mean 2, variance 2; group II: $n_2=n$, mean 3, variance 1). If the variance of the combined set of these two observations is $\dfrac{17}{9}$, then the value of n is equal to

Answer: 5

Accepted Answer

Group I ($n_1=10$): mean$=2 \implies \sum x_i=20$; variance$=2 \implies \sum x_i^2=60$

Group II ($n_2=n$): mean$=3 \implies \sum y_i=3n$; variance$=1 \implies \sum y_i^2=10n$

$$\sigma_{combined}^2 = \frac{60+10n}{n+10}-\left(\frac{20+3n}{n+10}\right)^2 = \frac{17}{9}$$

$$\frac{17}{9} = \frac{n^2+40n+200}{n^2+20n+100}$$

$$9n^2+360n+1800 = 9n^2... \implies n^2+340n+1700=9n^2+360n+1800$$

Simplifying: $8n^2-20n-100... $ wait, correctly: $2n^2-5n-25=0 \implies (2n+5)(n-5)=0 \implies n=5$

Solutions · 1