Let the mean and the variance of seven observations 2,4, ,8, ,12,14,

Question

Let the mean and the variance of seven observations $2,4,\alpha ,8,\beta ,12,14,\alpha <\beta $, be 8 and 16 respectively. Then the quadratic equation whose roots are $3\alpha +2$ and $2\beta +1$ is :

Accepted Answer

To find the required quadratic equation, we must first find the values of $\alpha $ and $\beta $.

Given that there are 7 observations: $2,4,\alpha ,8,\beta ,12,14$. Let $n=7$.

The mean of these observations ($\vec{x}$) is $8$.

According to the formula for the mean:

$\vec{x}=\frac{\sum {x}_{i}}{n}$

Substituting the given values:

$8=\frac{2+4+\alpha +8+\beta +12+14}{7}$

$56=40+\alpha +\beta $

$\alpha +\beta =16\,$

Next, the variance (${\sigma }^{2}$) is given as $16$. The formula for variance is:

${\sigma }^{2}=\frac{\sum {x}_{2}^{i}}{n}-(\vec{x}{)}^{2}$

Substituting the known values:

$16=\frac{{2}^{2}+{4}^{2}+{\alpha }^{2}+{8}^{2}+{\beta }^{2}+{12}^{2}+{14}^{2}}{7}-(8{)}^{2}$

$16=\frac{4+16+{\alpha }^{2}+64+{\beta }^{2}+144+196}{7}-64$

$16+64=\frac{424+{\alpha }^{2}+{\beta }^{2}}{7}$

$80	imes 7=424+{\alpha }^{2}+{\beta }^{2}$

$560=424+{\alpha }^{2}+{\beta }^{2}$

${\alpha }^{2}+{\beta }^{2}=136\,$

Now, using the algebraic identity $(\alpha +\beta {)}^{2}={\alpha }^{2}+{\beta }^{2}+2\alpha \beta $, we can find the product $\alpha \beta $:

$(16{)}^{2}=136+2\alpha \beta $

$256=136+2\alpha \beta $

$2\alpha \beta =120$

$\alpha \beta =60$

The numbers $\alpha $ and $\beta $ are the roots of the quadratic equation ${t}^{2}-(\alpha +\beta )t+\alpha \beta =0$:

${t}^{2}-16t+60=0$

${t}^{2}-10t-6t+60=0$

$t(t-10)-6(t-10)=0$

$(t-6)(t-10)=0$

Thus, the values are $6$ and $10$. Since we are given that $\alpha $\alpha =6$

$\beta =10$

We are asked to find the quadratic equation whose roots are $3\alpha +2$ and $2\beta +1$. Let's evaluate these roots:

$=3(6)+2=18+2=20$

$=2(10)+1=20+1=21$

Now, let's find the sum and the product of the roots for our new quadratic equation:

$=20+21=41$

$=20	imes 21=420$

The standard form of a quadratic equation is ${x}\^{2}-()x+()=0$.

Substituting the values:

$

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