Question

Coal-fired power plants used in the electrical industry have gained public attention because of the environmental problems associated with solid wastes generated by large-scale combustion. A study was conducted to measure the EC50 (Effective Concentration, in mg/L, that decreases 50% of the light in a luminescence bioassay) at four different locations. The experimental data found: Largest value = 3,400 Smallest value = 3,300 Sample mean = 3,350 s = 41.80111641 Find the values of the 2-middle sample observations. (Do not use successive guessing). Ans:

Answer 1

Answer:

$X_2 = 3339, X_3 = 3361$

Step-by-step explanation:

For this case we assume that w ehave the following observations:

$X_1 = 3300, X_2 = ? , X_3 = ?, X_4 = 3400$

The sample mean is given by this formula:

$\bar X = 3350 = \frac{3300 +X_2 + X_3 +3400}{4}$

And we have this after solve for X2+ X3 from the last equation.

$X_2 + X_3 = 13400-3300-3400 = 6700$

And if we solve for $X_2$ we got:

$X_2 = 6700-X_3$    (1)

The sample standard deviation is defined as:

$s^2 = \frac{\sum_{i=1}^4 (X_i -\bar X)^2}{4-1}$

And if we replace the values given we have:

$(41.80111641)^2 = 1747.333=\frac{(3300-3350)^2 +(X_2 -3350)^2 +(X_3 -3350)^2 +(3400-3350)^2}{3}$

And we can simplify this expression like this:

$5241.999 -2500-2500 = (X_2 -3350)^2 +(X_3 -3350)^2$

$241.999= (X_2 -3350)^2 +(X_3 -3350)^2$

Now we replace condition (1) and we got:

$241.999 = (6700-X_3 -3350)^2 +(X_3 -3350)^2$

$241.999 = (3350 -X_3)^2 + (X_3 -3350)^2$

And solving the squares we got:

$241.999 = 11222500 -6700X_3 +X^2_3 + X^2_3 -6700 X_3 [phone]$

And we have the following quadratic equation:

$2X^2_3 -13400 X_3 [phone]=0$

And using the quadratic equation given by:

$x = \frac{-b \pm \sqrt{b^2 -4ac}}{2a}$

Using a = 2, b= -13400, c= 22444758 we got for the solutions:

$X_3 = 3361$ and $X_3= 3339$

We can select for example $X_3 = 3361$

And solving for X2 using the equation (1) we got:

$X_2 = 6700-3361= 3339$

So then the answer for this case is:

$X_2 = 3339, X_3 = 3361$