The _______ of any statistical test represents the probability that the results were obtained by chance alone.

1 answer:

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ρ-Value, Probability Value, or Observed Level of Significance
is the definition of any statistical test represents the probability that the results were obtained by chance alone.

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Find the distance from the origin to the graph of 7x+9y+11=0

Cerrena [4.2K]

One way to do it is with calculus. The distance between any point $(x,y)=\left(x,-\dfrac{7x+11}9\right)$ on the line to the origin is given by

$d(x)=\sqrt{x^2+\left(-\dfrac{7x+11}9\right)^2}=\dfrac{\sqrt{130x^2+154x+121}}9$

Now, both $d(x)$ and $d(x)^2$ attain their respective extrema at the same critical points, so we can work with the latter and apply the derivative test to that.

$d(x)^2=\dfrac{130x^2+154x+121}{81}\implies\dfrac{\mathrm dd(x)^2}{\mathrm dx}=\dfrac{260}{81}x+\dfrac{154}{81}$

Solving for $(d(x)^2)'=0$ , you find a critical point of $x=-\dfrac{77}{130}$ .

Next, check the concavity of the squared distance to verify that a minimum occurs at this value. If the second derivative is positive, then the critical point is the site of a minimum.

You have

$\dfrac{\mathrm d^2d(x)^2}{\mathrm dx^2}=\dfrac{260}{81}>0$

so indeed, a minimum occurs at $x=-\dfrac{77}{130}$ .

The minimum distance is then

$d\left(-\dfrac{77}{130}\right)=\dfrac{11}{\sqrt{130}}$