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3 years ago

A bank's loan officer rates applicants for credit.

Mathematics

1 answer:

Artist 52 [7]3 years ago

5 0

Answer:

$a=200 +0.253*50=212.65$

So the value of height that separates the bottom 60% of data from the top 40% is 212.65. And rounded would be 212.7.

Step-by-step explanation:

1) Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".

2) Solution to the problem

Let X the random variable that represent the loan officer rates applicants for a credit of a population, and for this case we know the distribution for X is given by:

$X \sim N(200,50)$

Where $\mu=200$ and $\sigma=50$

And the best way to solve this problem is using the normal standard distribution and the z score given by:

$z=\frac{x-\mu}{\sigma}$

For this part we want to find a value a, such that we satisfy this condition:

$P(X>a)=0.40$ (a)

$P(X$ (b)

Both conditions are equivalent on this case.

As we can see on the figure attached the z value that satisfy the condition with 0.60 of the area on the left and 0.40 of the area on the right it's z=0.253. On this case P(Z<0.253)=0.60 and P(z>0.253)=0.4

If we use condition (b) from previous we have this:

$P(X$

$P(Z$

But we know which value of z satisfy the previous equation so then we can do this:

$z=0.253$

And if we solve for a we got

$a=200 +0.253*50=212.65$

So the value of height that separates the bottom 60% of data from the top 40% is 212.65. And rounded would be 212.7.

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Marine biologists have determined that when a shark detectsthe presence of blood in the water, it will swim in the directionin w

siniylev [52]

Solution :

a). The level curves of the function :

$C(x,y) = e^{-(x^2+2y^2)/10^4}$

are actually the curves

$e^{-(x^2+2y^2)/10^4}=k$

where k is a positive constant.

The equation is equivalent to

$x^2+2y^2=K$

$\Rightarrow \frac{x^2}{(\sqrt K)^2}+\frac{y^2}{(\sqrt {K/2})^2}=1, \text{ where}\ K = -10^4 \ln k$

which is a family of ellipses.

We sketch the level curves for K =1,2,3 and 4.

If the shark always swim in the direction of maximum increase of blood concentration, its direction at any point would coincide with the gradient vector.

Then we know the shark's path is perpendicular to the level curves it intersects.

b). We have :

$\triangledown C= \frac{\partial C}{\partial x}i+\frac{\partial C}{\partial y}j$

$\Rightarrow \triangledown C =-\frac{2}{10^4}e^{-(x^2+2y^2)/10^4}(xi+2yj),$ and

$\triangledown C$ points in the direction of most rapid increase in concentration, which means $\triangledown C$ is tangent to the most rapid increase curve.