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4 years ago

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An amount of 46,000 is borrowed for 9 years at 6.25% interest, compounded annually. If the loan is paid in full at the end of th

at period, how much must be paid back?
Use the calculator provided and round your answer to the nearest dollar.

1 answer:

o-na [289]4 years ago

5 0

Answer:

You must pay $79,381.3

Step-by-step explanation:

This is a problem of compound interest.

The formula used to solve these problems is:

$P = p_0(1 + r) ^ t$

Where P is the amount that must be paid at the end of t years,

$p_0$ is the initial amount

r is the compound interest rate

If the initial amount was 46,000

The compound interest rate is 0.0625 and the loan is for 9 years, then at the end of the 9 years, the amount that must be paid is:

$P = 46,000(1 + 0.0625) ^ 9$

We solve for P and obtain:

$P = 79,381.3$

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See below

Step-by-step explanation:

By using the table 1 attached (See Table 1 attached)

We can perform all the calculations to express both, y as a function of x or x as a function of y.

Let's make first the line relating y as a function of x.

<u>y as a function of x </u>

<em>(y=response variable, x=explanatory variable) </em>

$\bf y=m_{yx}x+b_{yx}$

where

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$\bf b_{yx}$ is the y-intercept

In this case we use these formulas:

$\bf m_{yx}=\frac{(\sum y)(\sum x)^2-(\sum x)(\sum xy)}{n\sum x^2-(\sum x)^2}$

$\bf b_{yx}=\frac{n\sum xy-(\sum x)(\sum y)}{n(\sum x^2)-(\sum x)^2}$

n = 10 is the number of observations taken (pairs x,y)

<u>Note:</u> <em>Be careful not to confuse </em>

$\bf \sum x^2$ with $\bf (\sum x)^2$

Performing our calculations we get:

$\bf m_{yx}=\frac{(83)(95)^2-(95)(923)}{10*1277-(95)^2}=176.6061$

$\bf b_{yx}=\frac{10*923-(95)(83)}{10(1277)-(95)^2}=0.3591$

So the equation of the line that relates y as a function of x is

<h3>y = 176.6061x + 0.3591 </h3>

In order to compute the standard error $\bf S_{yx}$ , we must use Table 2 (See Table 2 attached) and use the definition

$\bf s_{yx}=\sqrt{\frac{(y-y_{est})^2}{n}}$

and we have that standard error when y is a function of x is

$\bf s_{yx}=\sqrt{\frac{39515985}{10}}=1987.8628$

Now, to find the line that relates x as a function of y, we simply switch the roles of x and y in the formulas.

So now we have:

x as a function of y

(x=response variable, y=explanatory variable)

$\bf x=m_{xy}y+b_{xy}$

where

$\bf m_{xy}$ is the slope of the line

$\bf b_{xy}$ is the x-intercept

In this case we use these formulas:

$\bf m_{xy}=\frac{(\sum x)(\sum y)^2-(\sum y)(\sum xy)}{n\sum y^2-(\sum y)^2}$

$\bf b_{xy}=\frac{n\sum xy-(\sum x)(\sum y)}{n(\sum y^2)-(\sum y)^2}$

n = 10 is the number of observations taken (pairs x,y)

<u>Note:</u> <em>Be careful not to confuse </em>

$\bf \sum y^2$ with $\bf (\sum y)^2$

Remark:<em> </em><em>If you wanted to draw this line in the classical style (the independent variable on the horizontal axis), you would have to swap the axis X and Y) </em>

Computing our values, we get

$\bf m_{xy}=\frac{(95)(83)^2-(83)(923)}{10*743-(83)^2}=1068.1072$

$\bf b_{xy}=\frac{10*923-(95)(83)}{10(743)-(83)^2}=2.4861$

and the line that relates x as a function of y is

<h3>x = 1068.1072y + 2.4861 </h3>

To find the standard error $\bf S_{xy}$ we use Table 3 (See Table 3 attached) and the formula

$\bf s_{xy}=\sqrt{\frac{(x-x_{est})^2}{n}}$

and we have that standard error when y is a function of x is

$\bf s_{xy}=\sqrt{\frac{846507757}{10}}=9200.5856$

<em>In both cases the correlation coefficient r is the same and it can be computed with the formula: </em>

$\bf r=\frac{\sum xy}{\sqrt{(\sum x^2)(\sum y^2)}}$

Remark: <em>This formula for r is only true if we assume the correlation is linear. The formula does not hold for other kind of correlations like parabolic, exponential,..., etc. </em>

Computing the correlation coefficient :

$\bf r=\frac{923}{\sqrt{(1277)(743)}}=0.9478$

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