What is the means-to-MAD ratio of the two data sets, expressed as a decimal?

1 answer:

8 0

The correct answer for the question that is being presented above is this one:

To find the mean of a set, add up all of the numbers and divide by how many numbers you have. For the first set:
u = (10.7+2.4)/2 = 6.55
For the second set:
u = (13.7 + 2.5)/2 = 8.1

To find the mean absolute deviation, (MAD) find the mean of the distances of each number from the mean.
First set:
MAD = (|10.7-6.55| + |2.4-6.55|)/2 = 4.15
Second set:
MAD = (|13.7-8.1| + |2.5-8.1|)/2 = 5.6

The means-to-MAD ratio is simply that, divide one by the other and that's it.
For the first set:
6.55 / 4.15 = 1.57831325
For the second set:
8.1 / 5.6 = 1.44642857

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Find a linear approximation of the function f(x)=\sqrt[3]{1+x} at a=0, and use it to approximate the numbers \sqrt[3]{.96} and \

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$f(x)=\sqrt[3]{1+x}=(1+x)^{1/3}\implies f'(x)=\dfrac1{3(1+x)^{2/3}}$

The linear approximation to $f(x)$ around $x=a$ is then

$L(x)=f(a)+f'(a)(x-a)\approx f(x)$

So the approximation centered at $a=0$ will be

$L(x)=f(0)+f'(0)x=(1+0)^{1/3}+\dfrac x{3(1+0)^{2/3}}$
$L(x)=1+\dfrac x3$

which means we have

$\sqrt[3]{0.96}\approx L(0.96)=1+\dfrac{0.96}3=1.32$

$\sqrt[3]{1.02}\approx L(1.02)=1+\dfrac{1.02}3=1.34$

Compare to the actual values of

$\sqrt[3]{0.96}\approx0.9864$

$\sqrt[3]{1.02}\approx1.0066$