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azamat
3 years ago
13

In a normally distributed data set a mean of 55 where 99.7% of the data fall between 47.5 and 62.5, what would be the standard d

eviation of that data set? Devry course
Mathematics
1 answer:
NeX [460]3 years ago
3 0

Given that normally distributed data set has a mean of 55 and 99.7% of data fall between 47.5 and 62.5.

Let s be the standard deviation of data set.

Since 99.7% data fall within 3 standard deviations of mean, z-value for 47.5 and 62.5 has an absolute value of 3.

That is |z|=3

But z= \frac{x-mean}{standard deviation}

Let us plugin x=47.5 and mean =55 and equate it to 3.

That is |\frac{47.5-55}{s}|  = 3

             |\frac{-7.5}{s} | =3

Since x is always positive ( being standard deviation), |\frac{-7.5}{s} | = \frac{|-7.5|}{s} = \frac{7.5}{s}

Hence  \frac{7.5}{s}= 3

             s=\frac{7.5}{3}  = 2.5

We will get same value with 62.5 as well.

Hence standard deviation of data set is 2.5.

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Step-by-step explanation:

You can use this formula which relates the number of iterations, n, required by the bisection method to converge to within an absolute error tolerance of ε starting from the initial interval (a, b).

n\geq \frac{log(\frac{b-a}{\epsilon} )}{log(2)}

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n\geq \frac{log(\frac{1+2}{10^{-8}} )}{log(2)}\\n \geq 29

Thus, n = 29 iterations would be enough to obtain a root of f(x)=2x^2-3x-2 that is at most 10^{-8} away from the correct solution.

<u>You can prove this result by doing the computation as follows:</u>

From the information given we know:

  • f(x)=2x^2-3x-2
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This is the algorithm for the Bisection method:

  1. Find two numbers <em>a</em> and <em>b</em> at which <em>f</em> has different signs.
  2. Define c=\frac{a+b}{2}
  3. If b-c\leq \epsilon then accept c as the root and stop
  4. If f(a)f(c)\leq 0 then set <em>c </em>as the new<em> b</em>. Otherwise, set <em>c </em>as the new <em>a</em>. Return to step 1.

We know that f(-2)=2(-2)^2-3(-2)-2=12 and f(1)=2(1)^2-3(1)-2=-3 so we take a=-2 and b=1 then c=\frac{-2+1}{2} =-0.5

Because 1-(-0.5)\geq 10^{-8} we set c=-0.5 as the new <em>b.</em>

The bisection algorithm is detailed in the following table.

After the 29 steps we have that 6\cdot 10^{-9}\leq 10^{-8} hence the required root approximation is c = -0.50

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