Question 1151047
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{{{mu}}} = mu = greek letter representing the population mean
{{{sigma}}} = sigma = greek letter representing the population standard deviation
n = sample size
*[Tex \LARGE \overline{x}] = xbar = sample mean


In this case,
{{{mu = 23}}}
{{{sigma = 4.3}}}
{{{n = 15}}}
We're given two xbar values which are 20 and 30


Compute the z score for the raw score xbar = 20
{{{z = (xbar - mu)/(sigma/sqrt(n))}}}


{{{z = (20 - 23)/(4.3/sqrt(15))}}}


{{{z = -2.702081}}}


{{{z = -2.70}}}


Repeat for xbar = 30
{{{z = (xbar - mu)/(sigma/sqrt(n))}}}


{{{z = (30 - 23)/(4.3/sqrt(15))}}}


{{{z = 6.304856}}}


{{{z = 6.30}}}


We have
*[Tex \LARGE P\left(20 < \overline{x} < 30\right) \approx P\left(-2.70 < Z < 6.30\right)]


Use a Z table such as this one
<a href = "http://www.z-table.com/">http://www.z-table.com/</a>
to find that,
P(Z < -2.70) = 0.0035
P(Z < 6.30) = 1.00
note: if k is larger than 3.4, then P(Z < k) will be very very close to 1.00; this is especially true of z = 6.30 as its very distant from the center z = 0. So that is how I got P(Z < 6.30) = 1.00


Subtract the values to find the area between the z scores
P(A < Z < B) = P(Z < B) - P(Z < A)
P(-2.70 < Z < 6.30) = P(Z < 6.30) - P(Z < -2.70)
P(-2.70 < Z < 6.30) = 1 - 0.0035
P(-2.70 < Z < 6.30) = 0.9965


<font color=red size=4>Answer: 0.9965</font>
This answer is approximate.


side note:
You can use a calculator like this one
<a href = "http://davidmlane.com/hyperstat/z_table.html">http://davidmlane.com/hyperstat/z_table.html</a>
to compute the area under the Z curve. Leave the mean and standard deviation as 0 and 1 respectively. Click the "between" radio button and type in the values -2.70 and 6.30 into the boxes. Then hit "recalculate" to have the answer come up. This can also be done on TI83 and TI84 calculators as well using the normalcdf function.
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