Question 1127845
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There are a lot of problems here. I'll go over problem 1 to help get you started. If you still have trouble with the others, then please let me know. Thank you.


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Problem 1

*[Tex \LARGE \sigma] is the greek letter sigma for the population standard deviation. In this case, *[Tex \LARGE \sigma = 5]


n = 25 is the sample size


Use those two values to find the standard error.
*[Tex \LARGE SE = \text{Standard Error}]
*[Tex \LARGE SE = \frac{\sigma}{\sqrt{n}}]
*[Tex \LARGE SE = \frac{5}{\sqrt{25}}]
*[Tex \LARGE SE = \frac{5}{5}]
*[Tex \LARGE SE = 1]

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*[Tex \LARGE \mu] is the greek letter mu for the population mean. It looks like a letter U but it's not. In this case, *[Tex \LARGE \mu = 20]


xbar or *[Tex \LARGE \overline{x}] is the sample mean. In the case of problem 1, it would be xbar = 21.


Use the xbar, mu and SE values to find the z score
*[Tex \LARGE z = \frac{\overline{x} - \mu}{SE}]
*[Tex \LARGE z = \frac{21 - 20}{1}]
*[Tex \LARGE z = 1]



Now use a table such as <a href = "http://users.stat.ufl.edu/~athienit/Tables/Ztable.pdf">this one</a> to find that *[Tex \LARGE P(Z < 1) = 0.8413]; in other words, the area to the left of z = 1 is about 0.8413


note: look on page 2 for the row that starts with 1.0, then look at the column that has 0.00 at the top. This row and column intersect with the value 0.8413 inside


So this means,
*[Tex \LARGE P(Z < 1) = 0.8413]
*[Tex \LARGE P(Z > 1) = 1-P(Z < 1) = 1-0.8413 = 0.1587]
*[Tex \LARGE P(Z > 1) = 0.1587]
This is approximate to four decimal places


Therefore,
*[Tex \LARGE P(\overline{x} > 21) = 0.1587]
which is also approximate


If we use a calculator such as <a href = "http://onlinestatbook.com/2/calculators/normal_dist.html">this one</a> then we will find the answer is the same as found with the table
<img src = "https://i.imgur.com/l7PCWGX.png">


note: I typed in 
Mean = 0
SD = 1
then I clicked the "above" radio button and typed "1" into the box. 
After all that, I hit the "recalculate" button to produce the answer 0.1587
The image above shows the standard normal curve with the proper area shaded. That shaded area represents everything under the curve to the right of z = 1.



The final answer to problem 1 is approximately <b><font size=4 color=red>0.1587</font></b>
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