Question 878456
For this entire problem, I'm using <a href="http://www.math.upenn.edu/~chhays/zscoretable.pdf">this table</a>

a)


Compute the z-score when the raw score is x = 20580


z = (x - mu)/sigma
z = (20580 - 21900)/500
z = -2.64


Use the table (mentioned above) to find that the area to the left of z = -2.64 is 0.0041



Compute the z-score when the raw score is x = 21405


z = (x - mu)/sigma
z = (21405 - 21900)/500
z = -0.99


Use the table to find that the area to the left of z = -0.99 is 0.1611


Subtract the two areas: 0.1611 - 0.0041 = 0.1570


The area between z = -2.64 and z = -0.99 is 0.1570
The area between x = 20580 and x = 21405 is 0.1570
The two areas are the same because we applied a z-score transform.


That means the probability of being between $20,580.00 and $21,405.00 is <font color="red">0.1570</font>


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b)


Compute the z-score when the raw score is x = 22615


z = (x - mu)/sigma
z = (22615 - 21900)/500
z = 1.43


Use the table to find that the area to the left of z = 1.43 is 0.9236



Compute the z-score when the raw score is x = 23085


z = (x - mu)/sigma
z = (23085 - 21900)/500
z = 2.37


Use the table to find that the area to the left of z = 2.37 is 0.9911
Subtract that from 1 to get 1-0.9911 = 0.0089
The area to the right of z = 2.37 is 0.0089


Add the two areas: 0.9236+0.0089 = 0.9325


Final Answer: <font color="red">0.9325</font>


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c)


Compute the z-score when the raw score is x = 22455


z = (x - mu)/sigma
z = (22455 - 21900)/500
z = 1.11


Use the table to find that the area to the left of z = 1.11 is 0.8665


Final Answer: <font color="red">0.8665</font>


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d)


Compute the z-score when the raw score is x = 20525


z = (x - mu)/sigma
z = (20525 - 21900)/500
z = -2.75


Use the table (mentioned above) to find that the area to the left of z = -2.75 is 0.0030



Compute the z-score when the raw score is x = 22785


z = (x - mu)/sigma
z = (22785 - 21900)/500
z = 1.77


Use the table to find that the area to the left of z = 1.77 is 0.9616


Subtract the two areas: 0.9616 - 0.0030 = 0.9586


Final Answer: <font color="red">0.9586</font>


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e)

 
Compute the z-score when the raw score is x = 22825


z = (x - mu)/sigma
z = (22825 - 21900)/500
z = 1.85


Use the table (mentioned above) to find that the area to the left of z = 1.85 is 0.9678



Compute the z-score when the raw score is x = 23335


z = (x - mu)/sigma
z = (23335 - 21900)/500
z = 2.87


Use the table to find that the area to the left of z = 2.87 is 0.9979


Subtract the two areas: 0.9979 - 0.9678 = 0.0301


Final Answer: <font color="red">0.0301</font>