Question 1190385
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Part (a)


mu = 70 = population mean
sigma = 10 = population standard deviation


Convert x = 80 to its corresponding z score
z = (x-mu)/sigma
z = (80-70)/10
z = 10/10
z = 1


The question of asking P(X > 80) is identical to P(Z > 1) when we have the parameters of mu = 70 and sigma = 10.


Use a Z table in the back of your textbook to find that 
P(Z < 1) = 0.84134


Here's a free Z table if you don't have your stats textbook with you
https://www.ztable.net/


From that we could say:
P(Z > 1) = 1 - P(Z < 1)
P(Z > 1) = 1 - 0.84134
P(Z > 1) = 0.15866


Answer: Approximately 0.15866


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Part (b)


You'll follow the same steps as the previous part.
x = 90 converts to z = 2. I'm skipping steps a bit.
P(X > 90) is equivalent to P(Z > 2)


Use the table to find that
P(Z < 2) = 0.97725
So,
P(Z > 2) = 1 - P(Z < 2)
P(Z > 2) = 1 - 0.97725
P(Z > 2) = 0.02275


Answer: Approximately 0.02275


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Part (c)


x = 50 converts to z = -2
x = 90 converts to z = 2


Computing P(50 < X < 90) is equivalent to P(-2 < Z < 2)


The table shows that
P(Z < -2) = 0.02275
P(Z < 2) = 0.97725


This leads to:
P(a < Z < b) = P(Z < b) - P(Z < a)
P(-2 < Z < 2) = P(Z < 2) - P(Z < -2)
P(-2 < Z < 2) = 0.97725 - 0.02275
P(-2 < Z < 2) = 0.9545
This fits with the Empirical Rule which says roughly 95% of the normal distribution is within 2 standard deviations of the mean.


Answer: Approximately 0.9545


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Part (d)


x = 100 converts to z = 3
P(Z < 3) = 0.99865
P(Z > 3) = 1 - P(Z < 3)
P(Z > 3) = 1 - 0.99865
P(Z > 3) = 0.00135


Answer: Approximately 0.00135


You can use a normalCDF calculator to compute the more accurate versions of each approximate answer above.
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