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Assume that a random variable is normally distributed with a mean of 1,500 and a variance of 324. a. What is the probability that a randomly selected value will be greater than 1,550? b. What is the...

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Assume that a random variable is normally distributed with a mean of 1,500 and a variance of 324.

a. What is the probability that a randomly selected value will be greater than 1,550?

b. What is the probability that a randomly selected value will be less than 1,485?

c. What is the probability that a randomly selected value will be either less than 1,475 or greater than 1,535?

 

Answered Same Day Dec 27, 2021

Solution

Robert answered on Dec 27 2021
108 Votes
A​ ​random​ ​variable​ ​is​ ​normally​ ​distributed​ ​with​ ​a​ ​mean​ ​1500​ ​and​ ​a​ ​variance​ ​324.
(a)
We​ ​need​ ​to​ ​calculate​ ​the​ ​probability​ ​that​ ​a​ ​value​ ​randomly​ ​selected​ ​will​ ​be​ ​greater​ ​than​ ​1550.
Let​ ​us​ ​say​ ​x​ ​is​ ​a​ ​value​ ​randomly​ ​selected​ ​from​ ​the​ ​population.
Then,​ ​we​ ​need​ ​to​ ​calculate​ ​the​ ​probability.​ ​P(x>1550)
First,​ ​we​ ​need​ ​to​ ​convert​ ​the​ ​random​ ​variable​ ​x​ ​into​ ​co
esponding​ ​standardized​ ​z-value​ ​by
using
​ ​
where
​ ​
Since​ ​we​ ​have​ ​a​ ​variance​ ​as​ ​ ​ ​,​ ​the​ ​standard​ ​deviation​ ​is
​ ​
On​ ​substituting​ ​x=1550,​ ​ ​ ​​ ​in​ ​​ ​ ,​ ​we​ ​get


Thus,​ ​the​ ​co
esponding​ ​z-value​ ​for​ ​x=1550,​ ​z=2.78​ ​.​ ​And​ ​P(x>1550),​ ​​ ​can​ ​be​ ​written​ ​as
P(z>2.78).
Now,​ ​we​ ​need​ ​to​ ​find​ ​the​ ​probability​ ​associated​ ​with​ ​​ ​z=2.78​ ​​ ​by​ ​using​ ​the​ ​standard​ ​normal...
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