# a set of 50 data values has a mean of 18 & variance of 4 find the probability of a data value <20 find the standard score z for data value = 20 please explain and show work

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We are given a set with n=50, `bar(x)=18` , and `sigma=4` .

Find the probability that a data value is less than 20:

Convert 20 to a `z` score:

`z=(20-18)/4=0.5`

Then `P(x<20)=P(z<0.50)=.6915` or 69.15% will be less than 20

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Since you were given n, I wonder if your actual question is about a sample mean, not an actual data value. If that is the case, the question is what is the probability of obtaining a sample mean less than 20 if the sample size is 50:

Convert 20 to a z-score; note that since this is from a sample we use `z=(x-mu)/(sigma/sqrt(n))` :

`z=(20-18)/(4/sqrt(50))=3.54`

Then `P(x<20)=P(z<3.54)` From a standard normal table we find `P(z<.54)=.9998` (Most tables stop at z=3.5 -- this value is from a graphing calculator.)

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