Answer to Question #12597 in Statistics and Probability for Kolton
Carry your intermediate computations to at least four decimal places. Round your answer to at least three decimal places.
You can translate into standard normal units by:
Z = ( X - μ ) / σ
Where Z ~ Normal( μ = 0, σ = 1). You can then use the standard normal cdf tables to get probabilities.
If you are looking at the mean of a sample, then remember that for any sample with a large enough sample size the mean will be normally distributed. This is called the Central Limit Theorem.
If a sample of size is is drawn from a population with mean μ and standard deviation σ then the sample average xBar is normally distributed
with mean μ and standard deviation σ /√(n)
In this question we have
Xbar ~ Normal( μ = 32 , σ² = 56.25 / 36 )
Xbar ~ Normal( μ = 32 , σ² = 1.5625 )
Xbar ~ Normal( μ = 32 , σ = 7.5 / sqrt( 36 ) )
Xbar ~ Normal( μ = 32 , σ = 1.25 )
Find P( Xbar < 29.7 )
P( ( Xbar - μ ) / σ < ( 29.7 - 32 ) / 1.25 ) = P( Z < -1.84 ) = 0.03288412.
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