SOLUTION: Consider a normal population with µ = 25 and σ = 7.0. (A) Calculate the standard score for a value x of 27. (B) Calculate the standard score for a randomly selected

Algebra ->  Probability-and-statistics -> SOLUTION: Consider a normal population with µ = 25 and σ = 7.0. (A) Calculate the standard score for a value x of 27. (B) Calculate the standard score for a randomly selected       Log On


   

Question 701436: Consider a normal population with µ = 25 and σ = 7.0.
(A) Calculate the standard score for a value x of 27.
(B) Calculate the standard score for a randomly selected sample of 45 with xbar = 27.
(C) Explain why the standard scores of 27 are different between A and B above
please walk me through this
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
Consider a normal population with µ = 25 and σ = 7.0.
(A) Calculate the standard score for a value x of 27.
z(27) = (27-25)/0.7 = 2.857
-------
(B) Calculate the standard score for a randomly selected sample of 45 with xbar = 27.
z(27) = (27-25)/(0.7/sqrt(45)) = 2/(0.7/sqrt(45)) = 19.166
--------
(C) Explain why the standard scores of 27 are different between A and B above
The Central Limit Theorem says:
1. The mean of the sample means = the mean of the population
2. The std of the sample means = (the std of the population)/sqrt(sample size)
-------
So the population std is much smaller that the std of the population.
As a result the z-score (which is a count of std's) is greater for
sample means than it is for an equal sample element.
============================
Cheers,
Stan H.