SOLUTION: Assume the SAT Mathematics Level 2 test scores are normally distributed with a mean of 500 and a standard deviation of 100. Show all work. Just the answer, without supporting work,

Question 1111199: Assume the SAT Mathematics Level 2 test scores are normally distributed with a mean of 500 and a standard deviation of 100. Show all work. Just the answer, without supporting work, will receive no credit.
(a) If a random sample of 64 test scores is selected, what is the standard deviation of the sample mean?
(b) What is the probability that 64 randomly selected test scores will have a mean test score that is between 475 and 525?

x -2 0 1 2 3
P(x) 0.1 0.1 0.4 0.2 0.2
Answer by rothauserc(4718) (Show Source): You can put this solution on YOUR website!
The population mean is 500 and the population standard deviation is 100
:
(a) the sample size is 64
standard deviation of the sample mean is population standard deviation / the square root of the sample size
:
standard deviation of the sample mean = 100/square root(64) = 100/8 = 12.5
:
(b) let X be the sample mean variable
Probability (P) ( 475 < X < 525 ) = P ( X < 525 ) - P ( X < 475 )
:
calculate z-scores for each probability
:
z-score (525) = (525 - 500) / 12.5 = 2
z-score (475) = (475 - 500) / 12.5 = -2
:
lookup the P for each of the z-scores in the table of z-values
:
Probability (P) ( 475 < X < 525 ) = 0.9772 - 0.0228 = 0.9544
:

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