The population of SAT scores forms a normal distribution with a mean of mu = 500

Question

The population of SAT scores forms a normal distribution with a mean of mu = 500 and a standard deviation of sigma = 100. If the mean SAT score is calculated for a sample of n = 25 students: what is the probability that the sample mean will be greater than M = 510? In symbols, what is p (m > 510)? What is the probability that the sample mean will be greater than M = 420? In symbols, what is p (m > 420)? What is the probability that the sample mean will be less than or equal to M = 420? In symbols, what is p (M

Explanation / Answer

Mean ( u ) =500
Standard Deviation ( sd )= 100/ Sqrt(n) = 20
Number ( n ) = 25
Normal Distribution = Z= X- u / (sd/Sqrt(n) ~ N(0,1)                  
a.          
P(X > 510) = (510-500)/100
= 10/100 = 0.1
= P ( Z >0.1) From Standard Normal Table
= 0.4602                  
b.
P(X > 420) = (420-500)/100
= -80/100 = -0.8
= P ( Z >-0.8) From Standard Normal Table
= 0.7881                  
c.
P(X < = 420) = (1 - P(X > 420)
= 1 - 0.7881 = 0.2119                  
d.
P(X > 420) = (420-500)/100/ Sqrt ( 25 )
= -80/20= -4
= P ( Z >-4) From Standard Normal Table
= 1                  
P(X < = 420) = (1 - P(X > 420)
= 1 - 1 = 0                  
Yes, it gets the probability to lower

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