SAT scores are normally distributed with mean 1000 and stand

SAT scores are normally distributed with mean 1000 and standard deviation 115.

a) For an individual chosen at random from all SAT takers, what is the probability that their score is 924 or less? Give your answer to three decimal places.

Probability =

b) For a simple random sample of three individuals, what is the probability that their average SAT score is 924 or less?

Probability =

c) For a simple random sample of three individuals, what is the probability that all three have scores below 924?

Probability =

Solution

Normal Distribution
Mean ( u ) =1000
Standard Deviation ( sd )=115
Normal Distribution = Z= X- u / sd ~ N(0,1)                  
a)
P(X > 924) = (924-1000)/115
= -76/115 = -0.6609
= P ( Z >-0.661) From Standard Normal Table
= 0.7457                  
P(X < = 924) = (1 - P(X > 924)
= 1 - 0.7457 = 0.2543                  

b)
P(X > 924) = (924-1000)/115/ Sqrt ( 3 )
= -76/66.395= -1.1447
= P ( Z >-1.1447) From Standard Normal Table
= 0.8738                  
P(X < = 924) = (1 - P(X > 924)
= 1 - 0.8738 = 0.1262