The population of IQ scores forms a normal distribution with a mean of mu = 100

Question

The population of IQ scores forms a normal distribution with a mean of mu = 100 and a sigma = 15. For a random sample of n = 36 people, the probability of obtaining a sample mean greater than M = 105 is ___. (Make sure that you take your answer to four decimal places.) A population of scores forms a normal distribution with a mean of mu = 80 and a sigma = 10. For a sample of n = 4, the proportion of scores that will have values between 75 and 85 is __. (Make sure that you take your answer to four decimal places.) A population of scores forms a normal distribution with a mean of mu = 80 and a sigma = 10. For a sample of n = 16, the proportion of scores that will have values between 75 and 85 is __. (Make sure that you take your answer to four decimal places.)

Explanation / Answer

Q30.
Mean ( u ) =100
Standard Deviation ( sd )= 15/ Sqrt(n) = 2.5
Number ( n ) = 36
Normal Distribution = Z= X- u / (sd/Sqrt(n) ~ N(0,1)                  
P(X > 105) = (105-100)/15/ Sqrt ( 36 )
= 5/2.5= 2
= P ( Z >2) From Standard Normal Table
= 0.0228                  
Q31.
Mean ( u ) =80
Standard Deviation ( sd )= 10/ Sqrt(n) = 5
Number ( n ) = 4
Normal Distribution = Z= X- u / (sd/Sqrt(n) ~ N(0,1)                  
To find P(a <= Z <=b) = F(b) - F(a)
P(X < 75) = (75-80)/10/ Sqrt ( 4 )
= -5/5
= -1
= P ( Z <-1) From Standard Normal Table
= 0.15866
P(X < 85) = (85-80)/10/ Sqrt ( 4 )
= 5/5 = 1
= P ( Z <1) From Standard Normal Table
= 0.84134
P(75 < X < 85) = 0.84134-0.15866 = 0.6827                  

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