Suppose that X 1 ,X 2 ,....,X n are a random sample of size n taken from the nor

Question

Suppose that X1,X2,....,Xn are a random sample of size n taken from the normal distribution N(,2).

1. Find the moment generating function of the sample mean x=(sum i=1 to n of xi) /n

2. What is the distribution of x? (You don't need to give the pdf, just state the name and values of relevant parameters)

3. In the population of all U.S. adults, IQ scores are normally distributed with a mean of 100 and variance of 255. What is the probability that in a random sample of 25 U.S. adults, the mean IQ score is more than 110?

Explanation / Answer

3.

Mean ( u ) =100
Standard Deviation ( sd )= 15.96871/ Sqrt(n) = 3.1937
Number ( n ) = 25
Normal Distribution = Z= X- u / (sd/Sqrt(n) ~ N(0,1)                  
          
P(X > 110) = (110-100)/15.96871/ Sqrt ( 25 )
= 10/3.194= 3.1311
= P ( Z >3.1311) From Standard Normal Table
= 0.0009                  

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