The scores of 12thgrade students on the National Assessment

The scores of 12th-grade students on the National Assessment of Educational Progress year 2000 mathematics test have a distribution that is approximately Normal with mean = 273 and standard deviation = 38 .

Choose one 12th-grader at random. What is the probability (±0.1) that his or her score is higher than 273 ?   Higher than 387 (±0.001)?  

Now choose an SRS of 16 twelfth-graders and calculate their mean score x. If you did this many times, what would be the mean of all the x-values?  

What would be the standard deviation (±0.1) of all the x-values?  

What is the probability that the mean score for your SRS is higher
than 273 ? (±0.1)   Higher than 387 ? (±0.0001)  

Solution

Mean ( u ) =273
Standard Deviation ( sd )=38
Normal Distribution = Z= X- u / sd ~ N(0,1)                  
a)
P(X > 273) = (273-273)/38
= 0/38 = 0
= P ( Z >0) From Standard Normal Table
= 0.5                  
b)
P(X > 387) = (387-273)/38
= 114/38 = 3
= P ( Z >3) From Standard Normal Table
= 0.0013                  

WHEN n=16;
Mean ( u ) =273
Standard Deviation ( sd )=38/Sqrt(16) = 9.5
P(X > 273) = (273-273)/38/ Sqrt ( 16 )
= 0/9.5= 0
= P ( Z >0) From Standard Normal Table
= 0.5                  
P(X > 387) = (387-273)/38/ Sqrt ( 16 )
= 114/9.5= 12
= P ( Z >12) From Standard Normal Table
= 0