Suppose the weights of college textbooks are normally distributed with a populat

Question

Suppose the weights of college textbooks are normally distributed with a population mean of 4 pounds and a population standard deviation of 1.2 pounds. Suppose randomly sample and weigh 16 textbooks. Answer the following questions:

What is the probability that the sample average of the 16 textbooks is less than 3.5 pounds?

Imagine the distribution of all possible samples averages of textbook weights (with n=16). What is the weight that separates the lightest 25% of sample averages from the heaviest 75% of sample averages?

Explanation / Answer

Mean ( u ) =4
Standard Deviation ( sd )=1.2
Number ( n ) = 16
Normal Distribution = Z= X- u / (sd/Sqrt(n) ~ N(0,1)                  
a.
P(X < 3.5) = (3.5-4)/1.2/ Sqrt ( 16 )
= -0.5/0.3= -1.6667
= P ( Z <-1.6667) From Standard NOrmal Table
= 0.0478                  

b.
P ( Z < x ) = 0.25
Value of z to the cumulative probability of 0.25 from normal table is -0.674
P( x-u/s.d < x - 4/0.3 ) = 0.25
That is, ( x - 4/0.3 ) = -0.67
--> x = -0.67 * 0.3 + 4 = 3.7978                  

P ( Z < x ) = 0.75
Value of z to the cumulative probability of 0.75 from normal table is 0.674
P( x-u/s.d < x - 4/0.3 ) = 0.75
That is, ( x - 4/0.3 ) = 0.67
--> x = 0.67 * 0.3 + 4 = 4.2022                  
The weight is: 4.2022 - 3.7978= 0.4044

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