On the SAT, the average Writing score is 483 points and the average Mathematics

Question

On the SAT, the average Writing score is 483 points and the average Mathematics score is 536 points. The standard deviation is 67 points on each part of the test. Now suppose 64 test takers are selected. (See exercise 21 on page 326 of your textbook for a similar problem.) According to the National Center for Education Statistics, 56% of Texas students are eligible to receive free or reduced-price lunches. Suppose you randomly choose 315 Texas students. The weight of a 5th grader is normally distributed with a mean of 83 pounds and a variance of 78 pounds^2. Let weight, in pounds, be represented by random variable X. The Economic Policy institute reports that the average entry-level wage for male college graduates is $21.85 per hour and for female college graduates, is $19.67 per hour. The standard deviation for male graduates is $3.83 and for female graduates is $2.99. Assume wages are normally distributed. (See exercise 23 on page 326 of your textbook for a similar problem.)

Explanation / Answer

Q1.
Mean ( u ) =483
Standard Deviation ( sd )=67
Normal Distribution = Z= X- u / sd ~ N(0,1)                  
P ( Z < x ) = 0.11
Value of z to the cumulative probability of 0.11 from normal table is -1.227
P( x-u/s.d < x - 483/67 ) = 0.11
That is, ( x - 483/67 ) = -1.23
--> x = -1.23 * 67 + 483 = 400.8226                  
Q2.
Normal Distribution
Proportion ( P ) =0.56
Standard Deviation ( sd )= Sqrt (P*Q /n) = Sqrt(0.56*0.44/315)
Normal Distribution = Z= X- u / sd ~ N(0,1)                  
P(X > 0.61) = (0.61-0.56)/0.028
= 0.05/0.028 = 1.7857
= P ( Z >1.786) From Standard Normal Table
= 0.0371                  

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