Help please if you could show how you got the answers that w

Help please, if you could show how you got the answers that would be awesome

ACT and SAT scores are both known to be normally distributed. In 2010, The mean and standard deviation for The ACT were mu=21 and MU=7, respectively. The mean and standard deviation for The SAT were mu=1510 and a=310, respectively. a. What ACT score would place a student in The same percentile as a student who scored 2000 on The SAT in 2014? Round your answer to The nearest integer. What SAT score would place a student in The sample percentile as a student who scored 16 on The ACT in 2014? Round your answer to The nearest integer. Suppose Bertrand scored 32 on The ACT in 2014. Which of The following statements below are correct ways of describing his score? Select all that apply. Q Bertrand\'s score was about 1.57 standard deviations away from The mean score for everyone who took The ACT in 2014. Q About 94% of all students who took The ACT in 2014 scored lower than Bertrand. The probability that one randomly selected student who took The ACT in 2014 scored higher than Bertrand is about 0.0600000000000001. Q About 94% of all students who took The ACT in 2014 scored higher than Bertrand. Q Bertrand\'s score was about 10.99 standard deviations away from The mean score for everyone who took The ACT in 2014. Q Bertrand\'s score was about 0.94 standard deviations away from The mean score for everyone who took The ACT in 2014. The probability that one randomly selected student who took The ACT in 2014 scored higher than Bertrand is about 0.94.

Solution

Normal Distribution
Mean ( u ) =1510
Standard Deviation ( sd )=310
Normal Distribution = Z= X- u / sd ~ N(0,1)                  
a)
P(X < 2000) = (2000-1510)/310
= 490/310= 1.5806
= P ( Z <1.5806) From Standard Normal Table
= 0.943                  

94.3% is the SAT Score & the reserved ACT Score is
P ( Z < x ) = 0.943
Value of z to the cumulative probability of 0.943 from normal table is 1.58
P( x-u/s.d < x - 21/7 ) = 0.943
That is, ( x - 21/7 ) = 1.58
--> x = 1.58 * 7 + 21 = 32.06                  

The SAT Score is 32.06   ~ 32

b)
P(X < 16) = (16-21)/7
= -5/7= -0.7143
= P ( Z <-0.7143) From Standard Normal Table
= 0.2375                  

23.75 Percentile SAT Score is
P ( Z < x ) = 0.2375
Value of z to the cumulative probability of 0.2375 from normal table is -0.714
P( x-u/s.d < x - 1510/310 ) = 0.2375
That is, ( x - 1510/310 ) = -0.71
--> x = -0.71 * 310 + 1510 = 1288.66                  

c)
Option 2