SAT scores of appliants to UCLA are normally distributed wit

SAT scores of appliants to UCLA are normally distributed with mean=1200 and sd=100. Use R to find:

a. Probability of an applicant to have a score more than 1350.

b. Probability of an applicant to have a score between 1250 and 1300.

c.80th percentile for SAT Score.

d. 70th percentile for SAT Score.

e. Draw density function of SAT Score.

SAT scores ~ N (1200, 100)

a) P(Z > (1350 -1200)/100) = P(z > 1.5) = 1 - P(z <1.5)

R - command for P(z < 1.5 ) is pnorm (1.5) = 0.93319

Therefore 1 - P( z< 1.5) = 1 - 0.93319 = 0.06681

b) P (1250 < X < 1300) where X is a SAT Score variable

= P((1250 -1200)/100 < Z< (1300 -1200)/100)

= P(0.5 < Z < 1) = P(Z< 1) - P(Z<0.5)

R-command for P(Z<1) = pnorm(1) = 0.84135 and P(Z<0.5) = pnorm(0.5) = 0.69146

Therefore P(Z< 1) - P(Z<0.5) = 0.84135 - 0.69146 = 0.14989

c) 80th percentile for SAT score in R is

qnorm (0.8, mean, Std. deviation) = qnorm (0.8, 1200, 100) = 1284.162

d) 70th percentile for SAT score in R is

qnorm (0.7, mean, Std. deviation) = qnorm (0.7, 1200, 100) = 1252.44

e) To draw density function we use SAT score values as Xcompute density function at X using function dnorm with syntax dnorm(x,mean, std. deviation)

then plot(x,y, xlab = \"SAT Score\" , ylab = PDF\")