Question

et the random variable x follow a normal distribution with μ = 50 and σ² = 64.

a. find the probability that x is greater than 60.

b. find the probability that x is greater than 35 and less than 62 .

c. find the probability that x is less than 55.

d. the probability is 0.2 that x is greater than what number?

e. the probability is 0.05 that x is in the symmetric interval about the mean between which two numbers?

Answer 1

given mean = 50 standard deviation =sqrt(64)= 8

a)

P(Y > 60) = P(Y - mean > 60 - mean)
                  = P( (Y - mean)/SD > (60 - mean)/SD
                  = P(Z > (60 - mean)/SD)
                  = P(Z > (60 - 50)/8)
                  = P(Z > 1.25)
                  = 1 - P(Z <= 1.25)
                  = 0.106

b)

P(35 < Y < 62) = P(35 - mean < Y - mean < 62 - mean)
                  = P((35 - mean)/SD < (Y - mean)/SD < (62 - mean)/SD)
                  = P((35 - mean)/SD < Z < (62 - mean)/SD)
                  = P((35 - 50)/8< Z < (62 - 50)/8)
                  = P(-1.875 < Z < 1.5)
                  = P(Z < 1.5) - P(Z <-1.875)
                  = 0.903

c)

P(Y < 55) = P(Y - mean < 55 - mean)
                  = P( (Y - mean)/SD < (55 - mean)/SD
                  = P(Z < (55 - mean)/SD)
                  = P(Z < (55 - 50)/8)
                  = P(Z < 0.625)
                  = 0.734

d)

for top 20% values; at 80th percentile ; z=0.8416

corresponding value =+z* =50+0.8416*8=56.73

we are allowed to solve four sub parts only thank you.