Question

The score on an exam from a certain MAT 112 class, X, is normally distributed with μ=74.4 and σ=8.4. NOTE: Assume for the sake of this problem that the score is a continuous variable. A score can thus take on any value on the continuum. (In real life, scores are often treated as if they were continuous values but are actually discrete in most cases.)

IV'E ALREADY SOLVED PART A & B. I NEED HELP WITH C & D. (iv'e tried 0.0869 for C and 0.8756 for D) --> it says it is close but needs to be more accurate.

(a) Write the event ''a score less than 59.4'' in terms of XX: ANS= X<59.4

(b) Find the probability of this event: ANS=0.0375

(c) Find the probability that a randomly chosen score is greater than 85.9: ?

(d) Find the probability that a randomly chosen score is between 59.4 and 85.9: ?

Answer 1

Given that X follows a Normal distribution with Mean = μ = 74.4 Note thatz-(x-μ)/σ :(x-74.4)/8.4 (Eq. 1) follows standard Normal. When x-59.4, from (Eq1),z-1.79 PCX 59.4) P(Z-1.79)-0.0367 (if using Normal table) P(X 59.4)-0.0371 if using programming calculator) when x = 85.9, from (Eq.1) , z = 1.37 PCX> 85.9)-P(Z> 1.37)-1-P(Z1.37)-1-0.9147 0.0853 (if using Normal table) P(X> 85.9)-1-POX <85.9)-1-0.9145-0.0855 if using programming calculator) When x 59.4, from (Eq.1),z-1.79 When x- 85.9, from (Eq.1),z 1.37 P(59.4< X <85.9-P(-1.79<z-1.37)-P(Z <1.37). P(Z<-1.79-0.9 147-0.0367-0.878 (ifusing Normal table) P(59.4 < X < 85.9)-P(X<85.9)-P(X < 59.4-0.8774 (if using programming calculator)