Question

1000 students took an examination in Statistics and Probability. The mean score obtained is 71 and the standard deviation is 4. Assuming that the data are normally distributed, find the area and probability of students who obtained a score of:

a. at least 83

b. mean to 92

c. at most 70

Answer 1

X ~ N$(71, 4^2)$. The probability and corresponding area are equal

a) $P(X>83)=P(N(71,4^2)>83)=P(71+4Z>83)=P(Z>3)=0.001$

b) $P(71<X<92)=P(71<71+4Z<92)=P(0<Z<5.25)=P(Z<5.25)-P(Z<0)=1-0.5=0.5$

c) $P(X<70)=P(71+4Z<70)=P(Z<-0.25)=0.401$