Short Questions (Question 3 & 4)


Question 3 (4 marks):
Diagram shows a standard normal distribution graph.

Diagram

The probability represented by the area of the shaded region is 0.2881.
(a) Find the value of h.
(b) X is a continuous random variable which is normally distributed with a mean, μ and a variance of 16.
Find the value of μ if the z-score of X = 58.8 is h.


Solution:
(a)
P(X < h) = 0.5 – 0.2881
P(X < h) = 0.2119
P(X < –0.8) = 0.2119
h = –0.8

(b)

X=58.8Xμσ=58.8μσ   Z=58.8μ4   h=58.8μ40.8=58.8μ43.2=58.8μμ=58.8+3.2μ=62



Question 4 (4 marks):
A voluntary body organizes a first aid course 4 times per month, every Saturday from March until September.
[Assume there are four Saturdays in every month]

Salmah intends to join the course but she might need to spare a Saturday per month to accompany her mother to the hospital. The probability that Salmah will attend the course each Saturday is 0.8. Salmah will be given a certificate of monthly attendance if she can attend the course at least 3 times a month.

(a)
Find the probability that Salmah will be given the certificate of monthly attendance.

(b)
Salmah will qualify to sit for the first aid test if she obtains more than 5 certificates of monthly attendance.
Find the probability that Salmah qualifies to take the first aid test.


Solution:
(a)

P(X=r)=Cnrprqnrp=0.8, q=0.2, n=4, r=3, 4P(X3)=P(X=3)+P(X=4)=C43(0.8)3(0.2)1+C44(0.8)4(0.2)0=0.4096+0.4096=0.8192

(b)

P(X=r)=Cnrprqnrp=0.8192, q=0.1808, n=7, r=6, 7P(X>5)=P(X=6)+P(X=7)=C76(0.8192)6(0.1808)1+C77(0.8192)7(0.1808)0=0.3825+0.2476=0.6301