Probability Distribution, Short Questions (Question 3 & 4)


Question 3:
The masses of mangoes in a stall have a normal distribution with a mean of 200g and a standard deviation of 30g.
(a) Find the mass, in g, of a mango whose z-score is 0.5.
(b) If a mango is chosen at random, find the probability that the mango has a mass of at least 194g.

Solution:
µ = 200g
σ = 30g
Let X be the mass of a mango.
(a)
X200 30 =0.5 X=0.5(30)+200 X=215g

(b)
P( X194 ) =P( Z 194200 30 ) =P( Z0.2 ) =1P( Z>0.2 ) =10.4207 =0.5793


Question 4:
Diagram below shows a standard normal distribution graph.
The probability represented by the area of the shaded region is 0.3238.
(a) Find the value of k.
(b) X is a continuous random variable which is normally distributed with a mean of 80 and variance of 9.
Find the value of X when the z-score is k.

Solution:
(a)
P(Z > k) = 0.5 – 0.3238 
= 0.1762
k = 0.93

(b)
µ = 80,
σ2 = 9, σ = 3
X80 3 =0.93 X=3( 0.93 )+80 X=82.79