# 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)
$\begin{array}{l}\frac{X-200}{30}=0.5\\ X=0.5\left(30\right)+200\\ X=215g\end{array}$

(b)
$\begin{array}{l}P\left(X\ge 194\right)\\ =P\left(Z\ge \frac{194-200}{30}\right)\\ =P\left(Z\ge -0.2\right)\\ =1-P\left(Z>0.2\right)\\ =1-0.4207\\ =0.5793\end{array}$

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
$\begin{array}{l}\frac{X-80}{3}=0.93\\ X=3\left(0.93\right)+80\\ X=82.79\end{array}$