Long Questions (Question 5)

Posted on April 23, 2020 by user

Question 5:

The diameter of oranges harvested from a fruit orchard has a normal distribution with a mean of 3.2 cm and a variance of 2.25 cm.

Calculate

(a) the probability that an orange chosen at random from this fruit orchard has a diameter of more than 3.8 cm.

(b) the value of k if 30.5 % of the oranges have diameter less than k cm.

Solution:

µ = 3.2 cm

σ²= 2.25cm

σ = √2.25 = 1.5 cm

Let X represents the diameter of an orange.

X ~ N (3.2, 1.5²)

(a)

\begin{array}{l} P (X > 3.8) \\ = P (Z > \frac{3.8 - 3.2}{1.5}) \\ = P (Z > 0.4) \\ = 0.3446 \end{array}

(b)

\begin{array}{l} P (X < k) = 0.305 \\ P (Z < \frac{k - 3.2}{1.5}) = 0.305 \\ From the standard normal distribution table, \\ P (Z > 0.51) = 0.305 \\ P (Z < - 0.51) = 0.305 \\ \frac{k - 3.2}{1.5} = - 0.51 \\ k - 3.2 = - 0.765 \\ k = 2.435 \end{array}