If X is normally distributed with mean 10 and standard deviation 4, find x such that the probability of X between 10 and x is 0.4772.
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 18
Given $μ = 10, σ = 4$.
Also, $P(10 < x < x) = 0.4772,$
then $Z =\frac{X-10}{4}$
$P(10 <X < x)= 0.4772$
$⇒P\left(\frac{10-10}{4}<Z<\frac{x-10}{4}\right)= 0.4772$
$⇒P\left(0 <Z<\frac{x-10}{4}\right)= 0.4772$
$⇒F\left(\frac{x-10}{4}\right)- F(0) = 0.4772$
$⇒F\left(\frac{x-10}{4}\right)-0.5= 0.4772$ (using property 2)
$⇒F\left(\frac{x-10}{4}\right)= 0.9772$
$⇒\frac{x-10}{4}=2$ (using table)
$⇒x = 18$