Question

The length of time a full length movie runs from opening to credits is normally distributed with a mean of 1.9 hours and standard deviation of 0.3 hours. Calculate the following:

1. A random movie is between 1.5 and 2.0 hours.
2. A movie is longer than 2.4 hours.
3. The length of movie that is shorter than 94% of the movies.

Answer 1

Solution:

Given, $\mu=1.9,\sigma=0.3$

$X\sim N(\mu,\sigma)$

(i):

$P(1.5<X<2)=P(X<2)-P(X<1.5) \\=P(z<\dfrac{2-1.9}{0.3})-P(z<\dfrac{1.5-1.9}{0.3}) \\=P(z<0.33)-P(z<-1.33) \\=P(z<0.33)-[1-P(z<1.33)] \\=0.62930-[1-0.90824] \\=0.53754$

(ii):

$P(X>2.4)=1-P(X\le2.4) \\=1-P(z\le \dfrac{2.4-1.9}{0.3}) \\=1-P(z\le 1.67) \\=1-0.95254 \\=0.04746$

(iii):

$P(X<x)=0.94 \\\Rightarrow P(z<\dfrac{x-1.9}{0.3})=0.94 \\\Rightarrow \dfrac{x-1.9}{0.3}=1.55 \\\Rightarrow x=2.365$

Required length of movie is 2.365 hours.