Question

3. The average length of movies made in the US between 1994 and 2019 is 110 minutes. The length is approximately normally distributed with standard deviation of 10 minutes. Rose and Jack have decided to "binge watch 4 movies" during a recent holiday. Let X_i be the length of movie i, for I = 1, 2, 3 and 4. Let X be the sum of the X_i. Find the following probabilities.

a) The probability X is greater than 7 hours.

b) The probability X is between 8 and 10 hours.

c) The probability X is less than 6 hours.

Answer 1

expected total length of 4 movie =4*110=440

and standard deviation =10*√4 =20

a)

for normal distribution z score =(X-μ)/σ

probability X is greater than 7 hours (420 minutes):

probability =

P(X>420)

=

P(Z>-1)=

1-P(Z<-1)=

1-0.1587=

0.8413

b)

probability X is between 8 and 10 hours (480 minutes and 600 minutes):

probability =

P(480<X<600)

=

P(2<Z<8)=

1-0.9772=

0.0228

c)

  probability X is less than 6 hours( 360 minutes)

probability =

P(X<360)

=

P(Z<-4)=

0.0000