Question

17. The lengths of a population of certain HULU shows I watch are normally distributed with a mean running time of 38 minutes and a standard deviation of 11.5.

1. Find the percentage of shows with running times between 26 and 42 minutes.

2.Between what values would you expect to find the middle 80 %

3. Find the percentage of shows with running times below 47.5 minutes

4.Above what value would you expect to find the top 25 %?

5.Find the percentage of shows with running times above 18 minutes.

Answer 1

Part 1)
X ~ N ( µ = 38 , σ = 11.5 )
P ( 26 < X < 42 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 26 - 38 ) / 11.5
Z = -1.0435
Z = ( 42 - 38 ) / 11.5
Z = 0.3478
P ( -1.04 < Z < 0.35 )
P ( 26 < X < 42 ) = P ( Z < 0.35 ) - P ( Z < -1.04 )
P ( 26 < X < 42 ) = 0.636 - 0.1484
P ( 26 < X < 42 ) = 0.4876

Part 2)

X ~ N ( µ = 38 , σ = 11.5 )
P ( a < X < b ) = 0.8
Dividing the area 0.8 in two parts we get 0.8/2 = 0.4
since 0.5 area in normal curve is above and below the mean
Area below the mean is a = 0.5 - 0.4
Area above the mean is b = 0.5 + 0.4
Looking for the probability 0.1 in standard normal table to calculate critical value Z = -1.2816
Looking for the probability 0.9 in standard normal table to calculate critical value Z = 1.2816
Z = ( X - µ ) / σ
-1.2816 = ( X - 38 ) / 11.5
a = 23.2616
1.2816 = ( X - 38 ) / 11.5
b = 52.7384
P ( 23.2616 < X < 52.7384 ) = 0.8

Part 3)
X ~ N ( µ = 38 , σ = 11.5 )
P ( X < 47.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 47.5 - 38 ) / 11.5
Z = 0.8261
P ( ( X - µ ) / σ ) < ( 47.5 - 38 ) / 11.5 )
P ( X < 47.5 ) = P ( Z < 0.8261 )
P ( X < 47.5 ) = 0.7956
Percentage is 0.7956 * 100 = 79.56%

Part 4)
X ~ N ( µ = 38 , σ = 11.5 )
P ( X > x ) = 1 - P ( X < x ) = 1 - 0.25 = 0.75
To find the value of x
Looking for the probability 0.75 in standard normal table to calculate critical value Z = 0.6745
Z = ( X - µ ) / σ
0.6745 = ( X - 38 ) / 11.5
X = 45.7567
P ( X > 45.7567 ) = 0.25

Part 5)
X ~ N ( µ = 38 , σ = 11.5 )
P ( X > 18 ) = 1 - P ( X < 18 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 18 - 38 ) / 11.5
Z = -1.7391
P ( ( X - µ ) / σ ) > ( 18 - 38 ) / 11.5 )
P ( Z > -1.7391 )
P ( X > 18 ) = 1 - P ( Z < -1.7391 )
P ( X > 18 ) = 1 - 0.041
P ( X > 18 ) = 0.959

Percentage is 0.9590 * 100 = 95.90%