Question

The distribution of ages of Oscar winning actors is roughly symmetric and mound shaped (in other words, the ages for Oscar winning actors are normally distributed). The mean of this distribution is 44 years and the standard deviation of this distribution is 9 years.

The 68-95-99.7 Rule

Finding Percentiles

a.What percent of actors received the Oscar after the age of 70? Draw the picture of the area that represents this probability. (You will notice that it is a right-tailed area.)

b.What percentage of Oscar winning 10% of actors received the oscar after the age of---

c. What percentage of Osar winning actresses are at most 22 years old

d.68 percent of the actors win the Oscar between the ages of ---- and ----

e.Based on the data given do actresses win the Oscars at the younger age?

Answer 1

Let x be the ages of Oscar winning actors.

x follows normal distribution with mean µ = 44 and standard deviation σ = 9

Part a) We are asked to find P( x > 70 )

=  

= P ( z > 2.89 )

= 1 - P( z < 2.89 )

= 1 - 0.9981

= 0.0019 ~ 0.19%

So 0.19% of actors received the Oscar after the age of 70

Part b) We are asked to find x such that 10% area is to the right of x , that is 90% area is to the left of x

So first we need to find the z score corresponding to area 0.90 on z score table

So z = 1.28 has area to the left 0.90 on z score table

Therefore x = z*σ + µ  

x = ( 1.28*9 ) + 44

x = 55.52

Therefore 10% of actors received the Oscar after the age of 55.52

Part c) P( x ≤ 22 )

= P ( z ≤ -2.44 )

= 0.0073 ~ 0.73%

0.73% of Oscar winning actresses are at most 22 years old

Part d)

According to 68-95-99.7 Rule
Approximately 68% of the observations fall within 1 standard deviation of the mean . P ( µ - 1*σ < x < µ + 1*σ) = 0.68

µ - 1*σ = 44 - 9 = 35  

µ + 1*σ = 44 + 9 = 53

68 percent of the actors win the Oscar between the ages of 35 and 53

Part e) please provide the data set for actresses.