Question

Use the empirical rule to answer question. Please draw a picture with all the percentages as well. Suppose the average age for a professional soccer player is 25 years with a standard deviation of 4 years. Assume that this distribution follows a normal distribution.

a. Approximately what percentage of pro soccer players are between the ages of 21 and 25?

b. Approximately what percentage of pro soccer players are between the ages of 21 and 29?

c. Approximately what percentage of pro soccer players are greater than age 33?

d. Approximately what percentage of pro soccer players are between the ages of 17 and 21?

Answer 1

here we use standard normal variacte z=(x-mean)/sd

(a)required percentage=34.13

for x=21, z=(21-25)/4=-1

for x=25, z=(25-25)/4=0

P(21<X<25)=P(-1<Z<0)=P(Z<0)-P(Z<-1)=0.5-0.1587=0.3413

(b)required percentage=68.26

for x=21, z=(21-25)/4=-1

for x=29, z=(29-25)/4=1

P(21<X<29)=P(-1<Z<1)=P(Z<1)-P(Z<-1)=0.8413-0.1587=0.6826

(c) required percentage=2.27

for x=33, z=(33-25)/4=2

P(X>2)=1-P(Z<2)=1-0.9773=0.0227

(d) required percentage=13.59%

for x=21, z=(21-25)/4=-1

for x=17, z=(17-25)/4=-2

P(17<X<21)=P(-2<Z<-)=P(Z<-2)-P(Z<-1)=0.1587-0.0228=0.1359