Respuesta :
Answer:
P ( 23 < X < 64.7 ) = P ( -1 < Z < 2 ) = 0.8186
Step-by-step explanation:
Solution:
- Let X be a random variable that denotes the age of people who use smartphones.
- The random variable X follows a normal distribution with parameters mean (u) and standard deviation (s).
-The normal distribution can be expressed as:
X~ N ( u , s^2 )
X~ N ( 36.9 , 13.9^2 )
- The probability that a random smartphone user in the age range 13 to 55+ is between 23 and 64.7 years old can be expressed as:
P ( 23 < X < 64.7 )
- We will compute the Z-score values for the interval:
P ( 23 < X < 64.7 ) = P ( (x1 - u) / s < Z < (x2 - u) / s )
P ( 23 < X < 64.7 ) = P ( (23 - 36.9) / 13.9 < Z < (64.7 - 36.9) / 13.9 )
P ( 23 < X < 64.7 ) = P ( -1 < Z < 2 )
- We will use Z-table to evaluate:
P ( 23 < X < 64.7 ) = P ( -1 < Z < 2 ) = 0.8186
Answer:
The probability that a random user is between 23 an 64.7 years old is 0.8186
Step-by-step explanation:
1. Relevant data
[tex]\mu=36.9\\\sigma=13.9\\x_{1} =23\\x_{2}=64.9[/tex]
2. Calculate probabilty.
We want to find the probabilty that a random user is between 23 and 64.7 years old. We can note that condition with the expression:
[tex]P(x_{1}<X<x_{2})=P(X<x_{2}) -P(X>x_{1})[/tex]
Replacing [tex]x_{1}, x_{2} :[/tex]
[tex]P(23<X<64.7)=P(X<64.7) -P(X<23)[/tex]
3. Transform x in z-value.
Considering the ages follow a normal distribution, we need to transformx in z-value. We can calculate z with the equation:
[tex]z=\frac{x-\mu}{\sigma}[/tex]
Then, probability is given by the equation:
[tex]P(23<X<64.7)=P(\frac{23-\mu}{\sigma} <Z<\frac{64.7-\mu}{\sigma})\\P(23<X<64.7)=P(\frac{23-\36.9}{13.9} <Z<\frac{64.7-36.9}{13.9})\\P(23<X<64.7)=P(-1<Z<2)[/tex]
4. Estimate the probabilities in normal distribution table:
[tex]P(Z<1)=0.1586\\P(Z<2)=0.9772[/tex]
[tex]P(-1<Z<2)=P(Z<2)-P(Z<-1)\\P(-1<Z<2)=0.9772-0.1586\\P(-1<Z<2)=0.8186[/tex]
The probability that a random user is between 23 an 64.7 years old is 0.8186
