Question

In the United States the ages 13 to 55+ of smartphone users approximately follow a normal distribution with approximate mean and standard deviation of 36.9 years and 13.9 years, respectively. What is the probability that a random smartphone user in the age range 13 to 55+ is between 23 and 64.7 years old?

Answer 1

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 2

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

$\mu=36.9\\\sigma=13.9\\x_{1} =23\\x_{2}=64.9$

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:

$P(x_{1}$

Replacing $x_{1}, x_{2} :$

$P(23$

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:

$z=\frac{x-\mu}{\sigma}$

Then, probability is given by the equation:

$P(23$

4. Estimate the probabilities in normal distribution table:

$P(Z$

$P(-1$

The probability that a random user is between 23 an 64.7 years old is 0.8186