Question

In a recent study on world​ happiness, participants were asked to evaluate their current lives on a scale from 0 to​ 10, where 0 represents the worst possible life and 10 represents the best possible life. The responses were normally​ distributed, with a mean of 5.3 and a standard deviation of 2.5. Answer parts ​(a)dash​(d) below. ​(a) Find the probability that a randomly selected study​ participant's response was less than 4. The probability that a randomly selected study​ participant's response was less than 4 is nothing. ​(Round to four decimal places as​ needed.)

Answer 1

P ( X < 4 ) = 0.3015

Explanation:

Given:

- The ratings for current lives on a scale 0 - 10 were normally distributed with parameters mean (u) and standard deviation (s).

u = 5.3

s = 2.5

Find:

Find the probability that a randomly selected study​ participant's response was less than 4.

Solution:

- Declare a random variable X that follows a normally distribution with parameters u and s, mean and standard deviation respectively.

X~N( 5.3 , 2.5 )

- To determine the probability of the rating to be less than 4 for a randomly selected study participant's response we have:

P ( X < 4 )

- Compute the Z-score value for the limit given:

P ( Z < (4 - 5.3) / 2.5 )

P ( Z < -0.52 )

- Use the Z-Table to calculate the above probability as follows:

P ( Z < -0.52 ) = 0.3015

- Hence, the required probability is equivalent to Z-score value probability:

P ( X < 4 ) = P ( Z < -0.52 ) = 0.3015