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 mean response was

Complete 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 4.5 and a standard deviation of 2.3. Answer parts (a)-(d) below.

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

(b) Find the probability that a randomly selected study participant's response was between 4 and 6.

(c) Find the probability that a randomly selected study participant's response was more than 8.

1.) 0.4141

2.) 0.3288

3.) 0.0253

Step-by-step explanation:

Given a normal distribution :

Mean(m) = 4.5

Standard deviation (σ) = 2.3

A.) x < 4

Using :

Zscore = (4 - 4.5) / 2.3

Zscore = - 0.5 / 2.3

Zscore = - 0.2173913

P(Z < - 0.217) = 0.4141

B.) x between 4 and 6

From the above :

For x < 4;

P(Z < - 0.217) = 0.4141

For x < 6 ;

Using :

Zscore = (6 - 4.5) / 2.3

Zscore = 1.5 / 2.3

Zscore = 0.65217

P(Z < 0.65217) = 0.74285

0.74285 - 0.4141 = 0.3288

C.) X > 8

Using :

Zscore = (8 - 4.5) / 2.3

Zscore = 4.5 / 2.3

Zscore = 1.9565217

P(Z > 1.9565217 ) = 1 -P(Z < 1.9565217 )

P(Z < 1.9565217 ) = 0.9747

P(Z > 1.9565217 ) = 1 - 0.9747

= 0.0253