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.
Answer:
1.) 0.4141
2.) 0.3288
3.) 0.0253
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