Question

World Happiness 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 5.4 and a standard deviation of 2.2. Find the probability that a randomly selected study participant’s response was (a) less than 4, (b) between 4 and 6, and (c) more than 8. Identify any unusual events in parts (a)–(c). Explain your reasoning.

Answer 1

a)

µ =    5.4      
σ =    2.2      
          
P( X ≤    4   ) = P( (X-µ)/σ ≤ (4-5.4) /2.2)  
=P(Z ≤   -0.636   )

=   0.2623

......

b)

µ =    5.4                              
σ =    2.2                              
we need to calculate probability for ,                                  
P (   4   < X <   6   )                  
=P( (4-5.4)/2.2 < (X-µ)/σ < (6-5.4)/2.2 )                                  
                                  
P (    -0.636   < Z <    0.273   )                   
= P ( Z <    0.273   ) - P ( Z <   -0.636   )

=    0.6075   -    0.2623  

=    0.3452
.............

c)

µ =    5.4                  
σ =    2.2                  
                      
P ( X ≥   8.00   ) = P( (X-µ)/σ ≥ (8-5.4) / 2.2)              
= P(Z ≥   1.182   )

= P( Z <   -1.182   )

=    0.1186   (answer)

.........

thanks