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 mean response was 5.5 with a standard deviation of 2.2. ​(a) What response represents the 94th ​percentile? ​(b) What response represents the 59th ​percentile? ​(c) What response represents the first ​quartile?

Answer 1

Solution:-

Given that,

mean = = 5.5

standard deviation = = 2.2

a) Using standard normal table,

P(Z < z) = 94%

= P(Z < z ) = 0.94

= P(Z < 1.5548 ) = 0.94  

z = 1.5548

Using z-score formula,

x = z * +

x = 1.5548 * 2.2 + 5.5

x = 8.92

8.92 response represents the 94th ​percentile

b) Using standard normal table,

P(Z < z) = 59%

= P(Z < z ) = 0.59

= P(Z < 0.2275 ) = 0.59

z = 0.2275

Using z-score formula,

x = z * +

x = 0.2275 * 2.2 + 5.5

x = 6.00

6.00 response represents the 59th ​percentile

c) Using standard normal table,

The z dist'n First quartile is,

P(Z < z) = 25%

= P(Z < z) = 0.25  

= P(Z < -0.6745 ) = 0.25

z = -0.6745

Using z-score formula,

x = z * +

x = -0.6745 * 2.2 + 5.5

x = 4.02

First quartile =Q1 = 4.02