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 5.9 with a standard deviation of 2.2.

​(a) What response represents the 92nd ​percentile? ​

(b) What response represents the 62nd ​percentile?

​(c) What response represents the first ​quartile?

Answer 1

a) A response of 8.9 represents the 92nd ​percentile.

b) A response of 6.6 represents the 62nd ​percentile.

c) A response of 4.4 represents the first ​quartile.

Explanation:

We are given the following information in the question:

Mean, μ = 5.9

Standard Deviation, σ = 2.2

We assume that the distribution of response is a bell shaped distribution that is a normal distribution.

Formula:

`z_(score) = \displaystyle(x-\mu)/(\sigma)`

a) We have to find the value of x such that the probability is 0.92

P(X < x)

`P( X < x) = P( z < \displaystyle(x - 5.9)/(2.2))=0.92`

Calculation the value from standard normal z table, we have,

`P(z<1.405) = 0.92`

`\displaystyle(x - 5.9)/(2.2) = 1.405\\x = 8.991 \approx 8.9`

A response of 8.9 represents the 92nd ​percentile.

b) We have to find the value of x such that the probability is 0.62

P(X < x)

`P( X < x) = P( z < \displaystyle(x - 5.9)/(2.2))=0.62`

Calculation the value from standard normal z table, we have,

`P(z<0.305) = 0.92`

`\displaystyle(x - 5.9)/(2.2) = 0.305\\x = 6.571 \approx 6.6`

A response of 6.6 represents the 62nd ​percentile.

c) We have to find the value of x such that the probability is 0.25

P(X < x)

`P( X < x) = P( z < \displaystyle(x - 5.9)/(2.2))=0.25`

Calculation the value from standard normal z table, we have,

`P(z<0.305) = -0.674`

`\displaystyle(x - 5.9)/(2.2) = -0.674\\x = 4.4172 \approx 4.4`

A response of 4.4 represents the first ​quartile.