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 5.6 with a standard deviation of 2.6. ​(a) What response represents the 85th ​percentile? ​(b) What response represents the 62nd ​percentile? ​(c) What response represents the first ​quartile?

Answer 1

Answer:

a) 8.2962

b) 6.3956

c) 3.845

Step-by-step explanation:

Z-score:

In a set with mean $\mu$ and standard deviation $\sigma$, the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this question, we have that:

$\mu = 5.6, \sigma = 2.6$

(a) What response represents the 85th ​percentile? ​

This is X when Z has a pvalue of 0.85. So X when Z = 1.037.

$Z = \frac{X - \mu}{\sigma}$

$1.037 = \frac{X - 5.6}{2.6}$

$X - 5.6 = 2.6*1.037$

$X = 8.2962$

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

This is X when Z has a pvalue of 0.62. So X when Z = 0.306.

$Z = \frac{X - \mu}{\sigma}$

$0.306 = \frac{X - 5.6}{2.6}$

$X - 5.6 = 2.6*0.306$

$X = 6.3956$

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

The first quartile is the 100/4 = 25th percentile. So this is X when Z has a pvalue of 0.25, so X when Z = -0.675.

$Z = \frac{X - \mu}{\sigma}$

$-0.675 = \frac{X - 5.6}{2.6}$

$X - 5.6 = 2.6*(-0.675)$

$X = 3.845$