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?

a) 8.2962

b) 6.3956

c) 3.845

Step-by-step explanation:

Z-score:

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

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:

(a) What response represents the 85th percentile?

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

(b) What response represents the 62nd percentile?

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

(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.

or