Question

In a recent poll on world sadness, participants were asked to rate how sad they were on a scale of 0 to 10, with 0 being completely unhappy and 10 being incredibly unhappy. The standard deviation of responses was 1.9 and the mean response was 6.3.

What response would be the 90th percentile, rounded to one decimal place?

What response would be in the third quartile, rounded to one decimal place?

Please show your work! Any help is appreciated!

Answer 1

Let X is the random variable denoting rate of sadness.

Given

Mean ,

Standard deviation,

1) We have to find response that would be the 90th percentile,

First, we need to find the z-score associated to this percentile. We need to find the value z_p such​ that

P (Z < z_p​) = 0.9

From the normal table or using R we see that,

So,

P (Z < 1.282) = 0.9

The 90th percentile we are looking for is computed using the following formula:

          

          

2) We have to find response that would be the third quartile i.e 75th percentile

First, we need to find the z-score associated to this percentile. We need to find the value z_p such​ that

P (Z < z_p​) = 0.75

From the normal table or using R we see that,

So,

P(Z < 0.675) = 0.75

The 75th percentile (or third quartile) we are looking for is computed using the following formula: