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 5.8 with a standard deviation of 2.5 .
(a) What response represents the 89 th percentile?
(b) What response represents the 59 th percentile?
(c) What response represents the first quartile?

Step 1 ：We are given the mean and standard deviation of the responses. The mean is \(5.8\) and the standard deviation is \(2.5\).

Step 2 ：We are asked to find the responses that represent the 89th percentile, 59th percentile and the first quartile. Percentiles and quartiles are measures of position in a data set. The nth percentile is the value below which n% of the data falls. The first quartile, also known as the 25th percentile, is the value below which 25% of the data falls.

Step 3 ：To find these values, we can use the formula for a z-score, which is \((X - μ) / σ\), where X is the value, μ is the mean, and σ is the standard deviation. However, in this case, we are given the percentile and we need to find the corresponding value, so we need to rearrange the formula to solve for X: \(X = Zσ + μ\), where Z is the z-score corresponding to the given percentile.

Step 4 ：We can use a standard normal distribution table or a z-score calculator to find the z-scores for the 89th percentile, 59th percentile, and the first quartile. Then we can substitute these z-scores, along with the given mean and standard deviation, into the formula to find the corresponding values.

Step 5 ：The values corresponding to the 89th percentile, 59th percentile, and the first quartile are approximately 8.87, 6.37, and 4.11, respectively.

Step 6 ：Final Answer: The response that represents the 89th percentile is approximately \(\boxed{8.87}\), the response that represents the 59th percentile is approximately \(\boxed{6.37}\), and the response that represents the first quartile is approximately \(\boxed{4.11}\).