The mean household income in a country in a recent year was about ​$74 comma 31474,314 and the standard deviation was about ​$82 comma 00082,000.

Question

The mean household income in a country in a recent year was about

​$74 comma 31474,314

and the standard deviation was about

​$82 comma 00082,000.

​(The median income was

​$54 comma 66354,663​.)

​a) If a Normal model was used for these​ incomes, what would be the household income of the top

44​%?

​b) How confident should one be in the answer in part​ a?

​c) Why might the Normal model not be a good one for​ incomes?

Answer 1

The formatting of your numbers is not clear, but I think I can answer your questions in general.

(a) You need to consult a table with z-scores that tells you how many standard deviations above the mean the 56th percentile is. The table I found online says +0.151 standard deviations above the mean corresponds to the 56th percentile. So the predicted income of the top 44% would be above (Mean + 0.151*SD).

(b) We should not be confident in the answer to part (a)! It is almost certainly inaccurate.

(c) Because the mean and the median are quite far apart, we know this distribution is heavily skewed with a long tail into high incomes. So the normal distribution is not an appropriate model for these incomes.