Question

analyzing tax data
the table presents detailed tax data for 10 individuals, including their respective incomes, taxes paid, deductions, and tax refunds.
table with columns: tax id, income ($), tax paid ($), deductions ($), tax refund ($); rows for id001 to id010 with incomes: 78340, 51270, 114950, 88560, 47380, 104870, 67400, 83240, 91760, 57210 (and other columns)
if the income of id003 (originally $114,950) was entered incorrectly and should be $100,000, what would be the new mean income?
enter your answer in the box.

Explanation:

Step1: Calculate the sum of original incomes

First, we list all the original incomes: \(78340, 51270, 114950, 88560, 47380, 104870, 67400, 83240, 91760, 57210\)

Sum of original incomes \(S_{original}=\sum_{i = 1}^{10}x_i\)

Let's calculate the sum:

\(78340+51270 = 129610\)

\(129610+114950 = 244560\)

\(244560+88560 = 333120\)

\(333120+47380 = 380500\)

\(380500+104870 = 485370\)

\(485370+67400 = 552770\)

\(552770+83240 = 636010\)

\(636010+91760 = 727770\)

\(727770+57210 = 784980\)

Step2: Adjust the income for ID003

The original income for ID003 is \(114950\), and the correct income is \(100000\). So the change in total income \(\Delta S=100000 - 114950=- 14950\)

Step3: Calculate the new total income

New total income \(S_{new}=S_{original}+\Delta S=784980-14950 = 770030\)

Step4: Calculate the new mean income

Mean income is total income divided by the number of individuals (\(n = 10\)). So new mean \(\bar{x}_{new}=\frac{S_{new}}{n}=\frac{770030}{10}=77003\)

Answer:

\(77003\)