This exercise presents an example of Simpson's paradox. The total income and total taxes paid in each of five income categories are given for two years, 1974 and 1978, in the following table:
1974
Adjusted Gross Income | Income | Tax
Under $5,000 | 41,651,640 | 2,744,460
$5,000 to $9,999 | 146,400,740 | 13,646,348
$10,000 to $14,999 | 192,688,922 | 21,449,597
$15,000 to $99,999 | 470,010,790 | 75,038,230
$100,000 or more | 29,427,152 | 11,311,672
Total | 880,179,244 | 124,190,307
1978
Adjusted Gross Income | Income | Tax
Under $5,000 | 29,879,620 | 739,310
$5,000 to $9,999 | 122,853,315 | 8,819,461
$10,000 to $14,999 | 171,858,024 | 17,155,758
$15,000 to $99,999 | 865,037,814 | 137,860,951
$100,000 or more | 62,806,159 | 24,051,698
Total | 1,252,434,932 | 188,627,178
(a) The "tax rate" for any income category is calculated as Tax divided by Income. Calculate the tax rates for each income bracket in each year. (Round the answers to three decimal places.)
(b) Calculate the overall tax rate for each of the two years. (Round the answers to three decimal places.)
For 1974, the rate is
For 1978, the rate is