A village consists of twenty (20) households with the following annual incomes- (Incomes $00) 30 20 50 40 60 40 40 50 20 60 80 40 20 20 70 40 70 30 40 80 (a...

Answer Details

The twenty incomes (in \$00) are: 30, 20, 50, 40, 60, 40, 40, 50, 20, 60, 80, 40, 20, 20, 70, 40, 70, 30, 40, 80. Their total is \( \sum x = 900 \) (in \$00), that is \$90,000.

(a)(i) Mean income. \[ \bar{x} = \frac{\sum x}{n} = \frac{900}{20} = 45\;(\$00) = \$4{,}500 \]

(a)(ii) Modal income. Arranging the values by frequency, the income 40 (\$4,000) occurs six times, more than any other, so the mode = \$4,000.

(a)(iii) Median income. Arranging in order: 20, 20, 20, 20, 30, 30, 40, 40, 40, 40, 40, 40, 50, 50, 60, 60, 70, 70, 80, 80. With 20 items, the median is the average of the 10th and 11th values, both 40, so the median = \$4,000.

(b) Range. \[ \text{Range} = \text{highest} - \text{lowest} = 80 - 20 = 60\;(\$00) = \$6,000 \]

(c)(i) Flat tax of 7% on all households. Total income \( = \$90,000 \). \[ \text{Tax} = 7\% \times 90{,}000 = \frac{7}{100}\times 90{,}000 = \$6{,}300 \]

(c)(ii) Flat tax of 15% on households earning \$4,000 and above. Households below \$4,000 are those earning \$2,000 (four households) and \$3,000 (two households), a total of \( 4(20) + 2(30) = 80 + 60 = 140 \) (\$00) = \$14,000, which is exempt. Taxable income \( = 900 - 140 = 760 \) (\$00) = \$76,000. \[ \text{Tax} = 15\% \times 76{,}000 = \frac{15}{100}\times 76{,}000 = \$11{,}400 \]