Final HW 2

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2.4 2.93 (a) Range is difference of maximum data value and minimum data value. Highest amount of taxes paid per person is $7,764 and lowest amount of taxes paid per person is $2,782. So the range for the amount of taxes paid per person is $ 7764 – 2782 = 4982 (b) Range is difference of maximum data value and minimum data value. Highest percentage of personal income paid in taxes per person is 16.6% and lowest percentage of personal income paid in taxes per person is 9.1%. So the range for percentage of personal income paid in taxes per person is 16.6% - 9.1% = 7.5% 2.95 Let n is the number of data values. So = ∑ (x) - nxbar = ∑ (x) – n [∑x /n] = ∑ (x) - ∑x = 0 Hence, sum of deviation is always zero. 2.98 (a) The objective to calculate the range. The highest value is 9 and lowest value is 2. range = highest value – lowest value = 9 – 2

2.4 = 7 Therefore, the range is 7. (b) The objective is to calculate the variance . Sample data is 2, 4, 7, 8, 9. Here, the number of data values is 5. That is n = 5 The mean of the sample is calculated as follows: = 2+4+7+8 +9 / 5 = 30 /5 = 6 Thus, the sample mean is 6. X X – X̅ (X – X̅) ² 2 2 – 6 = -4 (2-6)2 = 16 4 4 – 6 = -2 (4-6)2 = 4 7 7 – 6 = 1 (7-6)2 = 1 8 8 – 6 = 2 (8-6)2 = 4 9 9 – 6 = 3 ( 9-6) 2 = 9 ∑x1 = 30 ∑( x1 – X̅) = 0 ∑( x1 – X̅)2 = 34 The variance is calculated as follows: s 2 = ∑ ( x 1 − ¯ x ) ² n − 1 = 34 / 5-1

2.4 = 34 /4 = 8.5 Therefore, the sample variance is 8.5 (c) The objective is to calculate the standard deviation. The standard deviation is calculated as follows: s = √ s 2 √ 8.5 = 2.9155 Therefore, the standard deviation is 2.9155 2.100 (a) Sample data is 7, 6, 10, 7, 5, 9, 3, 7, 5, 13. Here number of data values is 10 so. Therefore the mean of the sample is X̅ = 7+6+10+7+5+9+3+7+5+13 / 10 = 7.2 Now let us find out the variance of sample. Following table shows the calculations: X X – X̅ (X – X̅) ² 7 -0.2 0.04 6 -1.2 1.44 10 2.8 7.84 7 -0.2 0.04 5 -2.2 4.84 9 1.8 3.24 3 -4.2 17.64

2.4 7 -0.2 0.04 5 -2.2 4.84 13 5.8 33.64 Total 73.6 So the variance of the sample is = 73.6 / 10-1 = 8.18 Hence, the sample variance is 8.18. (b) Following is the calculation for variance using formula 2.9. x x² 7 49 6 36 10 100 7 49 5 25 9 81 3 9 7 49 5 25 13 169 ∑x = 72 ∑x² = 592

2.4 (c) Standard deviation of the data will be s = √ 8.18 s = 2.86 2.102 (a) Range is difference of highest and lowest value. So range will be range = highest value – lowest value = 48 - 21 = 27 Hence, the range of the data is 27 (b) Size of random sample is 10 so n = 10 Therefore the mean of the sample is x̄ͅ > = 36+26+48+28+45+21+21+38+27+32 / 10 = 322 / 10 = 32.2 Now let us find out the variance of sample. Following table shows the calculations:

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