1. **Question One: Travel Time Analysis** We are given travel times of 90 members of parliament and asked to: - i) Create a frequency distribution for classes 0-9, 10-19, ..., 90-99. - ii) Find the mean travel time. - iii) Find the median travel time. - iv) Find the mode travel time. --- **Step 1: Frequency Distribution** Classes: 0-9, 10-19, 20-29, 30-39, 40-49, 50-59, 60-69, 70-79, 80-89, 90-99 Count the number of data points in each class: - 0-9: 2,6,3,5,4,9,5,4,8 (9 values) - 10-19: 12,11,13,14,15,16,16,17,17,18,19,19 (12 values) - 20-29: 23,24,21,25,29,21,28,24,26,25,24,21,27,26,27,23,28,23,22,24,27,23,22,21 (24 values) - 30-39: 32,35,31,34,33,36,34,37,35,31,34,33,38,38,39 (15 values) - 40-49: 45,44,46,42,41,41,47,42,46,43,47,45,48,49,48,49 (16 values) - 50-59: 51,52,53,54,55,57,58,57,57,53,52,55 (12 values) - 60-69: 63,66,65,60,62,62,63,68,69 (9 values) - 70-79: 75,76,79,70,72,74,74,76 (8 values) - 80-89: 81,82,82,86,86,89 (6 values) - 90-99: None (0 values) --- **Step 2: Mean Travel Time** Mean formula: $$\bar{x} = \frac{\sum x_i}{n}$$ Sum all 90 travel times and divide by 90. Sum = 2+12+23+81+63+51+11+6+13+24+52+28+22+21+14+3+25+29+21+28+5+15+16+26+54+82+32+17+9+27+76+25+53+24+16+4+23+35+24+21+75+17+45+28+44+27+31+18+55+26+46+58+27+23+34+79+33+28+23+43+42+36+24+32+46+26+22+34+57+33+21+34+32+37+41+35+42+31+41+31+34+47+34+33+38+38+41+34+47+39 = 2823 Mean = $$\frac{2823}{90} = 31.37$$ minutes --- **Step 3: Median Travel Time** Sort data and find middle value (90 data points, median is average of 45th and 46th values). Sorted data median values are 28 and 28. Median = $$\frac{28 + 28}{2} = 28$$ minutes --- **Step 4: Mode Travel Time** Mode is the most frequent value. From frequency counts, 24 and 28 appear most frequently (each 6 times). Mode = 24 and 28 minutes (bimodal) --- 2. **Question Two: Lecturer Monthly Pay Analysis** Given monthly pay data, asked to: - i) Construct frequency distribution starting with class 5-9. - ii) Calculate variance. - iii) Calculate standard deviation. --- **Step 1: Frequency Distribution Classes:** 5-9, 10-19, 20-29, 30-39, 40-49, 50-59, 60-69, 70-79, 80-89, 90-99, 100-109, 110-119 Count frequencies: - 5-9: 5,8 (2) - 10-19: 12,14,16,19,19 (5) - 20-29: 24,24,27,29,24 (5) - 30-39: 31,37,34,39,39,34,34 (7) - 40-49: 45,48,45,48,49,49,43 (7) - 50-59: 51,57,57,53,52,58,57,50 (8) - 60-69: 60,62,62,63,65,66,68,69,63 (9) - 70-79: 70,72,73,74,74,76,76 (7) - 80-89: 82,84,86,86,89 (5) - 90-99: 90,91,92 (3) - 100-109: 101,107,109 (3) - 110-119: 114,116 (2) --- **Step 2: Variance Calculation** Calculate mean $$\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$$ where $f_i$ is frequency and $x_i$ is class midpoint. Class midpoints and frequencies: - 5-9 midpoint 7, freq 2 - 10-19 midpoint 14.5, freq 5 - 20-29 midpoint 24.5, freq 5 - 30-39 midpoint 34.5, freq 7 - 40-49 midpoint 44.5, freq 7 - 50-59 midpoint 54.5, freq 8 - 60-69 midpoint 64.5, freq 9 - 70-79 midpoint 74.5, freq 7 - 80-89 midpoint 84.5, freq 5 - 90-99 midpoint 94.5, freq 3 - 100-109 midpoint 104.5, freq 3 - 110-119 midpoint 114.5, freq 2 Calculate $$\bar{x} = \frac{\sum f_i x_i}{60} = \frac{7*2 + 14.5*5 + 24.5*5 + 34.5*7 + 44.5*7 + 54.5*8 + 64.5*9 + 74.5*7 + 84.5*5 + 94.5*3 + 104.5*3 + 114.5*2}{60} = \frac{3656.5}{60} = 60.94$$ Calculate variance $$\sigma^2 = \frac{\sum f_i (x_i - \bar{x})^2}{\sum f_i}$$ Calculate each term: - (7-60.94)^2*2 = 2920.6 - (14.5-60.94)^2*5 = 10402.3 - (24.5-60.94)^2*5 = 6643.3 - (34.5-60.94)^2*7 = 4833.3 - (44.5-60.94)^2*7 = 1833.3 - (54.5-60.94)^2*8 = 336.3 - (64.5-60.94)^2*9 = 114.3 - (74.5-60.94)^2*7 = 1231.3 - (84.5-60.94)^2*5 = 2923.3 - (94.5-60.94)^2*3 = 3303.3 - (104.5-60.94)^2*3 = 5233.3 - (114.5-60.94)^2*2 = 5943.3 Sum = 48473.7 Variance = $$\frac{48473.7}{60} = 807.9$$ --- **Step 3: Standard Deviation** Standard deviation $$\sigma = \sqrt{\sigma^2} = \sqrt{807.9} = 28.42$$ --- **Final answers:** - Question 1: - Frequency distribution as above - Mean = 31.37 minutes - Median = 28 minutes - Mode = 24 and 28 minutes - Question 2: - Frequency distribution as above - Variance = 807.9 - Standard deviation = 28.42