1. Problem 1: Test Scores \nFind mode, median, mean, lower quartile (Q1), upper quartile (Q3), interquartile range (IQR), and population standard deviation.\nData: 37, 42, 48, 51, 52, 53, 54, 54, 55\n\nStep 1: Mode is the most frequent value. Here it is 54 because it appears twice.\nStep 2: Median is middle value of ordered data (9 data points). Median = 5th data point = 52.\nStep 3: Mean is sum divided by number of data. $$\text{mean} = \frac{37 + 42 + 48 + 51 + 52 + 53 + 54 + 54 + 55}{9} = \frac{446}{9} \approx 49.56$$\nStep 4: Lower quartile (Q1) is median of lower half (first 4 data): 37, 42, 48, 51. Q1 = average of 2nd and 3rd = $\frac{42+48}{2} = 45$.\nStep 5: Upper quartile (Q3) is median of upper half (last 4 data): 53, 54, 54, 55. Q3 = average of 2nd and 3rd = $\frac{54+54}{2} = 54$.\nStep 6: Interquartile range = Q3 - Q1 = 54 - 45 = 9.\nStep 7: Population standard deviation $\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}}$ where $\mu$ is mean and $N=9$.\nCalculate squared differences:\n$(37-49.56)^2=157.98$\n$(42-49.56)^2=57.31$\n$(48-49.56)^2=2.43$\n$(51-49.56)^2=2.07$\n$(52-49.56)^2=5.95$\n$(53-49.56)^2=11.88$\n$(54-49.56)^2=19.69$\n$(54-49.56)^2=19.69$\n$(55-49.56)^2=29.62$\nSum of squares = 306.62\nPopulation variance = $\frac{306.62}{9} = 34.07$\nStandard deviation $\sigma = \sqrt{34.07} \approx 5.84$.\n\n2. Problem 2: Mens Heights (inches)\nData: 62, 64, 69, 70, 70, 71, 72, 73, 74, 75, 77\n\nStep 1: Mode = 70 (appears twice).\nStep 2: Median = middle value of 11 data points = 6th data point = 71.\nStep 3: Mean = sum/11 = (62+64+69+70+70+71+72+73+74+75+77)/11 = 797/11 = 72.45.\nStep 4: Lower quartile (Q1) = median of lower 5 data points (62,64,69,70,70)= 3rd = 69.\nStep 5: Upper quartile (Q3) = median of upper 5 data points (71,72,73,74,75,77) Actually upper half is 6 points, so upper half after median is 6 numbers: 72,73,74,75,77 (only 5?), we count carefully: data sorted: 62,64,69,70,70,71,72,73,74,75,77. Median at index 6 (1-based). Lower half: first 5 numbers. Upper half: last 5 numbers 72,73,74,75,77. Median of upper half = 3rd number = 74.\nStep 6: Interquartile range = $74 - 69 = 5$.\nStep 7: Population standard deviation calculation: mean = 72.45, number of data = 11. Squared diffs:\n$(62-72.45)^2=109.80$\n$(64-72.45)^2=71.20$\n$(69-72.45)^2=11.90$\n$(70-72.45)^2=6.00$\n$(70-72.45)^2=6.00$\n$(71-72.45)^2=2.10$\n$(72-72.45)^2=0.20$\n$(73-72.45)^2=0.30$\n$(74-72.45)^2=2.40$\n$(75-72.45)^2=6.50$\n$(77-72.45)^2=20.70$\nSum = 236.90\nVariance = 236.90 / 11 = 21.54\nStd dev = $\sqrt{21.54} \approx 4.64$.\n\n3. Problem 3: Age Assumed Office\nData: 34,39,40,43,44,47,50,50,52,53,54,55,56,58,60,65\n\nStep 1: Order data (already ordered). Count = 16\nStep 2: Median: average of 8th and 9th data points: $(50 + 52)/2 = 51$.\nStep 3: Q1: median of first 8 numbers (34,39,40,43,44,47,50,50): average of 4th and 5th values $(43 + 44)/2 = 43.5$.\nStep 4: Q3: median of last 8 numbers (52,53,54,55,56,58,60,65): average of 4th and 5th values $(55 + 56)/2 = 55.5$.\nStep 5: Interquartile range = $55.5 - 43.5 = 12$.\nStep 6: Mean = sum/16 = $(34+39+40+43+44+47+50+50+52+53+54+55+56+58+60+65)/16 = 800/16 = 50$.\nStep 7: Calculate population standard deviation:\nSquared differences sum $= (34-50)^2 + (39-50)^2 + ... + (65-50)^2 = 1240$.\nVariance = $1240/16 = 77.5$.\nStd dev = $\sqrt{77.5} \approx 8.8$.\n\n4. Problem 4: Sales Tax Percentages\nData: 2.9,4,4,4.5,5,5.125,5.5,6,6,6,6,6.15,6.25,6.5,7,7,7\n\nStep 1: Sort data (already sorted). Count = 17\nStep 2: Median = 9th data point = 6.\nStep 3: Mode = 6 and 7 both appear 3 times. Modes = 6 and 7 (bi-modal).\nStep 4: Q1 = median of lower 8 numbers: average of 4th and 5th values $(4.5 + 5)/2 = 4.75$.\nStep 5: Q3 = median of upper 8 numbers: average of 4th and 5th values $(6.15 + 6.25)/2 = 6.2$.\nStep 6: Interquartile range = $6.2 - 4.75 = 1.45$.\nStep 7: Mean = sum/17 = approx $\frac{2.9 + 4 + 4 + 4.5 + 5 + 5.125 + 5.5 + 6 + 6 + 6 + 6 + 6.15 + 6.25 + 6.5 + 7 + 7 + 7}{17} = 5.56$\nStep 8: Population standard deviation calculated via formula: approx 1.18.\n\nFinal answers:\n{\n"1": {"mode": 54, "median": 52, "mean": 49.56, "Q1": 45, "Q3": 54, "IQR": 9, "sd": 5.84},\n"2": {"mode": 70, "median": 71, "mean": 72.45, "Q1": 69, "Q3": 74, "IQR": 5, "sd": 4.64},\n"3": {"mode": "none", "median": 51, "mean": 50, "Q1": 43.5, "Q3": 55.5, "IQR": 12, "sd": 8.8},\n"4": {"mode": [6,7], "median": 6, "mean": 5.56, "Q1": 4.75, "Q3": 6.2, "IQR": 1.45, "sd": 1.18}\n}