1. **Problem Statement:** Construct a grouped frequency distribution for the leisure hours data of 80 college students, draw a histogram and describe skewness, then calculate median, mean, variance, standard deviation, coefficient of variation, and standard error of the mean. 2. **Step 1: Grouped Frequency Distribution** - Data range: minimum = 10, maximum = 38 - Choose class intervals (e.g., width 5): 10-14, 15-19, 20-24, 25-29, 30-34, 35-39 - Count frequencies in each class: - 10-14: 10,11,11,12,12,13,14,14 → 8 - 15-19: 15,15,15,15,15,16,16,16,16,16,17,17,17,17,17,18,18,18,18,18,18,19,19,19,19,19 → 26 - 20-24: 20,20,20,20,20,20,21,21,21,21,21,22,22,22,22,22,23,23,23,23,23,23,24,24,24,24 → 26 - 25-29: 25,25,25,26,26,27,27,28,28,29,29,29 → 12 - 30-34: 30,31,32,34 → 4 - 35-39: 38 → 1 3. **Step 2: Histogram and Skewness** - Histogram bars correspond to frequencies per class interval. - Skewness: Since the tail extends to the right (higher values), data is positively skewed. 4. **Step 3: Median Calculation** - Total n=80, median position = $\frac{80+1}{2} = 40.5$th value - Cumulative frequencies: - 10-14: 8 - 15-19: 8+26=34 - 20-24: 34+26=60 - Median class: 20-24 - Median formula: $$\text{Median} = L + \left(\frac{\frac{n}{2} - F}{f}\right) \times w$$ where $L=20$, $F=34$, $f=26$, $w=5$ - Calculate: $$20 + \left(\frac{40 - 34}{26}\right) \times 5 = 20 + \left(\frac{6}{26}\right) \times 5 = 20 + 1.15 = 21.15$$ 5. **Step 4: Mean Calculation** - Midpoints: 12,17,22,27,32,37 - Multiply midpoints by frequencies and sum: - $12 \times 8 = 96$ - $17 \times 26 = 442$ - $22 \times 26 = 572$ - $27 \times 12 = 324$ - $32 \times 4 = 128$ - $37 \times 1 = 37$ - Sum = $96 + 442 + 572 + 324 + 128 + 37 = 1599$ - Mean = $\frac{1599}{80} = 19.99$ 6. **Step 5: Variance and Standard Deviation** - Calculate $\sum f x^2$: - $12^2 \times 8 = 1152$ - $17^2 \times 26 = 7514$ - $22^2 \times 26 = 12616$ - $27^2 \times 12 = 8748$ - $32^2 \times 4 = 4096$ - $37^2 \times 1 = 1369$ - Sum = $1152 + 7514 + 12616 + 8748 + 4096 + 1369 = 45495$ - Variance formula: $$s^2 = \frac{\sum f x^2}{n} - \bar{x}^2 = \frac{45495}{80} - (19.99)^2 = 568.69 - 399.60 = 169.09$$ - Standard deviation: $$s = \sqrt{169.09} = 13.00$$ 7. **Step 6: Coefficient of Variation (CV)** $$CV = \frac{s}{\bar{x}} \times 100 = \frac{13.00}{19.99} \times 100 = 65.03\%$$ 8. **Step 7: Standard Error of the Mean (SEM)** $$SEM = \frac{s}{\sqrt{n}} = \frac{13.00}{\sqrt{80}} = \frac{13.00}{8.94} = 1.45$$ **Final answers:** - Median = 21.15 hours - Mean = 19.99 hours - Variance = 169.09 - Standard deviation = 13.00 - Coefficient of variation = 65.03% - Standard error of the mean = 1.45