1. **Problem Statement:** Construct a grouped frequency distribution for the leisure hours data of 80 college students, draw a histogram, describe skewness, and calculate median, mean, variance, standard deviation, coefficient of variation, and standard error of the mean. 2. **Step 1: Grouped Frequency Distribution** - Find the range: max = 38, min = 10, range = 38 - 10 = 28 - Choose class width: approximately $\frac{28}{6} \approx 5$ (6 classes) - Class intervals: 10-14, 15-19, 20-24, 25-29, 30-34, 35-39 - Count frequencies: - 10-14: 10,11,11,12,12,13,14,14 → 8 - 15-19: 15,15,15,15,16,16,16,16,16,17,17,17,17,18,18,18,18,18,19,19,19,19 → 22 - 20-24: 20,20,20,20,20,21,21,21,21,21,22,22,22,22,23,23,23,23,23,23,24,24,24,24 → 24 - 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 | Class Interval | Frequency | |---------------|-----------| | 10 - 14 | 8 | | 15 - 19 | 22 | | 20 - 24 | 24 | | 25 - 29 | 12 | | 30 - 34 | 4 | | 35 - 39 | 1 | 3. **Step 2: Histogram and Skewness** - Histogram bars correspond to frequencies of each class interval. - Skewness: Since the tail extends to the right (higher values less frequent), the distribution is **positively skewed**. 4. **Step 3: Calculate Median** - Total frequency $n=80$ - Median class is where cumulative frequency $\\geq \frac{n}{2} = 40$ - Cumulative frequencies: 8, 30, 54, 66, 70, 71 - Median class: 20-24 (cumulative frequency 54) - Median formula: $$\text{Median} = L + \left(\frac{\frac{n}{2} - F}{f}\right) \times w$$ where $L=20$, $F=30$ (cumulative freq before median class), $f=24$ (freq median class), $w=5$ $$\text{Median} = 20 + \left(\frac{40 - 30}{24}\right) \times 5 = 20 + \frac{10}{24} \times 5 = 20 + 2.08 = 22.08$$ 5. **Step 4: Calculate Mean** - Midpoints $x_i$: 12, 17, 22, 27, 32, 37 - Multiply midpoints by frequencies and sum: $$\sum f_i x_i = 8\times12 + 22\times17 + 24\times22 + 12\times27 + 4\times32 + 1\times37 = 96 + 374 + 528 + 324 + 128 + 37 = 1487$$ - Mean: $$\bar{x} = \frac{\sum f_i x_i}{n} = \frac{1487}{80} = 18.59$$ 6. **Step 5: Calculate Variance and Standard Deviation** - Calculate $\sum f_i x_i^2$: $$8\times12^2 + 22\times17^2 + 24\times22^2 + 12\times27^2 + 4\times32^2 + 1\times37^2 = 8\times144 + 22\times289 + 24\times484 + 12\times729 + 4\times1024 + 1\times1369 = 1152 + 6358 + 11616 + 8748 + 4096 + 1369 = 43239$$ - Variance formula: $$s^2 = \frac{\sum f_i x_i^2}{n} - \bar{x}^2 = \frac{43239}{80} - (18.59)^2 = 540.49 - 345.68 = 194.81$$ - Standard deviation: $$s = \sqrt{194.81} = 13.96$$ 7. **Step 6: Coefficient of Variation (CV)** $$CV = \frac{s}{\bar{x}} \times 100 = \frac{13.96}{18.59} \times 100 = 75.07\%$$ 8. **Step 7: Standard Error of the Mean (SEM)** $$SEM = \frac{s}{\sqrt{n}} = \frac{13.96}{\sqrt{80}} = \frac{13.96}{8.94} = 1.56$$ **Final answers:** - Median = 22.08 hours - Mean = 18.59 hours - Variance = 194.81 - Standard deviation = 13.96 - Coefficient of variation = 75.07% - Standard error of the mean = 1.56