1. **Problem Statement:** Calculate the average time spent to complete the exam, the range time used by the 1/4 fastest students, and the mean deviation about the mean for the given frequency distribution of exam times. 2. **Given Data:** | Exam Time | Frequency | |-----------|-----------| | 30-34 | 2 | | 35-39 | 4 | | 40-44 | 5 | | 45-49 | 4 | | 50-54 | 6 | | 55-59 | 9 | | 60-64 | 12 | | 65-69 | 13 | | 70-74 | 10 | | 75-79 | 12 | | 80-84 | 12 | | 85-89 | 11 | Total students $N=100$. 3. **Step 1: Calculate midpoints ($x_i$) for each class interval:** - $30-34 \to 32$ - $35-39 \to 37$ - $40-44 \to 42$ - $45-49 \to 47$ - $50-54 \to 52$ - $55-59 \to 57$ - $60-64 \to 62$ - $65-69 \to 67$ - $70-74 \to 72$ - $75-79 \to 77$ - $80-84 \to 82$ - $85-89 \to 87$ 4. **Step 2: Calculate the average (mean) time:** $$\bar{x} = \frac{\sum f_i x_i}{N}$$ Calculate $f_i x_i$ for each class and sum: $2\times32=64$, $4\times37=148$, $5\times42=210$, $4\times47=188$, $6\times52=312$, $9\times57=513$, $12\times62=744$, $13\times67=871$, $10\times72=720$, $12\times77=924$, $12\times82=984$, $11\times87=957$ Sum $=64+148+210+188+312+513+744+871+720+924+984+957=6635$ Mean: $$\bar{x} = \frac{6635}{100} = 66.35$$ 5. **Step 3: Find the range time used by the 1/4 fastest students (lowest 25 students):** Cumulative frequencies: - 30-34: 2 - 35-39: 6 - 40-44: 11 - 45-49: 15 - 50-54: 21 - 55-59: 30 The 25th student lies in the 55-59 interval. Range for 1/4 fastest students is from 30 (lowest class start) to 59 (upper limit of 55-59). 6. **Step 4: Calculate the mean deviation about the mean:** Formula: $$MD = \frac{\sum f_i |x_i - \bar{x}|}{N}$$ Calculate $|x_i - 66.35|$ and multiply by $f_i$: - $|32-66.35|=34.35 \to 2\times34.35=68.7$ - $|37-66.35|=29.35 \to 4\times29.35=117.4$ - $|42-66.35|=24.35 \to 5\times24.35=121.75$ - $|47-66.35|=19.35 \to 4\times19.35=77.4$ - $|52-66.35|=14.35 \to 6\times14.35=86.1$ - $|57-66.35|=9.35 \to 9\times9.35=84.15$ - $|62-66.35|=4.35 \to 12\times4.35=52.2$ - $|67-66.35|=0.65 \to 13\times0.65=8.45$ - $|72-66.35|=5.65 \to 10\times5.65=56.5$ - $|77-66.35|=10.65 \to 12\times10.65=127.8$ - $|82-66.35|=15.65 \to 12\times15.65=187.8$ - $|87-66.35|=20.65 \to 11\times20.65=227.15$ Sum $=68.7+117.4+121.75+77.4+86.1+84.15+52.2+8.45+56.5+127.8+187.8+227.15=1112.2$ Mean deviation: $$MD = \frac{1112.2}{100} = 11.122$$ **Final answers:** - Average time spent: $66.35$ minutes - Range time for 1/4 fastest students: $30$ to $59$ minutes - Mean deviation about mean: $11.122$ minutes