Chapter 19: Mean, Median, and Mode > HARD QUESTIONS (Questions 15-20)
HARD QUESTIONS (Questions 15-20)
Question 1 of 61 / 6
Step 1: Use the given information. Smallest number = 5 Range = 8, so Largest number = 5 + 8 = 13 Mode = 5, so 5 appears more than once. Step 2: Set up the five numbers in order. Since 5 is the mode, it must appear at least twice. Numbers so far: 5, 5, ?, ?, 13 Step 3: Use the median. The median (middle value) is 7. So the 3rd number must be 7. Numbers: 5, 5, 7, ?, 13 Step 4: Find the 4th number. The 4th number must be between 7 and 13. It cannot be 5 (that would affect the median). Any value from 7 to 13 works, but to keep 5 as the unique mode, it shouldn't repeat any number. One valid solution: 5, 5, 7, 10, 13 Check: Median = 7 ✓, Mode = 5 ✓, Range = 13 - 5 = 8 ✓ The five numbers are: 5, 5, 7, 10, 13 (other answers possible for the 4th number)Step 1: Find the sum of the original 8 numbers. Mean of 8 numbers = 15 Sum of 8 numbers = 15 × 8 = 120 Step 2: Find the sum of all 9 numbers. Mean of 9 numbers = 16 Sum of 9 numbers = 16 × 9 = 144 Step 3: Find the 9th number. 9th number = Sum of 9 numbers - Sum of 8 numbers 9th number = 144 - 120 = 24 Therefore, the 9th number is 24. Check: (120 + 24) ÷ 9 = 144 ÷ 9 = 16 ✓Step 1: Find the total marks for Class A. Class A: 20 students, mean = 72 Total marks for Class A = 20 × 72 = 1440 Step 2: Find the total marks for Class B. Class B: 30 students, mean = 68 Total marks for Class B = 30 × 68 = 2040 Step 3: Find the combined mean. Total students = 20 + 30 = 50 Total marks = 1440 + 2040 = 3480 Combined mean = 3480 ÷ 50 = 69.6 Explanation: The simple average of 72 and 68 is (72 + 68) ÷ 2 = 70. However, the combined mean is 69.6, not 70. This is because Class B has more students (30 vs 20). The combined mean is a weighted average, where Class B's lower score has more influence because it has more students. The combined mean is closer to 68 than to 72.Step 1: Calculate the mean. Sum = 25 + 28 + 30 + 32 + 35 + 38 + 40 + 45 + 50 + 150 = 473 Mean = 473 ÷ 10 = 47.3 thousand The manager used the mean. Step 2: Calculate the median. Ordered data: 25, 28, 30, 32, 35, 38, 40, 45, 50, 150 Middle values (5th and 6th): 35 and 38 Median = (35 + 38) ÷ 2 = 36.5 thousand The union representative used the median. Step 3: Discussion. Both are correct but using different measures of average. The mean (47.3) is heavily influenced by the outlier salary of 150 thousand. The median (36.5) better represents the typical salary as it is not affected by the extreme value. For salary discussions, the median is more representative because: - 9 out of 10 employees earn less than the mean - The median shows what a typical employee earns - The mean is distorted by the one very high salaryPart (a) - With all 15 students: Mean: Sum = 8+10+12+12+14+15+15+15+16+18+20+22+25+30+45 = 277 Mean = 277 ÷ 15 = 18.47 minutes (to 2 dp) Median: 8th value (middle of 15) = 15 minutes Median = 15 minutes Mode: 15 appears 3 times (the most) Mode = 15 minutes Part (b) - Without the 45-minute time (14 students): Mean: New sum = 277 - 45 = 232 New mean = 232 ÷ 14 = 16.57 minutes (to 2 dp) Median: Middle values are 7th and 8th: both 15 Median = 15 minutes Mode: 15 still appears 3 times Mode = 15 minutes Part (c) - Analysis: Mean changed from 18.47 to 16.57 (decreased by 1.9 minutes) Median stayed at 15 minutes (no change) Mode stayed at 15 minutes (no change) The mean changed the most because it uses all values in its calculation. The outlier (45) significantly increased the mean. Median and mode are resistant to outliers.Part (a) - Estimating the mean using mid-interval values: Mid-interval values: 44.5, 54.5, 64.5, 74.5, 84.5, 95 Method A: Total students = 2 + 4 + 8 + 10 + 5 + 1 = 30 Sum = (44.5×2) + (54.5×4) + (64.5×8) + (74.5×10) + (84.5×5) + (95×1) = 89 + 218 + 516 + 745 + 422.5 + 95 = 2085.5 Mean for Method A = 2085.5 ÷ 30 = 69.5 marks Method B: Total students = 1 + 3 + 5 + 12 + 7 + 2 = 30 Sum = (44.5×1) + (54.5×3) + (64.5×5) + (74.5×12) + (84.5×7) + (95×2) = 44.5 + 163.5 + 322.5 + 894 + 591.5 + 190 = 2206 Mean for Method B = 2206 ÷ 30 = 73.5 marks Part (b) - Modal class: Method A: Modal class is 70-79 (frequency 10) Method B: Modal class is 70-79 (frequency 12) Part (c) - Comparison and Conclusion: Method B appears more effective because: 1. Higher mean score (73.5 vs 69.5) - a difference of 4 marks 2. More students in higher score ranges (80-89: 7 vs 5, 90-100: 2 vs 1) 3. Fewer students in lower score ranges (40-49: 1 vs 2, 50-59: 3 vs 4) 4. Both have the same modal class (70-79), but Method B has more students there (12 vs 10) Method B produces consistently better results across the class.
