This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions for questions 1 to 4. First, let's create a frequency distribution table from the given cumulative frequency table: | Time (minutes) | Cumulative frequency (CF) | Frequency (f) | |:---------------|:--------------------------|:--------------| | < 20 | 0 | 0 | | 20 - < 30 | 18 | 18 - 0 = 18 | | 30 - < 40 | 45 | 45 - 18 = 27| | 40 - < 50 | 97 | 97 - 45 = 52| | 50 - < 60 | 163 | 163 - 97 = 66| | 60 - < 70 | 210 | 210 - 163 = 47| The total number of students, N = 210. Question 1: For the data in Table 1, state which interval the median race time lies in. (Assuming "median weight" is a typo and should be "median race time"). Step 1: Find the position of the median. The median position is given by (N)/(2). Median position = (210)/(2) = 105 Step 2: Identify the class interval containing the 105th value. From the cumulative frequency table: • The cumulative frequency for "< 50" is 97. • The cumulative frequency for "< 60" is 163. Since 105 is greater than 97 and less than or equal to 163, the 105th value falls into the 50 - < 60 interval. The median interval is C. 50 - 60. Question 2: Calculate the estimate for the median for the data in Table 1. The formula for the median of grouped data is: Median = L + ( (N)/(2) - CFf ) × w Where: • L = lower boundary of the median class = 50 • (N)/(2) = median position = 105 • CF = cumulative frequency of the class before the median class = 97 • f = frequency of the median class (50 - < 60) = 163 - 97 = 66 • w = width of the median class = 60 - 50 = 10 Step 1: Substitute the values into the formula. Median = 50 + ( (105 - 97)/(66) ) × 10 Step 2: Simplify the expression. Median = 50 + ( (8)/(66) ) × 10 Median = 50 + (80)/(66) Median = 50 + 1.2121... Median ≈ 51.21 The estimate for the median is B. 51.21. Question 3: Calculate the lower quartile for the data in Table 1. The formula for the lower quartile (Q_1) of grouped data is: Q_1 = L + ( (N)/(4) - CFf ) × w Step 1: Find the position of the lower quartile. Q_1 position = (N)/(4) = (210)/(4) = 52.5 Step 2: Identify the lower quartile class. From the cumulative frequency table: • The cumulative frequency for "< 40" is 45. • The cumulative frequency for "< 50" is 97. Since 52.5 is greater than 45 and less than or equal to 97, the lower quartile class is 40 - < 50. Step 3: Identify the values for the formula. • L = lower boundary of the lower quartile class = 40 • (N)/(4) = lower quartile position = 52.5 • CF = cumulative frequency of the class before the lower quartile class = 45 • f = frequency of the lower quartile class (40 - < 50) = 97 - 45 = 52 • w = width of the lower quartile class = 50 - 40 = 10 Step 4: Substitute the values into the formula. Q_1 = 40 + ( (52.5 - 45)/(52) ) × 10 Step 5: Simplify the expression. Q_1 = 40 + ( (7.5)/(52) ) × 10 Q_1 = 40 + (75)/(52) Q_1 = 40 + 1.4423... Q_1 ≈ 41.44 The lower quartile is B. 41.44. Question 4: Calculate the upper quartile for the data in Table 1. The formula for the upper quartile (Q_3) of grouped data is: Q_3 = L + ( (3N)/(4) - CFf ) × w Step 1: Find the position of the upper quartile. Q_3 position = (3N)/(4) = 3 × (210)/(4) = 3 × 52.5 = 157.5 Step 2: Identify the upper quartile class. From the cumulative frequency table: • The cumulative frequency for "< 50" is 97. • The cumulative frequency for "< 60" is 163. Since 157.5 is greater than 97 and less than or equal to 163, the upper quartile class is 50 - < 60. Step 3: Identify the values for the formula. • L = lower boundary of the upper quartile class = 50 • (3N)/(4) = upper quartile position = 157.5 • CF = cumulative frequency of the class before the upper quartile class = 97 • f = frequency of the upper quartile class (50 - < 60) = 163 - 97 = 66 • w = width of the upper quartile class = 60 - 50 = 10 Step 4: Substitute the values into the formula. Q_3 = 50 + ( (157.5 - 97)/(66) ) × 10 Step 5: Simplify the expression. Q_3 = 50 + ( (60.5)/(66) ) × 10 Q_3 = 50 + (605)/(66) Q_3 = 50 + 9.1666... Q_3 ≈ 59.17 The upper quartile is D. 59.17. That's 2 down. 3 left today — send the next one.