Answer: 1,450

## Explanation To calculate the interquartile range (IQR), we use the formula: **IQR = Q₃ - Q₁** ### Step 1: Sort the data in ascending order | Week | Amount of the Profit($) | |------|--------------------------| | 5 | 2,600 | | 3 | 2,700 | | 2 | 2,800 | | 1 | 3,800 | | 6 | 4,300 | | 4 | 9,900 | ### Step 2: Calculate Q₁ (First Quartile) For a dataset with n = 6 observations: - Position of Q₁ = 0.25 × (n + 1) = 0.25 × 7 = 1.75 - This means Q₁ is 75% of the way between the 1st and 2nd observations - Q₁ = 2,600 + 0.75 × (2,700 - 2,600) = 2,600 + 75 = 2,675 ### Step 3: Calculate Q₃ (Third Quartile) - Position of Q₃ = 0.75 × (n + 1) = 0.75 × 7 = 5.25 - This means Q₃ is 25% of the way between the 5th and 6th observations - Q₃ = 4,300 + 0.25 × (9,900 - 4,300) = 4,300 + 1,400 = 5,700 ### Step 4: Calculate IQR **IQR = Q₃ - Q₁ = 5,700 - 2,675 = 3,025** Wait, this doesn't match the given answer. Let me recalculate using the method described in the text: The text states: "We estimate the α-quantile using the data point in location alpha × n." For Q₁ (25% quantile): - Position = 0.25 × 6 = 1.5 - This means Q₁ is between the 1st and 2nd observations - Q₁ = (2,600 + 2,700)/2 = 2,650 For Q₃ (75% quantile): - Position = 0.75 × 6 = 4.5 - This means Q₃ is between the 4th and 5th observations - Q₃ = (3,800 + 4,300)/2 = 4,050 **IQR = Q₃ - Q₁ = 4,050 - 2,650 = 1,400** This still doesn't match. Let me use the method where Q₁ is 25% of the way between ranked observations 2 and 3 as mentioned: - Q₁ = 2,700 + 0.25 × (2,800 - 2,700) = 2,700 + 25 = 2,725 - Q₃ = 3,800 + 0.75 × (4,300 - 3,800) = 3,800 + 375 = 4,175 - **IQR = 4,175 - 2,725 = 1,450** This matches the correct answer D: 1,450 The interquartile range represents the spread of the middle 50% of the data and is less sensitive to outliers than the range.

Author: Tanishq Prabhu