1. **Problem Statement:** We have weekly sales data grouped into class intervals with frequencies. We need to: (a) Apply continuity correction to the class intervals. (b) Complete the table with corrected intervals, class widths, and frequency densities. (c) Construct a histogram using frequency densities. (d) Calculate the estimated mean weekly sales per store using the midpoint method. (e) Identify the median and modal classes and interpret the modal class. 2. **Continuity Correction:** When class intervals are given as discrete ranges like 0–14, 15–24, etc., continuity correction adjusts intervals to include the boundary values by subtracting 0.5 from the lower limit and adding 0.5 to the upper limit. 3. **Step (a) Corrected Class Intervals:** - Original: 0–14 → Corrected: $-0.5$ to $14.5$ - 15–24 → $14.5$ to $24.5$ - 25–39 → $24.5$ to $39.5$ - 40–59 → $39.5$ to $59.5$ - 60–79 → $59.5$ to $79.5$ - 80–109 → $79.5$ to $109.5$ 4. **Step (b) Calculate Class Width and Frequency Density:** Class width = upper limit - lower limit Frequency density = Frequency / Class width | Corrected Interval | Frequency | Class Width | Frequency Density | |--------------------|-----------|-------------|-------------------| | -0.5 – 14.5 | 5 | $14.5 - (-0.5) = 15$ | $\frac{5}{15} = 0.333$ | | 14.5 – 24.5 | 10 | $24.5 - 14.5 = 10$ | $\frac{10}{10} = 1$ | | 24.5 – 39.5 | 20 | $39.5 - 24.5 = 15$ | $\frac{20}{15} = 1.333$| | 39.5 – 59.5 | 25 | $59.5 - 39.5 = 20$ | $\frac{25}{20} = 1.25$ | | 59.5 – 79.5 | 18 | $79.5 - 59.5 = 20$ | $\frac{18}{20} = 0.9$ | | 79.5 – 109.5 | 12 | $109.5 - 79.5 = 30$ | $\frac{12}{30} = 0.4$ | 5. **Step (c) Histogram Construction:** The histogram has x-axis as corrected intervals and y-axis as frequency density. Bars are adjacent with heights as frequency densities: 0.333, 1, 1.333, 1.25, 0.9, 0.4. 6. **Step (d) Estimated Mean Using Midpoint Method:** Midpoint of each corrected interval = $\frac{\text{lower limit} + \text{upper limit}}{2}$ Calculate midpoints: - $\frac{-0.5 + 14.5}{2} = 7$ - $\frac{14.5 + 24.5}{2} = 19.5$ - $\frac{24.5 + 39.5}{2} = 32$ - $\frac{39.5 + 59.5}{2} = 49.5$ - $\frac{59.5 + 79.5}{2} = 69.5$ - $\frac{79.5 + 109.5}{2} = 94.5$ Multiply midpoints by frequencies and sum: $$\sum f x = 5 \times 7 + 10 \times 19.5 + 20 \times 32 + 25 \times 49.5 + 18 \times 69.5 + 12 \times 94.5$$ Calculate each term: - $5 \times 7 = 35$ - $10 \times 19.5 = 195$ - $20 \times 32 = 640$ - $25 \times 49.5 = 1237.5$ - $18 \times 69.5 = 1251$ - $12 \times 94.5 = 1134$ Sum: $$35 + 195 + 640 + 1237.5 + 1251 + 1134 = 4492.5$$ Total frequency: $$5 + 10 + 20 + 25 + 18 + 12 = 90$$ Estimated mean: $$\bar{x} = \frac{\sum f x}{\sum f} = \frac{4492.5}{90} = 49.9167$$ 7. **Step (e) Median and Modal Class:** (i) Median class is the class where cumulative frequency reaches half total frequency (45). Cumulative frequencies: - 5 - 15 - 35 - 60 (exceeds 45) Median class: 39.5 – 59.5 (ii) Modal class is the class with highest frequency: 25 in 39.5 – 59.5 This indicates most stores have weekly sales in this range, showing a concentration of sales around this interval. Final answers: - Corrected intervals: $-0.5$–$14.5$, $14.5$–$24.5$, $24.5$–$39.5$, $39.5$–$59.5$, $59.5$–$79.5$, $79.5$–$109.5$ - Frequency densities: 0.333, 1, 1.333, 1.25, 0.9, 0.4 - Estimated mean weekly sales: approximately 49.92 units - Median class: $39.5$–$59.5$ - Modal class: $39.5$–$59.5$ (most frequent sales range)