1. **State the problem:** We have a data set of 20 call counts and need to analyze measures of central tendency (mean, median, mode) and distribution shape. 2. **Original data:** 1, 4, 6, 7, 8, 9, 11, 11, 12, 12, 13, 13, 14, 14, 17, 17, 18, 19, 22, 23 3. **Calculate mean:** $$\text{Mean} = \frac{1 + 4 + 6 + 7 + 8 + 9 + 11 + 11 + 12 + 12 + 13 + 13 + 14 + 14 + 17 + 17 + 18 + 19 + 22 + 23}{20} = \frac{267}{20} = 13.35$$ 4. **Calculate median:** Sorted data has 20 values, median is average of 10th and 11th values. 10th value = 12, 11th value = 13 $$\text{Median} = \frac{12 + 13}{2} = 12.5$$ 5. **Calculate mode:** Values 11, 12, 13, 14, and 17 each appear twice, so multiple modes exist. 6. **Answer (a):** All measures exist: mean, median, and mode all exist. 7. **Change largest value 23 to 45:** New sum = 267 - 23 + 45 = 289 New mean = $$\frac{289}{20} = 14.45$$ (mean changes) Median remains between 12 and 13 (unchanged) Mode remains the same (unchanged) 8. **Answer (b):** Only mean is affected by changing 23 to 45. 9. **Remove smallest value 1:** New data has 19 values. New sum = 267 - 1 = 266 New mean = $$\frac{266}{19} \approx 14.00$$ (mean changes) New median is the 10th value in sorted 19 values, which is 13 (original median was 12.5), so median changes. Mode remains the same (unchanged) 10. **Answer (c):** Mean and median change; mode does not. 11. **Distribution shape:** Data has a longer tail on the right (larger values like 22, 23), so it is positively skewed. 12. **Answer (d):** Distribution is positively skewed. Final answers: (a) All of these measures exist (b) Mean (c) Mean, Median (d) Positively skewed