1. **Problem:** Find the mean, median, and mode of the staff salaries given the table. **Step 1:** Calculate the total number of staff. $$1+1+4+3+3+6+3+6+30+15+6=78$$ **Step 2:** Calculate the total salary sum. $$(1\times60000)+(1\times50000)+(4\times30000)+(3\times20000)+(3\times18000)+(6\times13000)+(3\times11500)+(6\times10000)+(30\times9000)+(15\times8000)+(6\times7000)$$ $$=60000+50000+120000+60000+54000+78000+34500+60000+270000+120000+42000=889500$$ **Step 3:** Calculate the mean salary. $$\text{Mean} = \frac{889500}{78} \approx 11404$$ **Step 4:** Find the median salary. The median is the middle value when salaries are ordered. Total staff = 78, median is average of 39th and 40th salaries. Ordering salaries from highest to lowest: - 1 at 60000 (1st) - 1 at 50000 (2nd) - 4 at 30000 (3rd to 6th) - 3 at 20000 (7th to 9th) - 3 at 18000 (10th to 12th) - 6 at 13000 (13th to 18th) - 3 at 11500 (19th to 21st) - 6 at 10000 (22nd to 27th) - 30 at 9000 (28th to 57th) The 39th and 40th staff fall in the 9000 salary group. So, median = 9000 **Step 5:** Find the mode salary. The mode is the salary with the highest frequency. Number of staff per salary: - 9000: 30 staff (highest frequency) So, mode = 9000 2. **Problem:** Raise salaries of 21 clerks (15 sales + 6 general) from 8000 and 7000 to 10000 and find new mean, median, mode. **Step 1:** Calculate new total salary sum. Original sum = 889500 Increase for sales clerks: (10000 - 8000) \times 15 = 2000 \times 15 = 30000 Increase for general clerks: (10000 - 7000) \times 6 = 3000 \times 6 = 18000 Total increase = 30000 + 18000 = 48000 New total salary sum = 889500 + 48000 = 937500 **Step 2:** Calculate new mean. $$\text{New mean} = \frac{937500}{78} \approx 12019$$ **Step 3:** Find new median. Salaries ordered with changes: - 1 at 60000 - 1 at 50000 - 4 at 30000 - 3 at 20000 - 3 at 18000 - 6 at 13000 - 3 at 11500 - 6 at 10000 (originally 10000) - 15 at 10000 (raised from 8000) - 6 at 10000 (raised from 7000) - 30 at 9000 Now, the 39th and 40th staff fall in the 10000 salary group. So, median = 10000 **Step 4:** Find new mode. Number of staff per salary: - 9000: 30 staff - 10000: 6 + 15 + 6 = 27 staff Mode remains 9000 (highest frequency) 3. **Problem:** If all salaries increase by the same amount, which central tendency measures change? **Answer:** - Mean increases by that amount because mean shifts by the constant added. - Median increases by that amount because the middle value shifts by the constant. - Mode increases by that amount because the most frequent salary shifts by the constant. **Justification:** Adding a constant shifts all data points equally, so all measures increase by that constant. 4. **Problem:** If only one or two clerks' salaries increase, which measure definitely changes? **Answer:** - The mean will definitely change because total sum changes. - The median may or may not change depending on which salaries increase. - The mode may or may not change depending on frequency changes. 5. **Problem:** Foremen's salary increase so mean salary increases by 5000. **Step 1:** Calculate total increase in sum needed. $$\text{Increase in total sum} = 5000 \times 78 = 390000$$ **Step 2:** Since only foremen's salary changes, total increase = number of foremen \times increase per foreman. Number of foremen = 6 $$6 \times \text{increase per foreman} = 390000$$ $$\text{increase per foreman} = \frac{390000}{6} = 65000$$ **Step 3:** New foreman salary: $$13000 + 65000 = 78000$$ 6. **Problem:** Company lays off 1 foreman and 2 workmen. Will mean salary increase, decrease, or remain same? **Step 1:** Calculate total salary and number of staff before layoff. Total salary = 889500 Number of staff = 78 **Step 2:** Calculate total salary and number of staff after layoff. Salaries removed: - 1 foreman at 13000 - 2 workmen at 9000 each = 18000 Total removed = 13000 + 18000 = 31000 New total salary = 889500 - 31000 = 858500 New number of staff = 78 - 3 = 75 **Step 3:** Calculate new mean salary. $$\text{New mean} = \frac{858500}{75} \approx 11446.67$$ **Step 4:** Compare new mean to old mean. Old mean = 11404 New mean = 11446.67 **Answer:** Mean salary increases because the laid off staff have salaries below the old mean, so removing them raises the average.