Question

The arithmetic mean of 10 items is 5. If 5 is added to each of the first seven items and 3 is subtracted from each of the last three items, then the mean of the new data series is:

• A. 5
• B. 7
• C. 7.6
• D. 6.7

Hint:

The new mean can also be calculated by finding the average change and adding it to the old mean. Average change = (Total Change) / N = \( (7 \times 5 - 3 \times 3) / 10 = 26/10 = 2.6 \). New Mean = Old Mean + Average Change = \( 5 + 2.6 = 7.6 \).

Answer 1

The Correct Option is C

Step 1: Compute the initial sum of the 10 values.\[ \text{Sum} = \text{Mean} \times \text{Number of items} \]\[ \text{Original Sum} = 5 \times 10 = 50 \]

Step 2: Determine the total change to the sum.- The first seven values each increase by 5. Total increase: \( 7 \times 5 = 35 \).- The last three values each decrease by 3. Total decrease: \( 3 \times 3 = 9 \).- The net change is \( +35 - 9 = 26 \).

Step 3: Calculate the new sum.\[ \text{New Sum} = \text{Original Sum} + \text{Net Change} \]\[ \text{New Sum} = 50 + 26 = 76 \]

Step 4: Find the new mean.The number of values remains 10.\[ \text{New Mean} = \frac{\text{New Sum}}{\text{Number of items}} = \frac{76}{10} = 7.6 \]