An agency interviewed 200 persons as part of an opinion poll. The following distribution represents the age of the persons interviewed. Calculate the average age.

Steps to Solve

1. Identify Age Midpoints

To calculate the average age, we first need the midpoints of each age group. The midpoints can be calculated as follows:

• 80-89: $ \frac{80 + 89}{2} = 84.5 $
• 70-79: $ \frac{70 + 79}{2} = 74.5 $
• 60-69: $ \frac{60 + 69}{2} = 64.5 $
• 50-59: $ \frac{50 + 59}{2} = 54.5 $
• 40-49: $ \frac{40 + 49}{2} = 44.5 $
• 30-39: $ \frac{30 + 39}{2} = 34.5 $
• 20-29: $ \frac{20 + 29}{2} = 24.5 $
• 10-19: $ \frac{10 + 19}{2} = 14.5 $

2. Create a Frequency Table with Midpoints

Now we can create a table listing the midpoints and their corresponding frequencies:

3. Calculate Weighted Frequencies

Multiply each midpoint by its corresponding frequency:

• For 80-89: $ 84.5 \times 2 = 169 $
• For 70-79: $ 74.5 \times 2 = 149 $
• For 60-69: $ 64.5 \times 6 = 387 $
• For 50-59: $ 54.5 \times 20 = 1090 $
• For 40-49: $ 44.5 \times 56 = 2492 $
• For 30-39: $ 34.5 \times 40 = 1380 $
• For 20-29: $ 24.5 \times 42 = 1029 $
• For 10-19: $ 14.5 \times 32 = 464 $

4. Sum of Weighted Frequencies and Total Frequencies

Now, we can sum the weighted frequencies and the total frequency:

• Total Weighted Sum: $$ 169 + 149 + 387 + 1090 + 2492 + 1380 + 1029 + 464 = 8650 $$

• Total Frequency: $$ 2 + 2 + 6 + 20 + 56 + 40 + 42 + 32 = 200 $$

5. Calculate Average Age

The average age can be calculated using the formula:

$$ \text{Average Age} = \frac{\text{Total Weighted Sum}}{\text{Total Frequency}} $$

Substituting in the values:

$$ \text{Average Age} = \frac{8650}{200} = 43.25 $$