The following data represent the number of people aged 25 to 64 years covered by health insurance​ (private or​ government) in 2018. Approximate the mean and standard deviation for age.
Age
​25-34
​35-44
​45-54
​55-64
 
Number​ (millions)
20.9
32.9
39.9
29.9                               please find the true mean of population

To calculate the true mean of the population given the data, we'll use a weighted average formula where the ages are approximated using the midpoints of the age ranges. The number of people in millions is used as the weight for each age group.

### Step 1: Determine the midpoints for each age group.
- 25-34: Midpoint = \( \frac{25 + 34}{2} = 29.5 \)
- 35-44: Midpoint = \( \frac{35 + 44}{2} = 39.5 \)
- 45-54: Midpoint = \( \frac{45 + 54}{2} = 49.5 \)
- 55-64: Midpoint = \( \frac{55 + 64}{2} = 59.5 \)

### Step 2: Multiply each midpoint by the number of people in millions (weight).
- For 25-34: \( 29.5 \times 20.9 = 616.55 \)
- For 35-44: \( 39.5 \times 32.9 = 1299.55 \)
- For 45-54: \( 49.5 \times 39.9 = 1975.05 \)
- For 55-64: \( 59.5 \times 29.9 = 1780.05 \)

### Step 3: Calculate the sum of the products and the sum of the weights.
- Sum of the products = \( 616.55 + 1299.55 + 1975.05 + 1780.05 = 5671.2 \)
- Sum of the weights (total population) = \( 20.9 + 32.9 + 39.9 + 29.9 = 123.6 \)

### Step 4: Calculate the mean.
\[
\text{Mean} = \frac{\text{Sum of the products}}{\text{Sum of the weights}} = \frac{5671.2}{123.6} \approx 45.9
\]

Thus, the true mean age of the population is approximately **45.9 years**.

Would you like help calculating the standard deviation next?

Here are five related questions:
1. How is the midpoint approach useful for approximating means?
2. What would happen to the mean if we added another age group?
3. How does the distribution of population across age groups affect the mean?
4. Why is it important to use weighted averages for grouped data?
5. How does a shift in the age range impact the mean?

**Tip:** Always ensure to use midpoints when calculating the mean for grouped data.

The following data represent the number of people aged 25 to 64 years covered by health insurance​ (private or​ government) in 2018. Approximate the mean and standard deviation for age.
Age​ 25-34​ 35-44​ 45-54​ 55-64
Number​ (millions) 20.9, 32.9, 39.9, 29.9. Please find the true mean of population.

Find the true mean age of a population based on health insurance data for ages 25 to 64. The solution uses the weighted average formula and midpoints for each age group, arriving at a mean of approximately 45.9 years. Ideal for students in Grades 10-12 studying statistics.

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