Recall that the population mean is given by the formula mu equals StartFraction Upper Sigma x Subscript i Baseline f Subscript i Over Upper Sigma f Subscript i EndFraction equals StartFraction x 1 f 1 plus x 2 f 2 plus ... plus x Subscript n Baseline f Subscript n Baseline Over f 1 plus f 2 plus ... plus f Subscript n EndFraction where x Subscript i is the midpoint or value of the ith​ class, f Subscript i is the frequency of the ith​ class, and n is the number of classes.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)

23.723.7

38.238.2

36.736.7

25.225.2muμ what does mean equal


The problem provides age groups and the corresponding number of people (in millions) covered by health insurance in 2018. To find the mean (μ) and standard deviation for age, we will follow these steps:

### Step 1: Midpoints for each age class
Since the data is grouped by age ranges, the midpoint (\(x_i\)) for each age group is calculated as:

\[
x_i = \frac{{\text{{Lower limit}} + \text{{Upper limit}}}}{2}
\]

For each age group:
- Age 25-34: Midpoint = \( \frac{25 + 34}{2} = 29.5 \)
- Age 35-44: Midpoint = \( \frac{35 + 44}{2} = 39.5 \)
- Age 45-54: Midpoint = \( \frac{45 + 54}{2} = 49.5 \)
- Age 55-64: Midpoint = \( \frac{55 + 64}{2} = 59.5 \)

### Step 2: Frequencies (\(f_i\)) for each age class
The frequencies (number of people in millions) are given as:
- Age 25-34: \(f_1 = 23.7\) million
- Age 35-44: \(f_2 = 38.2\) million
- Age 45-54: \(f_3 = 36.7\) million
- Age 55-64: \(f_4 = 25.2\) million

### Step 3: Calculate the Mean (\(μ\))
The formula for the population mean is:

\[
\mu = \frac{\sum (x_i f_i)}{\sum f_i}
\]

Where:
- \(x_i\) is the midpoint of the \(i\)-th class
- \(f_i\) is the frequency of the \(i\)-th class

We will compute the products \(x_i f_i\) and sum them up, then divide by the total frequency.

Let's calculate this.The approximate mean age of people covered by health insurance in 2018 is \( \mu = 44.62 \) years.

### Next Step: Standard Deviation Calculation

The formula for the population standard deviation is:

\[
\sigma = \sqrt{\frac{\sum f_i (x_i - \mu)^2}{\sum f_i}}
\]

Let's calculate the standard deviation next.The approximate standard deviation of the ages of people covered by health insurance in 2018 is \( \sigma = 10.20 \) years.

Would you like further details on any of the steps or calculations?

Here are 5 related questions you might find useful:
1. How is the variance related to the standard deviation?
2. How would these calculations change if you had unequal age intervals?
3. What are other measures of central tendency aside from the mean?
4. How can frequency distributions be used to estimate the median?
5. What are some methods for calculating weighted averages?

**Tip:** The standard deviation measures the spread of data around the mean, giving insight into the variability of the dataset.

Recall that the population mean is given by the formula mu equals StartFraction Upper Sigma x Subscript i Baseline f Subscript i Over Upper Sigma f Subscript i EndFraction equals StartFraction x 1 f 1 plus x 2 f 2 plus ... plus x Subscript n Baseline f Subscript n Baseline Over f 1 plus f 2 plus ... plus f Subscript n EndFraction where x Subscript i is the midpoint or value of the ith​ class, f Subscript i is the frequency of the ith​ class, and n is the number of classes.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.

Learn how to approximate the mean and standard deviation for age groups based on frequency data from 2018 health insurance coverage. This problem covers key statistical concepts, including frequency distributions, the calculation of midpoints, and the use of formulas for mean and standard deviation. Suitable for students in Grades 9-12.

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