Use the table to answer the questions in each part below.
| Year | 1963 | 1968 | 1994 | 2012 |
| :----------------------------- | :-------- | :-------- | :-------- | :-------- |
| Percent without Health insurance | [tex]$24.3 \%$[/tex] | [tex]$14.5 \%$[/tex] | [tex]$15.8 \%$[/tex] | [tex]$14.7 \%$[/tex] |
Part 1: To predict the percentage of people without health insurance in 2020, are you interpolating or extrapolating?
Part 2: What is the predicted percentage of people without health insurance in 2029? Round your result to two decimal places.
Answer (2)
Determine that predicting for 2029 is extrapolation since it's outside the given data range.
Calculate the rate of change between 1994 and 2012: r a t e = 2012 − 1994 14.7 − 15.8 = 18 − 1.1 .
Predict the percentage in 2029 using the rate of change: p erce n t a g e 2029 = 14.7 + 18 − 1.1 ∗ 17 .
Round the predicted percentage to two decimal places: 13.66 .
Explanation
Understanding the Problem We are given a table showing the percentage of people without health insurance for the years 1963, 1968, 1994, and 2012. The percentages are 24.3%, 14.5%, 15.8%, and 14.7% respectively. We need to predict the percentage of people without health insurance in 2029 and determine whether this prediction is interpolation or extrapolation.
Identifying Extrapolation Since 2029 is outside the range of the given years (1963-2012), predicting the percentage for 2029 is an example of extrapolation.
Calculating the Rate of Change To predict the percentage in 2029, we will use the data from the two most recent years, 1994 and 2012, to calculate the rate of change. The rate of change is calculated as follows: r a t e = 2012 − 1994 p erce n t a g e 2012 − p erce n t a g e 1994 = 2012 − 1994 14.7 − 15.8 = 18 − 1.1 This gives us the rate of change per year.
Predicting the Percentage in 2029 Now, we use the rate of change to predict the percentage in 2029. We can use the percentage in 2012 as our starting point: p erce n t a g e 2029 = p erce n t a g e 2012 + r a t e × ( 2029 − 2012 ) = 14.7 + 18 − 1.1 × 17 p erce n t a g e 2029 = 14.7 + 18 − 1.1 × 17 = 14.7 − 18 18.7 ≈ 14.7 − 1.0389 ≈ 13.6611 Rounding this to two decimal places, we get 13.66%.
Final Answer Therefore, the predicted percentage of people without health insurance in 2029 is approximately 13.66%.
Examples
Predicting trends, like the percentage of people without health insurance, is a common application of mathematical modeling in public health and economics. By analyzing historical data and using extrapolation techniques, we can forecast future outcomes. For example, governments and organizations use these predictions to plan healthcare policies, allocate resources, and assess the impact of various interventions. Understanding these trends helps in making informed decisions to improve public health and welfare.
Predicting the percentage of people without health insurance in 2029 is an example of extrapolation since it is outside the given data range. Using the rate of change calculated from the most recent years, the predicted percentage for 2029 is approximately 13.66%.
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