The bar graph shows the population of the United States, in...

The bar graph shows the population of the United States, in millions, for six selected years. (Graph cannot copy) Here are two functions that model the data: $$ \begin{array}{l}{M(x)=1.54 x+114.6} \\ {F(x)=1.48 x+120.6}\end{array} $$ Use the functions in the previous column to solve. a. Write a function that models the difference between the female U.S. population and the male U.S. population for the years shown in the bar graph. b. Use the function from part (a) to find how many more women than men there were in the U.S. population in $2005 .$ c. Does the result in part (b) overestimate or underestimate the actual difference between the female and male population in 2005 shown by the bar graph? By how much?

The bar graph shows the population of the United States, in millions, for six selected years.
(Graph cannot copy)
Here are two functions that model the data:
$$
\begin{array}{l}{M(x)=1.54 x+114.6} \\ {F(x)=1.48 x+120.6}\end{array}
$$
Use the functions in the previous column to solve.
a. Write a function that models the difference between the female U.S. population and the male U.S. population for the years shown in the bar graph.
b. Use the function from part (a) to find how many more women than men there were in the U.S. population in $2005 .$
c. Does the result in part (b) overestimate or underestimate the actual difference between the female and male population in 2005 shown by the bar graph? By how much?

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Key Concepts

Error Analysis

Error analysis involves comparing the model’s predictions with actual observed data to determine the accuracy of the model. This concept is essential for assessing whether a model tends to overestimate or underestimate real values and by what margin, thereby informing possible adjustments to improve predictive accuracy.

Linear Functions

Linear functions represent relationships with a constant rate of change, defined by a slope and an intercept. In population models, these functions are used to approximate trends over time, where the slope indicates the growth rate and the intercept provides the initial value.

Difference of Functions

The difference of functions involves subtracting one function from another to obtain a new function that models the discrepancy between the two sets of data. When comparing population models for different groups, this technique quantifies the relative difference between those groups.

Function Evaluation

Function evaluation is the process of substituting a specific value into a function to compute the corresponding output. This is crucial in modeling, as it allows for the determination of predicted values—such as the difference in population—at specific points in time.

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