Problem 68
Question
SOCIAL SCIENCE: Equal Pay for Equal Work Women's pay has often lagged behind men's, although Title VII of the Civil Rights Act requires equal pay for equal work. Based on data from \(2000-2008\), women's annual earnings as a percent of men's can be approximated by the formula \(y=0.51 x+77.2, \quad\) where \(x\) is the number of years since 2000 . (For example, \(x=10\) gives \(y=82.3\), so in 2010 women's wages were about \(82.3 \%\) of men's wages.) a. Graph this line on the window \([0,30]\) by \([0,100]\). b. Use this line to predict the percentage in the year 2020\. [Hint: Which \(x\) -value corresponds to 2020 ? Then use TRACE, EVALUATE, or TABLE.] c. Predict the percentage in the year 2025 .
Step-by-Step Solution
Verified Answer
In 2020, women's wages are predicted to be 87.2% of men's wages, and in 2025, 89.95%.
1Step 1: Identifying the Equation
The equation given is \( y = 0.51x + 77.2 \), which models women's earnings as a percentage of men's earnings over time, where \( x \) is the number of years since 2000.
2Step 2: Determine Value for Graphing
When graphing from \( x = 0 \) to \( x = 30 \), we are considering the years 2000 to 2030 since \( x \) represents the number of years since 2000.
3Step 3: Graph the Line
To graph the line \( y = 0.51x + 77.2 \) within the window \([0, 30]\) and \([0, 100]\), plot the y-intercept at \( y = 77.2 \) and use the slope of 0.51 to rise 0.51 units for every 1 unit run along the x-axis.
4Step 4: Calculate for 2020
For 2020, \( x = 20 \) (since 2020 is 20 years from 2000). Substitute \( x = 20 \) into the equation \( y = 0.51(20) + 77.2 \) to find \( y = 87.2 \). Thus, in 2020, women's wages are predicted to be 87.2% of men's wages.
5Step 5: Calculate for 2025
For 2025, \( x = 25 \) (since 2025 is 25 years from 2000). Substitute \( x = 25 \) into the equation \( y = 0.51(25) + 77.2 \) to find \( y = 89.95 \). Thus, in 2025, women's wages are predicted to be 89.95% of men's wages.
Key Concepts
Linear EquationsGraphingPredictive ModelingCivil Rights Act
Linear Equations
Linear equations are mathematical expressions that create straight lines when graphed. They typically follow the format of \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. In our example, the equation \( y = 0.51x + 77.2 \) helps us understand how women's earnings as a percentage of men's have changed over time.
The slope \( m = 0.51 \) indicates the rate of change, meaning for every additional year since 2000, women's earnings increase by 0.51% of men's earnings. The y-intercept \( b = 77.2 \) tells us that in the year 2000, women's earnings were approximately 77.2% of men's earnings.
Such equations allow for predictions about future values, enabling us to estimate that by substituting different values of \( x \). This predictability is essential for tracking and addressing issues like wage disparities.
The slope \( m = 0.51 \) indicates the rate of change, meaning for every additional year since 2000, women's earnings increase by 0.51% of men's earnings. The y-intercept \( b = 77.2 \) tells us that in the year 2000, women's earnings were approximately 77.2% of men's earnings.
Such equations allow for predictions about future values, enabling us to estimate that by substituting different values of \( x \). This predictability is essential for tracking and addressing issues like wage disparities.
Graphing
Graphing is a way of visually presenting data and mathematical relationships. When we graph the equation \( y = 0.51x + 77.2 \), we use a coordinate plane where the x-axis represents the number of years since 2000, and the y-axis displays women's wages as a percentage of men's wages.
It's essential to start with the y-intercept, \( y = 77.2 \), on the graph. From there, use the slope, \( 0.51 \), to determine the line's rise over run. For every unit you move right along the x-axis, you go up 0.51 units along the y-axis.
It's essential to start with the y-intercept, \( y = 77.2 \), on the graph. From there, use the slope, \( 0.51 \), to determine the line's rise over run. For every unit you move right along the x-axis, you go up 0.51 units along the y-axis.
- Using the plotting window from 0 to 30 on the x-axis (representing years) and 0 to 100 on the y-axis (percentage), it helps us see the upward trend.
- This visual tool aids in understanding and predicting future values directly by observing the line's path.
Predictive Modeling
Predictive modeling uses statistical techniques to forecast future data based on current and past information. By applying it to the equation \( y = 0.51x + 77.2 \), we anticipate future values for years like 2020 and 2025.
In this case, for the year 2020, we let \( x = 20 \). Substituting into the equation gives \( y = 0.51(20) + 77.2 = 87.2 \). Thus, by 2020, women's earnings are predicted to reach 87.2% of men's. For 2025, when \( x = 25 \), substituting gives \( y = 0.51(25) + 77.2 = 89.95 \). This means we're forecasting women's earnings to rise to 89.95% of men's by 2025.
Predictive modeling like this is vital for planning and policy-making, enabling us to foresee and address economic disparities efficiently.
In this case, for the year 2020, we let \( x = 20 \). Substituting into the equation gives \( y = 0.51(20) + 77.2 = 87.2 \). Thus, by 2020, women's earnings are predicted to reach 87.2% of men's. For 2025, when \( x = 25 \), substituting gives \( y = 0.51(25) + 77.2 = 89.95 \). This means we're forecasting women's earnings to rise to 89.95% of men's by 2025.
Predictive modeling like this is vital for planning and policy-making, enabling us to foresee and address economic disparities efficiently.
Civil Rights Act
The Civil Rights Act of 1964 was a landmark law aiming to end segregation and discrimination on various grounds, including gender. Title VII of this act specifically focuses on employment discrimination and mandates "equal pay for equal work," meaning individuals should receive the same pay for performing the same job, regardless of gender.
This legislation was pivotal in advancing gender equality in the workplace. However, as depicted by the gradual increase in women's earnings over time through our linear equation, achieving full equality has been an ongoing challenge.
This legislation was pivotal in advancing gender equality in the workplace. However, as depicted by the gradual increase in women's earnings over time through our linear equation, achieving full equality has been an ongoing challenge.
- Despite the Act's provisions, practice often lags behind policy, necessitating continuous monitoring and enforcement.
- The data modeled in this exercise shows that while progress is being made, further efforts are needed to close the gap entirely.
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