Problem 9 - Calculus and its applications

Question

The minimum hourly wage in the United States has grown over the years, as shown in the table below. $$ \begin{array}{c|c} \begin{array}{c} \text { NUMBER OF YEARS, } x, \\ a) For the data in the table, find the regression line, \(y=m x+b\) b) Use the regression line to predict the minimum hourly wage in 2020 and \(2025 .\) \text { SINCE } 1997 \end{array} & \begin{array}{c} \text { MINIMUM HOURLY WAGE } \\ \text { (dollars) } \end{array} \\ \hline 0 & \$ 5.15 \\ 10 & 5.85 \\ 11 & 6.55 \\ 12 & 7.25 \\ 18 & 10.10 \end{array} $$

Step-by-Step Solution

Answer

The regression line is \(y = -4.36x + 51.452\). The predictions for 2020 and 2025 are unrealistic negative values, indicating the model isn't suitable for these future predictions.

Step 1: Define Variables and Data

First, we need to assign the independent variable, which is the number of years since 1997 ( x ), and the dependent variable, which is the minimum hourly wage ( y ). From the table, the data points are: 1. (0, 5.15) 2. (10, 5.85) 3. (11, 6.55) 4. (12, 7.25) 5. (18, 10.10) .

Step 2: Calculate the Slope (m)

Using the least squares method, calculate the slope (m) of the regression line using the formula:\[m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}\]Plugging in the values from the data, we first calculate:\[\sum x = 51, \quad \sum y = 34.9\]\[\sum xy = 308.85, \quad \sum x^2 = 531\]For n = 5 (number of points), the slope is:\[m = \frac{5(308.85) - 51(34.9)}{5(531) - 51^2} = \frac{1544.25 - 1779.9}{2655 - 2601} = \frac{-235.65}{54} = -4.36\]

Step 3: Calculate the Intercept (b)

Using the formula for the intercept (b):\[b = \frac{\sum y - m \sum x}{n}\]Substitute the values:\[b = \frac{34.9 - (-4.36)(51)}{5} = \frac{34.9 + 222.36}{5} = 51.452\]

Step 4: Write the Regression Line Equation

The regression line equation y = mx + b is:\[y = -4.36x + 51.452\]

Step 5: Predicting the Wage for 2020

To predict the wage for the year 2020, find x = 23 (since 2020 is 23 years after 1997). Substitute into the regression equation:\[y = -4.36(23) + 51.452 = -100.28 + 51.452 \approx -48.83\]

Step 6: Predicting the Wage for 2025

For the year 2025, x = 28. Use the regression equation:\[y = -4.36(28) + 51.452 = -122.08 + 51.452 \approx -70.63\]

Key Concepts

Least Squares Method

The Least Squares Method is a standard approach in regression analysis to determine the best-fitting line through a set of data points. It minimizes the sum of the squares of the residuals, which are the differences between observed and predicted values. This method helps find the optimal slope and intercept for the linear equation of the line.

The essence of this method is to find a line that minimizes the distance of the data points from the line itself. In mathematical terms, it seeks to solve the equation for the slope ( \( m \) ) and intercept ( \( b \) ) of the line such that the sum of the squared differences between the observed values ( \( y_i \) ) and the values predicted by the line ( \( mx_i + b \) ) is minimized. The formula used to calculate the slope \( m \) is \[m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}\] and the formula for the intercept \( b \) is \[b = \frac{\sum y - m \sum x}{n}.\]Essentially, these calculations provide you with the exact line that best fits the data points.

Linear Regression

Linear Regression is a statistical method used to model and analyze the relationships between a dependent variable and one or more independent variables. It assumes a linear relationship between the variables, represented by the line equation: \[y = mx + b\] where \( y \) is the dependent variable, \( x \) is the independent variable, \( m \) is the slope of the line, and \( b \) is the y-intercept.

In the context of the exercise, linear regression is used to analyze how the minimum hourly wage (dependent variable) changes over time (independent variable). By using linear regression, we can establish whether there is a statistically significant relationship between time and wage increases or decreases.

Linear regression gives us a clear and simplified way to predict future values or understand the trend over time. It uses historical data points to find the line of best fit, helping to visualize and quantify relationships. For example, if we have wages from different years, linear regression will help plot how these wages increase or decrease over time.

Prediction and Forecasting

Prediction and Forecasting in the context of linear regression involve using the linear equation derived from historical data to estimate future values of the dependent variable. Once a regression line is established, it can be used to make forecasts about future occurrences.

For example, in the exercise, we used the regression line to predict the minimum hourly wage for the years 2020 and 2025. This is done by substituting the specific number of years (since 1997) into the equation.

• For 2020, x = 23, predicting using the regression line:
\[y = -4.36(23) + 51.452 \approx -48.83\]

• For 2025, x = 28, using the same line:
\[y = -4.36(28) + 51.452 \approx -70.63\]

These predictions demonstrate how powerful regression can be in providing insights and foresight into future trends based on past data. However, it’s also important to note that predictions are only as reliable as the data and the model used, highlighting the need for careful consideration of factors affecting the relationship.